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libPARI(3pm) User Contributed Perl Documentation libPARI(3pm)
NAME
libPARI - Functions and Operations Available in PARI and GP
DESCRIPTION
The functions and operators available in PARI and in the GP/PARI calculator are numerous
and everexpanding. Here is a description of the ones available in version 2.2.0. It should
be noted that many of these functions accept quite different types as arguments, but
others are more restricted. The list of acceptable types will be given for each function
or class of functions. Except when stated otherwise, it is understood that a function or
operation which should make natural sense is legal. In this chapter, we will describe the
functions according to a rough classification. The general entry looks something like:
foo"(x,{flag = 0})": short description.
The library syntax is foo"(x,flag)".
This means that the GP function "foo" has one mandatory argument "x", and an optional one,
"flag", whose default value is 0 (the "{}" should never be typed, it is just a convenient
notation we will use throughout to denote optional arguments). That is, you can type
"foo(x,2)", or foo(x), which is then understood to mean "foo(x,0)". As well, a comma or
closing parenthesis, where an optional argument should have been, signals to GP it should
use the default. Thus, the syntax "foo(x,)" is also accepted as a synonym for our last
expression. When a function has more than one optional argument, the argument list is
filled with user supplied values, in order. And when none are left, the defaults are used
instead. Thus, assuming that "foo"'s prototype had been
" foo({x = 1},{y = 2},{z = 3}), "
typing in "foo(6,4)" would give you "foo(6,4,3)". In the rare case when you want to set
some far away flag, and leave the defaults in between as they stand, you can use the
``empty arg'' trick alluded to above: "foo(6,,1)" would yield "foo(6,2,1)". By the way,
"foo()" by itself yields "foo(1,2,3)" as was to be expected. In this rather special case
of a function having no mandatory argument, you can even omit the "()": a standalone "foo"
would be enough (though we don't really recommend it for your scripts, for the sake of
clarity). In defining GP syntax, we strove to put optional arguments at the end of the
argument list (of course, since they would not make sense otherwise), and in order of
decreasing usefulness so that, most of the time, you will be able to ignore them.
Binary Flags. For some of these optional flags, we adopted the customary binary notation
as a compact way to represent many toggles with just one number. Letting "(p_0,...,p_n)"
be a list of switches (i.e. of properties which can be assumed to take either the value 0
or 1), the number "2^3 + 2^5 = 40" means that "p_3" and "p_5" have been set (that is, set
to 1), and none of the others were (that is, they were set to 0). This will usually be
announced as ``The binary digits of "flag" mean 1: "p_0", 2: "p_1", 4: "p_2"'', and so on,
using the available consecutive powers of 2.
Pointers. If a parameter in the function prototype is prefixed with a & sign, as in
foo"(x,&e)"
it means that, besides the normal return value, the variable named "e" may be set as a
side effect. When passing the argument, the & sign has to be typed in explicitly. As of
version 2.2.0, this "pointer" argument is optional for all documented functions, hence the
& will always appear between brackets as in "issquare""(x,{&e})".
About library programming. To finish with our generic simple-minded example, the library
function "foo", as defined above, is seen to have two mandatory arguments, "x" and flag
(no PARI mathematical function has been implemented so as to accept a variable number of
arguments). When not mentioned otherwise, the result and arguments of a function are
assumed implicitly to be of type "GEN". Most other functions return an object of type
"long" integer in C (see Chapter 4). The variable or parameter names prec and flag always
denote "long" integers.
The "entree" type is used by the library to implement iterators (loops, sums, integrals,
etc.) when a formal variable has to successively assume a number of values in a given set.
When programming with the library, it is easier and much more efficient to code loops and
the like directly. Hence this type is not documented, although it does appear in a few
library function prototypes below. See "Label se:sums" for more details.
Standard monadic or dyadic operators
+"/"-
The expressions "+""x" and "-""x" refer to monadic operators (the first does nothing, the
second negates "x").
The library syntax is gneg"(x)" for "-""x".
+, "-"
The expression "x" "+" "y" is the sum and "x" "-" "y" is the difference of "x" and "y".
Among the prominent impossibilities are addition/subtraction between a scalar type and a
vector or a matrix, between vector/matrices of incompatible sizes and between an
integermod and a real number.
The library syntax is gadd"(x,y)" "x" "+" "y", "gsub(x,y)" for "x" "-" "y".
*
The expression "x" "*" "y" is the product of "x" and "y". Among the prominent
impossibilities are multiplication between vector/matrices of incompatible sizes, between
an integermod and a real number. Note that because of vector and matrix operations, "*" is
not necessarily commutative. Note also that since multiplication between two column or two
row vectors is not allowed, to obtain the scalar product of two vectors of the same
length, you must multiply a line vector by a column vector, if necessary by transposing
one of the vectors (using the operator "~" or the function "mattranspose", see "Label
se:linear_algebra").
If "x" and "y" are binary quadratic forms, compose them. See also "qfbnucomp" and
"qfbnupow".
The library syntax is gmul"(x,y)" for "x" "*" "y". Also available is "gsqr(x)" for "x" "*"
"x" (faster of course!).
/
The expression "x" "/" "y" is the quotient of "x" and "y". In addition to the
impossibilities for multiplication, note that if the divisor is a matrix, it must be an
invertible square matrix, and in that case the result is "x*y^{-1}". Furthermore note that
the result is as exact as possible: in particular, division of two integers always gives a
rational number (which may be an integer if the quotient is exact) and not the Euclidean
quotient (see "x" "\" "y" for that), and similarly the quotient of two polynomials is a
rational function in general. To obtain the approximate real value of the quotient of two
integers, add 0. to the result; to obtain the approximate "p"-adic value of the quotient
of two integers, add "O(p^k)" to the result; finally, to obtain the Taylor series
expansion of the quotient of two polynomials, add "O(X^k)" to the result or use the
"taylor" function (see "Label se:taylor").
The library syntax is gdiv"(x,y)" for "x" "/" "y".
\
The expression "x" "\" "y" is the
Euclidean quotient of "x" and "y". The types must be either both integer or both
polynomials. The result is the Euclidean quotient. In the case of integer division, the
quotient is such that the corresponding remainder is non-negative.
The library syntax is gdivent"(x,y)" for "x" "\" "y".
\/
The expression "x" "\/" "y" is the Euclidean quotient of "x" and "y". The types must be
either both integer or both polynomials. The result is the rounded Euclidean quotient. In
the case of integer division, the quotient is such that the corresponding remainder is
smallest in absolute value and in case of a tie the quotient closest to "+ oo " is chosen.
The library syntax is gdivround"(x,y)" for "x" "\/" "y".
%
The expression "x" "%" "y" is the
Euclidean remainder of "x" and "y". The modulus "y" must be of type integer or polynomial.
The result is the remainder, always non-negative in the case of integers. Allowed dividend
types are scalar exact types when the modulus is an integer, and polynomials, polmods and
rational functions when the modulus is a polynomial.
The library syntax is gmod"(x,y)" for "x" "%" "y".
divrem"(x,y)"
creates a column vector with two components, the first being the Euclidean quotient, the
second the Euclidean remainder, of the division of "x" by "y". This avoids the need to do
two divisions if one needs both the quotient and the remainder. The arguments must be both
integers or both polynomials; in the case of integers, the remainder is non-negative.
The library syntax is gdiventres"(x,y)".
^
The expression "x^n" is powering. If the exponent is an integer, then exact operations
are performed using binary (left-shift) powering techniques. In particular, in this case
"x" cannot be a vector or matrix unless it is a square matrix (and moreover invertible if
the exponent is negative). If "x" is a "p"-adic number, its precision will increase if
"v_p(n) > 0". PARI is able to rewrite the multiplication "x * x" of two identical objects
as "x^2", or sqr(x) (here, identical means the operands are two different labels
referencing the same chunk of memory; no equality test is performed). This is no longer
true when more than two arguments are involved.
If the exponent is not of type integer, this is treated as a transcendental function (see
"Label se:trans"), and in particular has the effect of componentwise powering on vector or
matrices.
As an exception, if the exponent is a rational number "p/q" and "x" an integer modulo a
prime, return a solution "y" of "y^q = x^p" if it exists. Currently, "q" must not have
large prime factors.
Beware that
? Mod(7,19)^(1/2)
%1 = Mod(11, 19)/*is any square root*/
? sqrt(Mod(7,19))
%2 = Mod(8, 19)/*is the smallest square root*/
? Mod(7,19)^(3/5)
%3 = Mod(1, 19)
? %3^(5/3)
%4 = Mod(1, 19)/*Mod(7,19) is just another cubic root*/
The library syntax is gpow"(x,n,prec)" for "x^n".
shift"(x,n)" or "x" "<< " "n" ( = "x" ">> " "(-n)")
shifts "x" componentwise left by "n" bits if "n >= 0" and right by "|n|" bits if "n < 0".
A left shift by "n" corresponds to multiplication by "2^n". A right shift of an integer
"x" by "|n|" corresponds to a Euclidean division of "x" by "2^{|n|}" with a remainder of
the same sign as "x", hence is not the same (in general) as "x \ 2^n".
The library syntax is gshift"(x,n)" where "n" is a "long".
shiftmul"(x,n)"
multiplies "x" by "2^n". The difference with "shift" is that when "n < 0", ordinary
division takes place, hence for example if "x" is an integer the result may be a fraction,
while for "shift" Euclidean division takes place when "n < 0" hence if "x" is an integer
the result is still an integer.
The library syntax is gmul2n"(x,n)" where "n" is a "long".
Comparison and boolean operators
The six standard comparison operators "<= ", "< ", ">= ", "> ", " == ", "! = " are
available in GP, and in library mode under the names gle, glt, gge, ggt, geq, gne
respectively. The library syntax is "co(x,y)", where co is the comparison operator. The
result is 1 (as a "GEN") if the comparison is true, 0 (as a "GEN") if it is false.
The standard boolean functions "||" (inclusive or), "&&" (and) and "!" (not) are also
available, and the library syntax is "gor(x,y)", "gand(x,y)" and "gnot(x)" respectively.
In library mode, it is in fact usually preferable to use the two basic functions which are
"gcmp(x,y)" which gives the sign (1, 0, or -1) of "x-y", where "x" and "y" must be in R,
and "gegal(x,y)" which can be applied to any two PARI objects "x" and "y" and gives 1
(i.e. true) if they are equal (but not necessarily identical), 0 (i.e. false) otherwise.
Particular cases of gegal which should be used are "gcmp0(x)" ("x == 0" ?), "gcmp1(x)" ("x
== 1" ?), and "gcmp_1(x)" ("x == -1" ?).
Note that "gcmp0(x)" tests whether "x" is equal to zero, even if "x" is not an exact
object. To test whether "x" is an exact object which is equal to zero, one must use
"isexactzero".
Also note that the "gcmp" and "gegal" functions return a C-integer, and not a "GEN" like
"gle" etc.
GP accepts the following synonyms for some of the above functions: since we thought it
might easily lead to confusion, we don't use the customary C operators for bitwise "and"
or bitwise "or" (use "bitand" or "bitor"), hence "|" and "&" are accepted as synonyms of
"||" and "&&" respectively. Also, "< > " is accepted as a synonym for "! = ". On the
other hand, " = " is definitely not a synonym for " == " since it is the assignment
statement. and bitwise or"
lex"(x,y)"
gives the result of a lexicographic comparison between "x" and "y". This is to be
interpreted in quite a wide sense. For example, the vector "[1,3]" will be considered
smaller than the longer vector "[1,3,-1]" (but of course larger than "[1,2,5]"),
i.e. "lex([1,3], [1,3,-1])" will return "-1".
The library syntax is lexcmp"(x,y)".
sign"(x)"
sign (0, 1 or "-1") of "x", which must be of type integer, real or fraction.
The library syntax is gsigne"(x)". The result is a "long".
max"(x,y)" and min"(x,y)"
creates the maximum and minimum of "x" and "y" when they can be compared.
The library syntax is gmax"(x,y)" and "gmin(x,y)".
vecmax"(x)"
if "x" is a vector or a matrix, returns the maximum of the elements of "x", otherwise
returns a copy of "x". Returns "- oo " in the form of "-(2^{31}-1)" (or "-(2^{63}-1)" for
64-bit machines) if "x" is empty.
The library syntax is vecmax"(x)".
vecmin"(x)"
if "x" is a vector or a matrix, returns the minimum of the elements of "x", otherwise
returns a copy of "x". Returns "+ oo " in the form of "2^{31}-1" (or "2^{63}-1" for 64-bit
machines) if "x" is empty.
The library syntax is vecmin"(x)".
Conversions and similar elementary functions or commands
Many of the conversion functions are rounding or truncating operations. In this case, if
the argument is a rational function, the result is the Euclidean quotient of the numerator
by the denominator, and if the argument is a vector or a matrix, the operation is done
componentwise. This will not be restated for every function.
List"({x = []})"
transforms a (row or column) vector "x" into a list. The only other way to create a
"t_LIST" is to use the function "listcreate".
This is useless in library mode.
Mat"({x = []})"
transforms the object "x" into a matrix. If "x" is not a vector or a matrix, this creates
a "1 x 1" matrix. If "x" is a row (resp. column) vector, this creates a 1-row (resp.
1-column) matrix. If "x" is already a matrix, a copy of "x" is created.
This function can be useful in connection with the function "concat" (see there).
The library syntax is gtomat"(x)".
Mod"(x,y,{flag = 0})"
creates the PARI object "(x mod y)", i.e. an integermod or a polmod. "y" must be an
integer or a polynomial. If "y" is an integer, "x" must be an integer, a rational number,
or a "p"-adic number compatible with the modulus "y". If "y" is a polynomial, "x" must be
a scalar (which is not a polmod), a polynomial, a rational function, or a power series.
This function is not the same as "x" "%" "y", the result of which is an integer or a
polynomial.
If "flag" is equal to 1, the modulus of the created result is put on the heap and not on
the stack, and hence becomes a permanent copy which cannot be erased later by garbage
collecting (see "Label se:garbage"). Functions will operate faster on such objects and
memory consumption will be lower. On the other hand, care should be taken to avoid
creating too many such objects.
Under GP, the same effect can be obtained by assigning the object to a GP variable (the
value of which is a permanent object for the duration of the relevant library function
call, and is treated as such). This value is subject to garbage collection, since it will
be deleted when the value changes. This is preferable and the above flag is only retained
for compatibility reasons (it can still be useful in library mode).
The library syntax is Mod0"(x,y,flag)". Also available are
"*" for "flag = 1": "gmodulo(x,y)".
"*" for "flag = 0": "gmodulcp(x,y)".
Pol"(x,{v = x})"
transforms the object "x" into a polynomial with main variable "v". If "x" is a scalar,
this gives a constant polynomial. If "x" is a power series, the effect is identical to
"truncate" (see there), i.e. it chops off the "O(X^k)". If "x" is a vector, this function
creates the polynomial whose coefficients are given in "x", with "x[1]" being the leading
coefficient (which can be zero).
Warning: this is not a substitution function. It is intended to be quick and dirty. So if
you try "Pol(a,y)" on the polynomial "a = x+y", you will get "y+y", which is not a valid
PARI object.
The library syntax is gtopoly"(x,v)", where "v" is a variable number.
Polrev"(x,{v = x})"
transform the object "x" into a polynomial with main variable "v". If "x" is a scalar,
this gives a constant polynomial. If "x" is a power series, the effect is identical to
"truncate" (see there), i.e. it chops off the "O(X^k)". If "x" is a vector, this function
creates the polynomial whose coefficients are given in "x", with "x[1]" being the constant
term. Note that this is the reverse of "Pol" if "x" is a vector, otherwise it is identical
to "Pol".
The library syntax is gtopolyrev"(x,v)", where "v" is a variable number.
Qfb"(a,b,c,{D = 0.})"
creates the binary quadratic form "ax^2+bxy+cy^2". If "b^2-4ac > 0", initialize Shanks'
distance function to "D".
The library syntax is Qfb0"(a,b,c,D,prec)". Also available are "qfi(a,b,c)" (when "b^2-4ac
< 0"), and "qfr(a,b,c,d)" (when "b^2-4ac > 0").
Ser"(x,{v = x})"
transforms the object "x" into a power series with main variable "v" ("x" by default). If
"x" is a scalar, this gives a constant power series with precision given by the default
"serieslength" (corresponding to the C global variable "precdl"). If "x" is a polynomial,
the precision is the greatest of "precdl" and the degree of the polynomial. If "x" is a
vector, the precision is similarly given, and the coefficients of the vector are
understood to be the coefficients of the power series starting from the constant term
(i.e. the reverse of the function "Pol").
The warning given for "Pol" applies here: this is not a substitution function.
The library syntax is gtoser"(x,v)", where "v" is a variable number (i.e. a C integer).
Set"({x = []})"
converts "x" into a set, i.e. into a row vector with strictly increasing entries. "x" can
be of any type, but is most useful when "x" is already a vector. The components of "x" are
put in canonical form (type "t_STR") so as to be easily sorted. To recover an ordinary
"GEN" from such an element, you can apply "eval" to it.
The library syntax is gtoset"(x)".
Str"({x = ""},{flag = 0})"
converts "x" into a character string (type "t_STR", the empty string if "x" is omitted).
To recover an ordinary "GEN" from a string, apply "eval" to it. The arguments of "Str" are
evaluated in string context (see "Label se:strings"). If flag is set, treat "x" as a
filename and perform environment expansion on the string. This feature can be used to read
environment variable values.
? i = 1; Str("x" i)
%1 = "x1"
? eval(%)
%2 = x1;
? Str("$HOME", 1)
%3 = "/home/pari"
The library syntax is strtoGENstr"(x,flag)". This function is mostly useless in library
mode. Use the pair "strtoGEN"/"GENtostr" to convert between "char*" and "GEN".
Vec"({x = []})"
transforms the object "x" into a row vector. The vector will be with one component only,
except when "x" is a vector/matrix or a quadratic form (in which case the resulting vector
is simply the initial object considered as a row vector), but more importantly when "x" is
a polynomial or a power series. In the case of a polynomial, the coefficients of the
vector start with the leading coefficient of the polynomial, while for power series only
the significant coefficients are taken into account, but this time by increasing order of
degree.
The library syntax is gtovec"(x)".
binary"(x)"
outputs the vector of the binary digits of "|x|". Here "x" can be an integer, a real
number (in which case the result has two components, one for the integer part, one for the
fractional part) or a vector/matrix.
The library syntax is binaire"(x)".
bitand"(x,y)"
bitwise "and" of two integers "x" and "y", that is the integer
"sum (x_i and y_i) 2^i"
Negative numbers behave as if modulo a huge power of 2.
The library syntax is gbitand"(x,y)".
bitneg"(x,{n = -1})"
bitwise negation of an integer "x", truncated to "n" bits, that is the integer
"sum_{i = 0}^n not(x_i) 2^i"
The special case "n = -1" means no truncation: an infinite sequence of leading 1 is then
represented as a negative number.
Negative numbers behave as if modulo a huge power of 2.
The library syntax is gbitneg"(x)".
bitnegimply"(x,y)"
bitwise negated imply of two integers "x" and "y" (or "not" "(x ==> y)"), that is the
integer
"sum (x_i and not(y_i)) 2^i"
Negative numbers behave as if modulo a huge power of 2.
The library syntax is gbitnegimply"(x,y)".
bitor"(x,y)"
bitwise (inclusive) "or" of two integers "x" and "y", that is the integer inclusive or"
"sum (x_i or y_i) 2^i"
Negative numbers behave as if modulo a huge power of 2.
The library syntax is gbitor"(x,y)".
bittest"(x,n)"
outputs the "n^{th}" bit of "|x|" starting from the right (i.e. the coefficient of "2^n"
in the binary expansion of "x"). The result is 0 or 1. To extract several bits at once as
a vector, pass a vector for "n".
The library syntax is bittest"(x,n)", where "n" and the result are "long"s.
bitxor"(x,y)"
bitwise (exclusive) "or" of two integers "x" and "y", that is the integer exclusive or"
"sum (x_i xor y_i) 2^i"
Negative numbers behave as if modulo a huge power of 2.
The library syntax is gbitxor"(x,y)".
ceil"(x)"
ceiling of "x". When "x" is in R, the result is the smallest integer greater than or equal
to "x". Applied to a rational function, ceil(x) returns the euclidian quotient of the
numerator by the denominator.
The library syntax is gceil"(x)".
centerlift"(x,{v})"
lifts an element "x = a mod n" of "Z/nZ" to "a" in Z, and similarly lifts a polmod to a
polynomial. This is the same as "lift" except that in the particular case of elements of
"Z/nZ", the lift "y" is such that "-n/2 < y <= n/2". If "x" is of type fraction, complex,
quadratic, polynomial, power series, rational function, vector or matrix, the lift is done
for each coefficient. Real and "p"-adics are forbidden.
The library syntax is centerlift0"(x,v)", where "v" is a "long" and an omitted "v" is
coded as "-1". Also available is centerlift"(x)" = "centerlift0(x,-1)".
changevar"(x,y)"
creates a copy of the object "x" where its variables are modified according to the
permutation specified by the vector "y". For example, assume that the variables have been
introduced in the order "x", "a", "b", "c". Then, if "y" is the vector "[x,c,a,b]", the
variable "a" will be replaced by "c", "b" by "a", and "c" by "b", "x" being unchanged.
Note that the permutation must be completely specified, e.g. "[c,a,b]" would not work,
since this would replace "x" by "c", and leave "a" and "b" unchanged (as well as "c" which
is the fourth variable of the initial list). In particular, the new variable names must be
distinct.
The library syntax is changevar"(x,y)".
components of a PARI object
There are essentially three ways to extract the components from a PARI object.
The first and most general, is the function "component(x,n)" which extracts the
"n^{th}"-component of "x". This is to be understood as follows: every PARI type has one or
two initial code words. The components are counted, starting at 1, after these code words.
In particular if "x" is a vector, this is indeed the "n^{th}"-component of "x", if "x" is
a matrix, the "n^{th}" column, if "x" is a polynomial, the "n^{th}" coefficient (i.e. of
degree "n-1"), and for power series, the "n^{th}" significant coefficient. The use of the
function "component" implies the knowledge of the structure of the different PARI types,
which can be recalled by typing "\t" under GP.
The library syntax is compo"(x,n)", where "n" is a "long".
The two other methods are more natural but more restricted. The function " polcoeff(x,n)"
gives the coefficient of degree "n" of the polynomial or power series "x", with respect to
the main variable of "x" (to check variable ordering, or to change it, use the function
"reorder", see "Label se:reorder"). In particular if "n" is less than the valuation of "x"
or in the case of a polynomial, greater than the degree, the result is zero (contrary to
"compo" which would send an error message). If "x" is a power series and "n" is greater
than the largest significant degree, then an error message is issued.
For greater flexibility, vector or matrix types are also accepted for "x", and the meaning
is then identical with that of "compo".
Finally note that a scalar type is considered by "polcoeff" as a polynomial of degree
zero.
The library syntax is truecoeff"(x,n)".
The third method is specific to vectors or matrices under GP. If "x" is a (row or column)
vector, then "x[n]" represents the "n^{th}" component of "x", i.e. "compo(x,n)". It is
more natural and shorter to write. If "x" is a matrix, "x[m,n]" represents the coefficient
of row "m" and column "n" of the matrix, "x[m,]" represents the "m^{th}" row of "x", and
"x[,n]" represents the "n^{th}" column of "x".
Finally note that in library mode, the macros coeff and mael are available to deal with
the non-recursivity of the "GEN" type from the compiler's point of view. See the
discussion on typecasts in Chapter 4.
conj"(x)"
conjugate of "x". The meaning of this is clear, except that for real quadratic numbers, it
means conjugation in the real quadratic field. This function has no effect on integers,
reals, integermods, fractions or "p"-adics. The only forbidden type is polmod (see
"conjvec" for this).
The library syntax is gconj"(x)".
conjvec"(x)"
conjugate vector representation of "x". If "x" is a polmod, equal to "Mod""(a,q)", this
gives a vector of length degree(q) containing the complex embeddings of the polmod if "q"
has integral or rational coefficients, and the conjugates of the polmod if "q" has some
integermod coefficients. The order is the same as that of the "polroots" functions. If "x"
is an integer or a rational number, the result is "x". If "x" is a (row or column) vector,
the result is a matrix whose columns are the conjugate vectors of the individual elements
of "x".
The library syntax is conjvec"(x,prec)".
denominator"(x)"
lowest denominator of "x". The meaning of this is clear when "x" is a rational number or
function. When "x" is an integer or a polynomial, the result is equal to 1. When "x" is a
vector or a matrix, the lowest common denominator of the components of "x" is computed.
All other types are forbidden.
The library syntax is denom"(x)".
floor"(x)"
floor of "x". When "x" is in R, the result is the largest integer smaller than or equal to
"x". Applied to a rational function, floor(x) returns the euclidian quotient of the
numerator by the denominator.
The library syntax is gfloor"(x)".
frac"(x)"
fractional part of "x". Identical to "x-floor(x)". If "x" is real, the result is in
"[0,1[".
The library syntax is gfrac"(x)".
imag"(x)"
imaginary part of "x". When "x" is a quadratic number, this is the coefficient of "omega"
in the ``canonical'' integral basis "(1,omega)".
The library syntax is gimag"(x)".
length"(x)"
number of non-code words in "x" really used (i.e. the effective length minus 2 for
integers and polynomials). In particular, the degree of a polynomial is equal to its
length minus 1. If "x" has type "t_STR", output number of letters.
The library syntax is glength"(x)" and the result is a C long.
lift"(x,{v})"
lifts an element "x = a mod n" of "Z/nZ" to "a" in Z, and similarly lifts a polmod to a
polynomial if "v" is omitted. Otherwise, lifts only polmods with main variable "v" (if
"v" does not occur in "x", lifts only intmods). If "x" is of type fraction, complex,
quadratic, polynomial, power series, rational function, vector or matrix, the lift is done
for each coefficient. Forbidden types for "x" are reals and "p"-adics.
The library syntax is lift0"(x,v)", where "v" is a "long" and an omitted "v" is coded as
"-1". Also available is lift"(x)" = "lift0(x,-1)".
norm"(x)"
algebraic norm of "x", i.e. the product of "x" with its conjugate (no square roots are
taken), or conjugates for polmods. For vectors and matrices, the norm is taken
componentwise and hence is not the "L^2"-norm (see "norml2"). Note that the norm of an
element of R is its square, so as to be compatible with the complex norm.
The library syntax is gnorm"(x)".
norml2"(x)"
square of the "L^2"-norm of "x". "x" must be a (row or column) vector.
The library syntax is gnorml2"(x)".
numerator"(x)"
numerator of "x". When "x" is a rational number or function, the meaning is clear. When
"x" is an integer or a polynomial, the result is "x" itself. When "x" is a vector or a
matrix, then numerator(x) is defined to be "denominator(x)*x". All other types are
forbidden.
The library syntax is numer"(x)".
numtoperm"(n,k)"
generates the "k"-th permutation (as a row vector of length "n") of the numbers 1 to "n".
The number "k" is taken modulo "n!", i.e. inverse function of "permtonum".
The library syntax is permute"(n,k)", where "n" is a "long".
padicprec"(x,p)"
absolute "p"-adic precision of the object "x". This is the minimum precision of the
components of "x". The result is "VERYBIGINT" ("2^{31}-1" for 32-bit machines or
"2^{63}-1" for 64-bit machines) if "x" is an exact object.
The library syntax is padicprec"(x,p)" and the result is a "long" integer.
permtonum"(x)"
given a permutation "x" on "n" elements, gives the number "k" such that "x =
numtoperm(n,k)", i.e. inverse function of "numtoperm".
The library syntax is permuteInv"(x)".
precision"(x,{n})"
gives the precision in decimal digits of the PARI object "x". If "x" is an exact object,
the largest single precision integer is returned. If "n" is not omitted, creates a new
object equal to "x" with a new precision "n". This is to be understood as follows:
For exact types, no change. For "x" a vector or a matrix, the operation is done
componentwise.
For real "x", "n" is the number of desired significant decimal digits. If "n" is smaller
than the precision of "x", "x" is truncated, otherwise "x" is extended with zeros.
For "x" a "p"-adic or a power series, "n" is the desired number of significant "p"-adic or
"X"-adic digits, where "X" is the main variable of "x".
Note that the function "precision" never changes the type of the result. In particular it
is not possible to use it to obtain a polynomial from a power series. For that, see
"truncate".
The library syntax is precision0"(x,n)", where "n" is a "long". Also available are
"ggprecision(x)" (result is a "GEN") and "gprec(x,n)", where "n" is a "long".
random"({N = 2^{31}})"
gives a random integer between 0 and "N-1". "N" can be arbitrary large. This is an
internal PARI function and does not depend on the system's random number generator. Note
that the resulting integer is obtained by means of linear congruences and will not be well
distributed in arithmetic progressions.
The library syntax is genrand"(N)".
real"(x)"
real part of "x". In the case where "x" is a quadratic number, this is the coefficient of
1 in the ``canonical'' integral basis "(1,omega)".
The library syntax is greal"(x)".
round"(x,{&e})"
If "x" is in R, rounds "x" to the nearest integer and sets "e" to the number of error
bits, that is the binary exponent of the difference between the original and the rounded
value (the ``fractional part''). If the exponent of "x" is too large compared to its
precision (i.e. "e > 0"), the result is undefined and an error occurs if "e" was not
given.
Important remark: note that, contrary to the other truncation functions, this function
operates on every coefficient at every level of a PARI object. For example
"truncate((2.4*X^2-1.7)/(X)) = 2.4*X,"
whereas
"round((2.4*X^2-1.7)/(X)) = (2*X^2-2)/(X)."
An important use of "round" is to get exact results after a long approximate computation,
when theory tells you that the coefficients must be integers.
The library syntax is grndtoi"(x,&e)", where "e" is a "long" integer. Also available is
"ground(x)".
simplify"(x)"
this function tries to simplify the object "x" as much as it can. The simplifications do
not concern rational functions (which PARI automatically tries to simplify), but type
changes. Specifically, a complex or quadratic number whose imaginary part is exactly equal
to 0 (i.e. not a real zero) is converted to its real part, and a polynomial of degree zero
is converted to its constant term. For all types, this of course occurs recursively. This
function is useful in any case, but in particular before the use of arithmetic functions
which expect integer arguments, and not for example a complex number of 0 imaginary part
and integer real part (which is however printed as an integer).
The library syntax is simplify"(x)".
sizebyte"(x)"
outputs the total number of bytes occupied by the tree representing the PARI object "x".
The library syntax is taille2"(x)" which returns a "long". The function taille returns the
number of words instead.
sizedigit"(x)"
outputs a quick bound for the number of decimal digits of (the components of) "x", off by
at most 1. If you want the exact value, you can use "length(Str(x))", which is much
slower.
The library syntax is sizedigit"(x)" which returns a "long".
truncate"(x,{&e})"
truncates "x" and sets "e" to the number of error bits. When "x" is in R, this means that
the part after the decimal point is chopped away, "e" is the binary exponent of the
difference between the original and the truncated value (the ``fractional part''). If the
exponent of "x" is too large compared to its precision (i.e. "e > 0"), the result is
undefined and an error occurs if "e" was not given. The function applies componentwise on
rational functions and vector / matrices; "e" is then the maximal number of error bits.
Note a very special use of "truncate": when applied to a power series, it transforms it
into a polynomial or a rational function with denominator a power of "X", by chopping away
the "O(X^k)". Similarly, when applied to a "p"-adic number, it transforms it into an
integer or a rational number by chopping away the "O(p^k)".
The library syntax is gcvtoi"(x,&e)", where "e" is a "long" integer. Also available is
gtrunc"(x)".
valuation"(x,p)"
computes the highest exponent of "p" dividing "x". If "p" is of type integer, "x" must be
an integer, an integermod whose modulus is divisible by "p", a fraction, a "q"-adic number
with "q = p", or a polynomial or power series in which case the valuation is the minimum
of the valuation of the coefficients.
If "p" is of type polynomial, "x" must be of type polynomial or rational function, and
also a power series if "x" is a monomial. Finally, the valuation of a vector, complex or
quadratic number is the minimum of the component valuations.
If "x = 0", the result is "VERYBIGINT" ("2^{31}-1" for 32-bit machines or "2^{63}-1" for
64-bit machines) if "x" is an exact object. If "x" is a "p"-adic numbers or power series,
the result is the exponent of the zero. Any other type combinations gives an error.
The library syntax is ggval"(x,p)", and the result is a "long".
variable"(x)"
gives the main variable of the object "x", and "p" if "x" is a "p"-adic number. Gives an
error if "x" has no variable associated to it. Note that this function is useful only in
GP, since in library mode the function "gvar" is more appropriate.
The library syntax is gpolvar"(x)". However, in library mode, this function should not be
used. Instead, test whether "x" is a "p"-adic (type "t_PADIC"), in which case "p" is in
"x[2]", or call the function "gvar(x)" which returns the variable number of "x" if it
exists, "BIGINT" otherwise.
Transcendental functions
As a general rule, which of course in some cases may have exceptions, transcendental
functions operate in the following way:
"*" If the argument is either an integer, a real, a rational, a complex or a quadratic
number, it is, if necessary, first converted to a real (or complex) number using the
current precision held in the default "realprecision". Note that only exact arguments are
converted, while inexact arguments such as reals are not.
Under GP this is transparent to the user, but when programming in library mode, care must
be taken to supply a meaningful parameter prec as the last argument of the function if the
first argument is an exact object. This parameter is ignored if the argument is inexact.
Note that in library mode the precision argument prec is a word count including codewords,
i.e. represents the length in words of a real number, while under GP the precision (which
is changed by the metacommand "\p" or using "default(realprecision,...)") is the number of
significant decimal digits.
Note that some accuracies attainable on 32-bit machines cannot be attained on 64-bit
machines for parity reasons. For example the default GP accuracy is 28 decimal digits on
32-bit machines, corresponding to prec having the value 5, but this cannot be attained on
64-bit machines.
After possible conversion, the function is computed. Note that even if the argument is
real, the result may be complex (e.g. "acos(2.0)" or "acosh(0.0)"). Note also that the
principal branch is always chosen.
"*" If the argument is an integermod or a "p"-adic, at present only a few functions like
"sqrt" (square root), "sqr" (square), "log", "exp", powering, "teichmuller" (Teichmueller
character) and "agm" (arithmetic-geometric mean) are implemented.
Note that in the case of a 2-adic number, sqr(x) may not be identical to "x*x": for
example if "x = 1+O(2^5)" and "y = 1+O(2^5)" then "x*y = 1+O(2^5)" while "sqr(x) =
1+O(2^6)". Here, "x * x" yields the same result as sqr(x) since the two operands are known
to be identical. The same statement holds true for "p"-adics raised to the power "n",
where "v_p(n) > 0".
Remark: note that if we wanted to be strictly consistent with the PARI philosophy, we
should have "x*y = (4 mod 8)" and "sqr(x) = (4 mod 32)" when both "x" and "y" are
congruent to 2 modulo 4. However, since integermod is an exact object, PARI assumes that
the modulus must not change, and the result is hence "(0 mod 4)" in both cases. On the
other hand, "p"-adics are not exact objects, hence are treated differently.
"*" If the argument is a polynomial, power series or rational function, it is, if
necessary, first converted to a power series using the current precision held in the
variable "precdl". Under GP this again is transparent to the user. When programming in
library mode, however, the global variable "precdl" must be set before calling the
function if the argument has an exact type (i.e. not a power series). Here "precdl" is not
an argument of the function, but a global variable.
Then the Taylor series expansion of the function around "X = 0" (where "X" is the main
variable) is computed to a number of terms depending on the number of terms of the
argument and the function being computed.
"*" If the argument is a vector or a matrix, the result is the componentwise evaluation of
the function. In particular, transcendental functions on square matrices, which are not
implemented in the present version 2.2.0 (see Appendix B however), will have a slightly
different name if they are implemented some day.
^
If "y" is not of type integer, "x^y" has the same effect as "exp(y*ln(x))". It can be
applied to "p"-adic numbers as well as to the more usual types.
The library syntax is gpow"(x,y,prec)".
Euler
Euler's constant 0.57721.... Note that "Euler" is one of the few special reserved names
which cannot be used for variables (the others are "I" and "Pi", as well as all function
names).
The library syntax is mpeuler"(prec)" where "prec" must be given. Note that this creates
"gamma" on the PARI stack, but a copy is also created on the heap for quicker computations
next time the function is called.
I
the complex number " sqrt {-1}".
The library syntax is the global variable "gi" (of type "GEN").
Pi
the constant "Pi" (3.14159...).
The library syntax is mppi"(prec)" where "prec" must be given. Note that this creates "Pi"
on the PARI stack, but a copy is also created on the heap for quicker computations next
time the function is called.
abs"(x)"
absolute value of "x" (modulus if "x" is complex). Power series and rational functions
are not allowed. Contrary to most transcendental functions, an exact argument is not
converted to a real number before applying "abs" and an exact result is returned if
possible.
? abs(-1)
%1 = 1
? abs(3/7 + 4/7*I)
%2 = 5/7
? abs(1 + I)
%3 = 1.414213562373095048801688724
If "x" is a polynomial, returns "-x" if the leading coefficient is real and negative else
returns "x". For a power series, the constant coefficient is considered instead.
The library syntax is gabs"(x,prec)".
acos"(x)"
principal branch of "cos^{-1}(x)", i.e. such that "Re(acos(x)) belongs to [0,Pi]". If "x
belongs to R" and "|x| > 1", then acos(x) is complex.
The library syntax is gacos"(x,prec)".
acosh"(x)"
principal branch of "cosh^{-1}(x)", i.e. such that "Im(acosh(x)) belongs to [0,Pi]". If "x
belongs to R" and "x < 1", then acosh(x) is complex.
The library syntax is gach"(x,prec)".
agm"(x,y)"
arithmetic-geometric mean of "x" and "y". In the case of complex or negative numbers, the
principal square root is always chosen. "p"-adic or power series arguments are also
allowed. Note that a "p"-adic agm exists only if "x/y" is congruent to 1 modulo "p"
(modulo 16 for "p = 2"). "x" and "y" cannot both be vectors or matrices.
The library syntax is agm"(x,y,prec)".
arg"(x)"
argument of the complex number "x", such that "-Pi < arg(x) <= Pi".
The library syntax is garg"(x,prec)".
asin"(x)"
principal branch of "sin^{-1}(x)", i.e. such that "Re(asin(x)) belongs to [-Pi/2,Pi/2]".
If "x belongs to R" and "|x| > 1" then asin(x) is complex.
The library syntax is gasin"(x,prec)".
asinh"(x)"
principal branch of "sinh^{-1}(x)", i.e. such that "Im(asinh(x)) belongs to [-Pi/2,Pi/2]".
The library syntax is gash"(x,prec)".
atan"(x)"
principal branch of "tan^{-1}(x)", i.e. such that "Re(atan(x)) belongs to ]-Pi/2,Pi/2[".
The library syntax is gatan"(x,prec)".
atanh"(x)"
principal branch of "tanh^{-1}(x)", i.e. such that "Im(atanh(x)) belongs to
]-Pi/2,Pi/2]". If "x belongs to R" and "|x| > 1" then atanh(x) is complex.
The library syntax is gath"(x,prec)".
bernfrac"(x)"
Bernoulli number "B_x", where "B_0 = 1", "B_1 = -1/2", "B_2 = 1/6",..., expressed as a
rational number. The argument "x" should be of type integer.
The library syntax is bernfrac"(x)".
bernreal"(x)"
Bernoulli number "B_x", as "bernfrac", but "B_x" is returned as a real number (with the
current precision).
The library syntax is bernreal"(x,prec)".
bernvec"(x)"
creates a vector containing, as rational numbers, the Bernoulli numbers "B_0", "B_2",...,
"B_{2x}". These Bernoulli numbers can then be used as follows. Assume that this vector has
been put into a variable, say "bernint". Then you can define under GP:
bern(x) =
{
if (x == 1, return(-1/2));
if (x < 0 || x % 2, return(0));
bernint[x/2+1]
}
and then bern(k) gives the Bernoulli number of index "k" as a rational number, exactly as
bernreal(k) gives it as a real number. If you need only a few values, calling bernfrac(k)
each time will be much more efficient than computing the huge vector above.
The library syntax is bernvec"(x)".
besseljh"(n,x)"
"J"-Bessel function of half integral index. More precisely, "besseljh(n,x)" computes
"J_{n+1/2}(x)" where "n" must be of type integer, and "x" is any element of C. In the
present version 2.2.0, this function is not very accurate when "x" is small.
The library syntax is jbesselh"(n,x,prec)".
besselk"(nu,x,{flag = 0})"
"K"-Bessel function of index nu (which can be complex) and argument "x". Only real and
positive arguments "x" are allowed in the present version 2.2.0. If "flag" is equal to 1,
uses another implementation of this function which is often faster.
The library syntax is kbessel"(nu,x,prec)" and "kbessel2(nu,x,prec)" respectively.
cos"(x)"
cosine of "x".
The library syntax is gcos"(x,prec)".
cosh"(x)"
hyperbolic cosine of "x".
The library syntax is gch"(x,prec)".
cotan"(x)"
cotangent of "x".
The library syntax is gcotan"(x,prec)".
dilog"(x)"
principal branch of the dilogarithm of "x", i.e. analytic continuation of the power series
" log _2(x) = sum_{n >= 1}x^n/n^2".
The library syntax is dilog"(x,prec)".
eint1"(x,{n})"
exponential integral "int_x^ oo (e^{-t})/(t)dt" ("x belongs to R")
If "n" is present, outputs the "n"-dimensional vector "[eint1(x),...,eint1(nx)]" ("x >=
0"). This is faster than repeatedly calling "eint1(i * x)".
The library syntax is veceint1"(x,n,prec)". Also available is "eint1(x,prec)".
erfc"(x)"
complementary error function "(2/ sqrt Pi)int_x^ oo e^{-t^2}dt".
The library syntax is erfc"(x,prec)".
eta"(x,{flag = 0})"
Dedekind's "eta" function, without the "q^{1/24}". This means the following: if "x" is a
complex number with positive imaginary part, the result is "prod_{n = 1}^ oo (1-q^n)",
where "q = e^{2iPi x}". If "x" is a power series (or can be converted to a power series)
with positive valuation, the result is "prod_{n = 1}^ oo (1-x^n)".
If "flag = 1" and "x" can be converted to a complex number (i.e. is not a power series),
computes the true "eta" function, including the leading "q^{1/24}".
The library syntax is eta"(x,prec)".
exp"(x)"
exponential of "x". "p"-adic arguments with positive valuation are accepted.
The library syntax is gexp"(x,prec)".
gammah"(x)"
gamma function evaluated at the argument "x+1/2". When "x" is an integer, this is much
faster than using "gamma(x+1/2)".
The library syntax is ggamd"(x,prec)".
gamma"(x)"
gamma function of "x". In the present version 2.2.0 the "p"-adic gamma function is not
implemented.
The library syntax is ggamma"(x,prec)".
hyperu"(a,b,x)"
"U"-confluent hypergeometric function with parameters "a" and "b". The parameters "a" and
"b" can be complex but the present implementation requires "x" to be positive.
The library syntax is hyperu"(a,b,x,prec)".
incgam"(s,x,{y})"
incomplete gamma function.
"x" must be positive and "s" real. The result returned is "int_x^ oo e^{-t}t^{s-1}dt".
When "y" is given, assume (of course without checking!) that "y = Gamma(s)". For small
"x", this will tremendously speed up the computation.
The library syntax is incgam"(s,x,prec)" and "incgam4(s,x,y,prec)", respectively. There
exist also the functions incgam1 and incgam2 which are used for internal purposes.
incgamc"(s,x)"
complementary incomplete gamma function.
The arguments "s" and "x" must be positive. The result returned is "int_0^x
e^{-t}t^{s-1}dt", when "x" is not too large.
The library syntax is incgam3"(s,x,prec)".
log"(x,{flag = 0})"
principal branch of the natural logarithm of "x", i.e. such that "Im(ln(x)) belongs to
]-Pi,Pi]". The result is complex (with imaginary part equal to "Pi") if "x belongs to R"
and "x < 0".
"p"-adic arguments are also accepted for "x", with the convention that " ln (p) = 0".
Hence in particular " exp ( ln (x))/x" will not in general be equal to 1 but to a
"(p-1)"-th root of unity (or "+-1" if "p = 2") times a power of "p".
If "flag" is equal to 1, use an agm formula suggested by Mestre, when "x" is real,
otherwise identical to "log".
The library syntax is glog"(x,prec)" or "glogagm(x,prec)".
lngamma"(x)"
principal branch of the logarithm of the gamma function of "x". Can have much larger
arguments than "gamma" itself. In the present version 2.2.0, the "p"-adic "lngamma"
function is not implemented.
The library syntax is glngamma"(x,prec)".
polylog"(m,x,{flag = 0})"
one of the different polylogarithms, depending on flag:
If "flag = 0" or is omitted: "m^th" polylogarithm of "x", i.e. analytic continuation of
the power series "Li_m(x) = sum_{n >= 1}x^n/n^m". The program uses the power series when
"|x|^2 <= 1/2", and the power series expansion in " log (x)" otherwise. It is valid in a
large domain (at least "|x| < 230"), but should not be used too far away from the unit
circle since it is then better to use the functional equation linking the value at "x" to
the value at "1/x", which takes a trivial form for the variant below. Power series,
polynomial, rational and vector/matrix arguments are allowed.
For the variants to follow we need a notation: let " Re _m" denotes " Re " or " Im "
depending whether "m" is odd or even.
If "flag = 1": modified "m^th" polylogarithm of "x", called "~ D_m(x)" in Zagier, defined
for "|x| <= 1" by
" Re _m(sum_{k = 0}^{m-1} ((- log |x|)^k)/(k!)Li_{m-k}(x) +((- log |x|)^{m-1})/(m!) log
|1-x|)."
If "flag = 2": modified "m^th" polylogarithm of "x", called D_m(x) in Zagier, defined for
"|x| <= 1" by
" Re _m(sum_{k = 0}^{m-1}((- log |x|)^k)/(k!)Li_{m-k}(x) -(1)/(2)((- log |x|)^m)/(m!))."
If "flag = 3": another modified "m^th" polylogarithm of "x", called P_m(x) in Zagier,
defined for "|x| <= 1" by
" Re _m(sum_{k = 0}^{m-1}(2^kB_k)/(k!)( log |x|)^kLi_{m-k}(x) -(2^{m-1}B_m)/(m!)( log
|x|)^m)."
These three functions satisfy the functional equation "f_m(1/x) = (-1)^{m-1}f_m(x)".
The library syntax is polylog0"(m,x,flag,prec)".
psi"(x)"
the "psi"-function of "x", i.e. the logarithmic derivative "Gamma'(x)/Gamma(x)".
The library syntax is gpsi"(x,prec)".
sin"(x)"
sine of "x".
The library syntax is gsin"(x,prec)".
sinh"(x)"
hyperbolic sine of "x".
The library syntax is gsh"(x,prec)".
sqr"(x)"
square of "x". This operation is not completely straightforward, i.e. identical to "x *
x", since it can usually be computed more efficiently (roughly one-half of the elementary
multiplications can be saved). Also, squaring a 2-adic number increases its precision. For
example,
? (1 + O(2^4))^2
%1 = 1 + O(2^5)
? (1 + O(2^4)) * (1 + O(2^4))
%2 = 1 + O(2^4)
Note that this function is also called whenever one multiplies two objects which are known
to be identical, e.g. they are the value of the same variable, or we are computing a
power.
? x = (1 + O(2^4)); x * x
%3 = 1 + O(2^5)
? (1 + O(2^4))^4
%4 = 1 + O(2^6)
(note the difference between %2 and %3 above).
The library syntax is gsqr"(x)".
sqrt"(x)"
principal branch of the square root of "x", i.e. such that "Arg(sqrt(x)) belongs to
]-Pi/2, Pi/2]", or in other words such that " Re (sqrt(x)) > 0" or " Re (sqrt(x)) = 0" and
" Im (sqrt(x)) >= 0". If "x belongs to R" and "x < 0", then the result is complex with
positive imaginary part.
Integermod a prime and "p"-adics are allowed as arguments. In that case, the square root
(if it exists) which is returned is the one whose first "p"-adic digit (or its unique
"p"-adic digit in the case of integermods) is in the interval "[0,p/2]". When the argument
is an integermod a non-prime (or a non-prime-adic), the result is undefined.
The library syntax is gsqrt"(x,prec)".
sqrtn"(x,n,{&z})"
principal branch of the "n"th root of "x", i.e. such that "Arg(sqrt(x)) belongs to
]-Pi/n, Pi/n]".
Integermod a prime and "p"-adics are allowed as arguments.
If "z" is present, it is set to a suitable root of unity allowing to recover all the other
roots. If it was not possible, z is set to zero.
The following script computes all roots in all possible cases:
sqrtnall(x,n)=
{
local(V,r,z,r2);
r = sqrtn(x,n, &z);
if (!z, error("Impossible case in sqrtn"));
if (type(x) == "t_INTMOD" || type(x)=="t_PADIC" ,
r2 = r*z; n = 1;
while (r2!=r, r2*=z;n++));
V = vector(n); V[1] = r;
for(i=2, n, V[i] = V[i-1]*z);
V
}
addhelp(sqrtnall,"sqrtnall(x,n):compute the vector of nth-roots of x");
The library syntax is gsqrtn"(x,n,&z,prec)".
tan"(x)"
tangent of "x".
The library syntax is gtan"(x,prec)".
tanh"(x)"
hyperbolic tangent of "x".
The library syntax is gth"(x,prec)".
teichmuller"(x)"
Teichmueller character of the "p"-adic number "x".
The library syntax is teich"(x)".
theta"(q,z)"
Jacobi sine theta-function.
The library syntax is theta"(q,z,prec)".
thetanullk"(q,k)"
"k"-th derivative at "z = 0" of "theta(q,z)".
The library syntax is thetanullk"(q,k,prec)", where "k" is a "long".
weber"(x,{flag = 0})"
one of Weber's three "f" functions. If "flag = 0", returns
"f(x) = exp (-iPi/24).eta((x+1)/2)/eta(x) such that j = (f^{24}-16)^3/f^{24},"
where "j" is the elliptic "j"-invariant (see the function "ellj"). If "flag = 1",
returns
"f_1(x) = eta(x/2)/eta(x) such that j = (f_1^{24}+16)^3/f_1^{24}."
Finally, if "flag = 2", returns
"f_2(x) = sqrt {2}eta(2x)/eta(x) such that j = (f_2^{24}+16)^3/f_2^{24}."
Note the identities "f^8 = f_1^8+f_2^8" and "ff_1f_2 = sqrt 2".
The library syntax is weber0"(x,flag,prec)", or "wf(x,prec)", "wf1(x,prec)" or
"wf2(x,prec)".
zeta"(s)"
Riemann's zeta function "zeta(s) = sum_{n >= 1}n^{-s}", computed using the Euler-Maclaurin
summation formula, except when "s" is of type integer, in which case it is computed using
Bernoulli numbers for "s <= 0" or "s > 0" and even, and using modular forms for "s > 0"
and odd.
The library syntax is gzeta"(s,prec)".
Arithmetic functions
These functions are by definition functions whose natural domain of definition is either Z
(or "Z_{ > 0}"), or sometimes polynomials over a base ring. Functions which concern
polynomials exclusively will be explained in the next section. The way these functions are
used is completely different from transcendental functions: in general only the types
integer and polynomial are accepted as arguments. If a vector or matrix type is given, the
function will be applied on each coefficient independently.
In the present version 2.2.0, all arithmetic functions in the narrow sense of the word ---
Euler's totient function, the Moebius function, the sums over divisors or powers of
divisors etc.--- call, after trial division by small primes, the same versatile factoring
machinery described under "factorint". It includes Shanks SQUFOF, Pollard Rho, ECM and
MPQS stages, and has an early exit option for the functions moebius and (the integer
function underlying) issquarefree. Note that it relies on a (fairly strong) probabilistic
primality test: numbers found to be strong pseudo-primes after 10 successful trials of the
Rabin-Miller test are declared primes.
addprimes"({x = []})"
adds the primes contained in the vector "x" (or the single integer "x") to the table
computed upon GP initialization (by "pari_init" in library mode), and returns a row vector
whose first entries contain all primes added by the user and whose last entries have been
filled up with 1's. In total the returned row vector has 100 components. Whenever
"factor" or "smallfact" is subsequently called, first the primes in the table computed by
"pari_init" will be checked, and then the additional primes in this table. If "x" is empty
or omitted, just returns the current list of extra primes.
The entries in "x" are not checked for primality. They need only be positive integers not
divisible by any of the pre-computed primes. It's in fact a nice trick to add composite
numbers, which for example the function "factor(x,0)" was not able to factor. In case the
message ``impossible inverse modulo "<"some integermod">"'' shows up afterwards, you have
just stumbled over a non-trivial factor. Note that the arithmetic functions in the narrow
sense, like eulerphi, do not use this extra table.
The present PARI version 2.2.0 allows up to 100 user-specified primes to be appended to
the table. This limit may be changed by altering "NUMPRTBELT" in file "init.c". To remove
primes from the list use "removeprimes".
The library syntax is addprimes"(x)".
bestappr"(x,k)"
if "x belongs to R", finds the best rational approximation to "x" with denominator at most
equal to "k" using continued fractions.
The library syntax is bestappr"(x,k)".
bezout"(x,y)"
finds "u" and "v" minimal in a natural sense such that "x*u+y*v = gcd(x,y)". The arguments
must be both integers or both polynomials, and the result is a row vector with three
components "u", "v", and "gcd(x,y)".
The library syntax is vecbezout"(x,y)" to get the vector, or "gbezout(x,y, &u, &v)" which
gives as result the address of the created gcd, and puts the addresses of the
corresponding created objects into "u" and "v".
bezoutres"(x,y)"
as "bezout", with the resultant of "x" and "y" replacing the gcd.
The library syntax is vecbezoutres"(x,y)" to get the vector, or "subresext(x,y, &u, &v)"
which gives as result the address of the created gcd, and puts the addresses of the
corresponding created objects into "u" and "v".
bigomega"(x)"
number of prime divisors of "|x|" counted with multiplicity. "x" must be an integer.
The library syntax is bigomega"(x)", the result is a "long".
binomial"(x,y)"
binomial coefficient "\binom x y". Here "y" must be an integer, but "x" can be any PARI
object.
The library syntax is binome"(x,y)", where "y" must be a "long".
chinese"(x,y)"
if "x" and "y" are both integermods or both polmods, creates (with the same type) a "z" in
the same residue class as "x" and in the same residue class as "y", if it is possible.
This function also allows vector and matrix arguments, in which case the operation is
recursively applied to each component of the vector or matrix. For polynomial arguments,
it is applied to each coefficient. Finally "chinese(x,x) = x" regardless of the type of
"x"; this allows vector arguments to contain other data, so long as they are identical in
both vectors.
The library syntax is chinois"(x,y)".
content"(x)"
computes the gcd of all the coefficients of "x", when this gcd makes sense. If "x" is a
scalar, this simply returns "x". If "x" is a polynomial (and by extension a power series),
it gives the usual content of "x". If "x" is a rational function, it gives the ratio of
the contents of the numerator and the denominator. Finally, if "x" is a vector or a
matrix, it gives the gcd of all the entries.
The library syntax is content"(x)".
contfrac"(x,{b},{lmax})"
creates the row vector whose components are the partial quotients of the continued
fraction expansion of "x", the number of partial quotients being limited to "lmax". If
"x" is a real number, the expansion stops at the last significant partial quotient if
"lmax" is omitted. "x" can also be a rational function or a power series.
If a vector "b" is supplied, the numerators will be equal to the coefficients of "b". The
length of the result is then equal to the length of "b", unless a partial remainder is
encountered which is equal to zero. In which case the expansion stops. In the case of real
numbers, the stopping criterion is thus different from the one mentioned above since, if
"b" is too long, some partial quotients may not be significant.
If "b" is an integer, the command is understood as "contfrac(x,lmax)".
The library syntax is contfrac0"(x,b,lmax)". Also available are "gboundcf(x,lmax)",
"gcf(x)", or "gcf2(b,x)", where "lmax" is a C integer.
contfracpnqn"(x)"
when "x" is a vector or a one-row matrix, "x" is considered as the list of partial
quotients "[a_0,a_1,...,a_n]" of a rational number, and the result is the 2 by 2 matrix
"[p_n,p_{n-1};q_n,q_{n-1}]" in the standard notation of continued fractions, so "p_n/q_n =
a_0+1/(a_1+...+1/a_n)...)". If "x" is a matrix with two rows "[b_0,b_1,...,b_n]" and
"[a_0,a_1,...,a_n]", this is then considered as a generalized continued fraction and we
have similarly "p_n/q_n = 1/b_0(a_0+b_1/(a_1+...+b_n/a_n)...)". Note that in this case one
usually has "b_0 = 1".
The library syntax is pnqn"(x)".
core"(n,{flag = 0})"
if "n" is a non-zero integer written as "n = df^2" with "d" squarefree, returns "d". If
"flag" is non-zero, returns the two-element row vector "[d,f]".
The library syntax is core0"(n,flag)". Also available are "core(n)" ( = core"(n,0)") and
"core2(n)" ( = core"(n,1)").
coredisc"(n,{flag})"
if "n" is a non-zero integer written as "n = df^2" with "d" fundamental discriminant
(including 1), returns "d". If "flag" is non-zero, returns the two-element row vector
"[d,f]". Note that if "n" is not congruent to 0 or 1 modulo 4, "f" will be a half integer
and not an integer.
The library syntax is coredisc0"(n,flag)". Also available are "coredisc(n)" ( =
coredisc"(n,0)") and "coredisc2(n)" ( = coredisc"(n,1)").
dirdiv"(x,y)"
"x" and "y" being vectors of perhaps different lengths but with "y[1] ! = 0" considered as
Dirichlet series, computes the quotient of "x" by "y", again as a vector.
The library syntax is dirdiv"(x,y)".
direuler"(p = a,b,expr,{c})"
computes the Dirichlet series to "b" terms of the Euler product of expression expr as "p"
ranges through the primes from "a" to "b". expr must be a polynomial or rational function
in another variable than "p" (say "X") and "expr(X)" is understood as the Dirichlet series
(or more precisely the local factor) "expr(p^{-s})". If "c" is present, output only the
first "c" coefficients in the series.
The library syntax is direuler"(entree *ep, GEN a, GEN b, char *expr)"
dirmul"(x,y)"
"x" and "y" being vectors of perhaps different lengths considered as Dirichlet series,
computes the product of "x" by "y", again as a vector.
The library syntax is dirmul"(x,y)".
divisors"(x)"
creates a row vector whose components are the positive divisors of the integer "x" in
increasing order. The factorization of "x" (as output by "factor") can be used instead.
The library syntax is divisors"(x)".
eulerphi"(x)"
Euler's "phi" (totient) function of "|x|", in other words "|(Z/xZ)^*|". "x" must be of
type integer.
The library syntax is phi"(x)".
factor"(x,{lim = -1})"
general factorization function. If "x" is of type integer, rational, polynomial or
rational function, the result is a two-column matrix, the first column being the
irreducibles dividing "x" (prime numbers or polynomials), and the second the exponents.
If "x" is a vector or a matrix, the factoring is done componentwise (hence the result is a
vector or matrix of two-column matrices). By definition, 0 is factored as "0^1".
If "x" is of type integer or rational, an argument lim can be added, meaning that we look
only for factors up to lim, or to "primelimit", whichever is lowest (except when "lim = 0"
where the effect is identical to setting "lim = primelimit"). Hence in this case, the
remaining part is not necessarily prime. See factorint for more information about the
algorithms used.
The polynomials or rational functions to be factored must have scalar coefficients. In
particular PARI does not know how to factor multivariate polynomials.
Note that PARI tries to guess in a sensible way over which ring you want to factor. Note
also that factorization of polynomials is done up to multiplication by a constant. In
particular, the factors of rational polynomials will have integer coefficients, and the
content of a polynomial or rational function is discarded and not included in the
factorization. If you need it, you can always ask for the content explicitly:
? factor(t^2 + 5/2*t + 1)
%1 =
[2*t + 1 1]
[t + 2 1]
? content(t^2 + 5/2*t + 1)
%2 = 1/2
See also factornf.
The library syntax is factor0"(x,lim)", where lim is a C integer. Also available are
"factor(x)" ( = "factor0(x,-1)"), "smallfact(x)" ( = "factor0(x,0)").
factorback"(f,{nf})"
"f" being any factorization, gives back the factored object. If a second argument "nf" is
supplied, "f" is assumed to be a prime ideal factorization in the number field "nf". The
resulting ideal is given in HNF form.
The library syntax is factorback"(f,nf)", where an omitted "nf" is entered as "NULL".
factorcantor"(x,p)"
factors the polynomial "x" modulo the prime "p", using distinct degree plus Cantor-
Zassenhaus. The coefficients of "x" must be operation-compatible with "Z/pZ". The result
is a two-column matrix, the first column being the irreducible polynomials dividing "x",
and the second the exponents. If you want only the degrees of the irreducible polynomials
(for example for computing an "L"-function), use "factormod(x,p,1)". Note that the
"factormod" algorithm is usually faster than "factorcantor".
The library syntax is factcantor"(x,p)".
factorff"(x,p,a)"
factors the polynomial "x" in the field "F_q" defined by the irreducible polynomial "a"
over "F_p". The coefficients of "x" must be operation-compatible with "Z/pZ". The result
is a two-column matrix, the first column being the irreducible polynomials dividing "x",
and the second the exponents. It is recommended to use for the variable of "a" (which will
be used as variable of a polmod) a name distinct from the other variables used, so that a
"lift()" of the result will be legible. If all the coefficients of "x" are in "F_p", a
much faster algorithm is applied, using the computation of isomorphisms between finite
fields.
The library syntax is factmod9"(x,p,a)".
factorial"(x)" or "x!"
factorial of "x". The expression "x!" gives a result which is an integer, while
factorial(x) gives a real number.
The library syntax is mpfact"(x)" for "x!" and "mpfactr(x,prec)" for factorial(x). "x"
must be a "long" integer and not a PARI integer.
factorint"(n,{flag = 0})"
factors the integer n using a combination of the Shanks SQUFOF and Pollard Rho method
(with modifications due to Brent), Lenstra's ECM (with modifications by Montgomery), and
MPQS (the latter adapted from the LiDIA code with the kind permission of the LiDIA
maintainers), as well as a search for pure powers with exponents" <= 10". The output is a
two-column matrix as for "factor".
This gives direct access to the integer factoring engine called by most arithmetical
functions. flag is optional; its binary digits mean 1: avoid MPQS, 2: skip first stage ECM
(we may still fall back to it later), 4: avoid Rho and SQUFOF, 8: don't run final ECM (as
a result, a huge composite may be declared to be prime). Note that a (strong)
probabilistic primality test is used; thus composites might (very rarely) not be detected.
The machinery underlying this function is still in a somewhat experimental state, but
should be much faster on average than pure ECM as used by all PARI versions up to 2.0.8,
at the expense of heavier memory use. You are invited to play with the flag settings and
watch the internals at work by using GP's "debuglevel" default parameter (level 3 shows
just the outline, 4 turns on time keeping, 5 and above show an increasing amount of
internal details). If you see anything funny happening, please let us know.
The library syntax is factorint"(n,flag)".
factormod"(x,p,{flag = 0})"
factors the polynomial "x" modulo the prime integer "p", using Berlekamp. The coefficients
of "x" must be operation-compatible with "Z/pZ". The result is a two-column matrix, the
first column being the irreducible polynomials dividing "x", and the second the exponents.
If "flag" is non-zero, outputs only the degrees of the irreducible polynomials (for
example, for computing an "L"-function). A different algorithm for computing the mod "p"
factorization is "factorcantor" which is sometimes faster.
The library syntax is factormod"(x,p,flag)". Also available are "factmod(x,p)" (which is
equivalent to "factormod(x,p,0)") and "simplefactmod(x,p)" ( = "factormod(x,p,1)").
fibonacci"(x)"
"x^{th}" Fibonacci number.
The library syntax is fibo"(x)". "x" must be a "long".
ffinit"(p,n,{v = x})"
computes a monic polynomial of degree "n" which is irreducible over "F_p". For instance if
"P = ffinit(3,2,y)", you can represent elements in "F_{3^2}" as polmods modulo "P".
The library syntax is ffinit"(p,n,v)", where "v" is a variable number.
gcd"(x,y,{flag = 0})"
creates the greatest common divisor of "x" and "y". "x" and "y" can be of quite general
types, for instance both rational numbers. Vector/matrix types are also accepted, in which
case the GCD is taken recursively on each component. Note that for these types, "gcd" is
not commutative.
If "flag = 0", use Euclid's algorithm.
If "flag = 1", use the modular gcd algorithm ("x" and "y" have to be polynomials, with
integer coefficients).
If "flag = 2", use the subresultant algorithm.
The library syntax is gcd0"(x,y,flag)". Also available are "ggcd(x,y)", "modulargcd(x,y)",
and "srgcd(x,y)" corresponding to "flag = 0", 1 and 2 respectively.
hilbert"(x,y,{p})"
Hilbert symbol of "x" and "y" modulo "p". If "x" and "y" are of type integer or fraction,
an explicit third parameter "p" must be supplied, "p = 0" meaning the place at infinity.
Otherwise, "p" needs not be given, and "x" and "y" can be of compatible types integer,
fraction, real, integermod a prime (result is undefined if the modulus is not prime), or
"p"-adic.
The library syntax is hil"(x,y,p)".
isfundamental"(x)"
true (1) if "x" is equal to 1 or to the discriminant of a quadratic field, false (0)
otherwise.
The library syntax is gisfundamental"(x)", but the simpler function "isfundamental(x)"
which returns a "long" should be used if "x" is known to be of type integer.
isprime"(x,{flag = 0})"
if "flag = 0" (default), true (1) if "x" is a strong pseudo-prime for 10 randomly chosen
bases, false (0) otherwise.
If "flag = 1", use Pocklington-Lehmer ``P-1'' test. true (1) if "x" is prime, false (0)
otherwise.
If "flag = 2", use Pocklington-Lehmer ``P-1'' test and output a primality certificate as
follows: return 0 if "x" is composite, 1 if "x" is a small prime (currently strictly less
than "341 550 071 728 321"), and a matrix if "x" is a large prime. The matrix has three
columns. The first contains the prime factors "p", the second the corresponding elements
"a_p" as in Proposition 8.3.1 in GTM 138, and the third the output of isprime(p,2).
In the two last cases, the algorithm fails if one of the (strong pseudo-)prime factors is
not prime, but it should be exceedingly rare.
The library syntax is gisprime"(x,flag)", but the simpler function "isprime(x)" which
returns a "long" should be used if "x" is known to be of type integer. Also available is
"plisprime(N,flag)", corresponding to "gisprime(x,flag+1)" if "x" is known to be of type
integer.
ispseudoprime"(x)"
true (1) if "x" is a strong pseudo-prime for a randomly chosen base, false (0) otherwise.
The library syntax is gispsp"(x)", but the simpler function "ispsp(x)" which returns a
"long" should be used if "x" is known to be of type integer.
issquare"(x,{&n})"
true (1) if "x" is square, false (0) if not. "x" can be of any type. If "n" is given and
an exact square root had to be computed in the checking process, puts that square root in
"n". This is in particular the case when "x" is an integer or a polynomial. This is not
the case for intmods (use quadratic reciprocity) or series (only check the leading
coefficient).
The library syntax is gcarrecomplet"(x,&n)". Also available is "gcarreparfait(x)".
issquarefree"(x)"
true (1) if "x" is squarefree, false (0) if not. Here "x" can be an integer or a
polynomial.
The library syntax is gissquarefree"(x)", but the simpler function "issquarefree(x)" which
returns a "long" should be used if "x" is known to be of type integer. This issquarefree
is just the square of the Moebius function, and is computed as a multiplicative arithmetic
function much like the latter.
kronecker"(x,y)"
Kronecker (i.e. generalized Legendre) symbol "((x)/(y))". "x" and "y" must be of type
integer.
The library syntax is kronecker"(x,y)", the result (0 or "+- 1") is a "long".
lcm"(x,y)"
least common multiple of "x" and "y", i.e. such that "lcm(x,y)*gcd(x,y) = abs(x*y)".
The library syntax is glcm"(x,y)".
moebius"(x)"
Moebius "mu"-function of "|x|". "x" must be of type integer.
The library syntax is mu"(x)", the result (0 or "+- 1") is a "long".
nextprime"(x)"
finds the smallest prime greater than or equal to "x". "x" can be of any real type. Note
that if "x" is a prime, this function returns "x" and not the smallest prime strictly
larger than "x".
The library syntax is nextprime"(x)".
numdiv"(x)"
number of divisors of "|x|". "x" must be of type integer, and the result is a "long".
The library syntax is numbdiv"(x)".
omega"(x)"
number of distinct prime divisors of "|x|". "x" must be of type integer.
The library syntax is omega"(x)", the result is a "long".
precprime"(x)"
finds the largest prime less than or equal to "x". "x" can be of any real type. Returns 0
if "x <= 1". Note that if "x" is a prime, this function returns "x" and not the largest
prime strictly smaller than "x".
The library syntax is precprime"(x)".
prime"(x)"
the "x^{th}" prime number, which must be among the precalculated primes.
The library syntax is prime"(x)". "x" must be a "long".
primes"(x)"
creates a row vector whose components are the first "x" prime numbers, which must be among
the precalculated primes.
The library syntax is primes"(x)". "x" must be a "long".
qfbclassno"(x,{flag = 0})"
class number of the quadratic field of discriminant "x". In the present version 2.2.0, a
simple algorithm is used for "x > 0", so "x" should not be too large (say "x < 10^7") for
the time to be reasonable. On the other hand, for "x < 0" one can reasonably compute
classno("x") for "|x| < 10^{25}", since the method used is Shanks' method which is in
"O(|x|^{1/4})". For larger values of "|D|", see "quadclassunit".
If "flag = 1", compute the class number using Euler products and the functional equation.
However, it is in "O(|x|^{1/2})".
Important warning. For "D < 0", this function often gives incorrect results when the class
group is non-cyclic, because the authors were too lazy to implement Shanks' method
completely. It is therefore strongly recommended to use either the version with "flag =
1", the function "qfbhclassno(-x)" if "x" is known to be a fundamental discriminant, or
the function "quadclassunit".
The library syntax is qfbclassno0"(x,flag)". Also available are "classno(x)" ( =
"qfbclassno(x)"), "classno2(x)" ( = "qfbclassno(x,1)"), and finally there exists the
function "hclassno(x)" which computes the class number of an imaginary quadratic field by
counting reduced forms, an "O(|x|)" algorithm. See also "qfbhclassno".
qfbcompraw"(x,y)"
composition of the binary quadratic forms "x" and "y", without reduction of the result.
This is useful e.g. to compute a generating element of an ideal.
The library syntax is compraw"(x,y)".
qfbhclassno"(x)"
Hurwitz class number of "x", where "x" is non-negative and congruent to 0 or 3 modulo 4.
See also "qfbclassno".
The library syntax is hclassno"(x)".
qfbnucomp"(x,y,l)"
composition of the primitive positive definite binary quadratic forms "x" and "y" using
the NUCOMP and NUDUPL algorithms of Shanks (a la Atkin). "l" is any positive constant, but
for optimal speed, one should take "l = |D|^{1/4}", where "D" is the common discriminant
of "x" and "y". When "x" and "y" do not have the same discriminant, the result is
undefined.
The library syntax is nucomp"(x,y,l)". The auxiliary function "nudupl(x,l)" should be used
instead for speed when "x = y".
qfbnupow"(x,n)"
"n"-th power of the primitive positive definite binary quadratic form "x" using the NUCOMP
and NUDUPL algorithms (see "qfbnucomp").
The library syntax is nupow"(x,n)".
qfbpowraw"(x,n)"
"n"-th power of the binary quadratic form "x", computed without doing any reduction
(i.e. using "qfbcompraw"). Here "n" must be non-negative and "n < 2^{31}".
The library syntax is powraw"(x,n)" where "n" must be a "long" integer.
qfbprimeform"(x,p)"
prime binary quadratic form of discriminant "x" whose first coefficient is the prime
number "p". By abuse of notation, "p = 1" is a valid special case which returns the unit
form. Returns an error if "x" is not a quadratic residue mod "p". In the case where "x >
0", the ``distance'' component of the form is set equal to zero according to the current
precision.
The library syntax is primeform"(x,p,prec)", where the third variable "prec" is a "long",
but is only taken into account when "x > 0".
qfbred"(x,{flag = 0},{D},{isqrtD},{sqrtD})"
reduces the binary quadratic form "x" (updating Shanks's distance function if "x" is
indefinite). The binary digits of "flag" are toggles meaning
1: perform a single reduction step
2: don't update Shanks's distance
"D", isqrtD, sqrtD, if present, supply the values of the discriminant, "\lfloor sqrt
{D}\rfloor", and " sqrt {D}" respectively (no checking is done of these facts). If "D < 0"
these values are useless, and all references to Shanks's distance are irrelevant.
The library syntax is qfbred0"(x,flag,D,isqrtD,sqrtD)". Use "NULL" to omit any of "D",
isqrtD, sqrtD.
Also available are
"redimag(x)" ( = "qfbred(x)" where "x" is definite),
and for indefinite forms:
"redreal(x)" ( = "qfbred(x)"),
"rhoreal(x)" ( = "qfbred(x,1)"),
"redrealnod(x,sq)" ( = "qfbred(x,2,,isqrtD)"),
"rhorealnod(x,sq)" ( = "qfbred(x,3,,isqrtD)").
quadclassunit"(D,{flag = 0},{tech = []})"
Buchmann-McCurley's sub-exponential algorithm for computing the class group of a quadratic
field of discriminant "D". If "D" is not fundamental, the function may or may not be
defined, but usually is, and often gives the right answer (a warning is issued). The more
general function "bnrinit" should be used to compute the class group of an order.
This function should be used instead of "qfbclassno" or "quadregula" when "D < -10^{25}",
"D > 10^{10}", or when the structure is wanted.
If "flag" is non-zero and "D > 0", computes the narrow class group and regulator, instead
of the ordinary (or wide) ones. In the current version 2.2.0, this doesn't work at all :
use the general function "bnfnarrow".
Optional parameter tech is a row vector of the form "[c_1,c_2]", where "c_1" and "c_2" are
positive real numbers which control the execution time and the stack size. To get maximum
speed, set "c_2 = c". To get a rigorous result (under GRH) you must take "c_2 = 6".
Reasonable values for "c" are between 0.1 and 2.
The result of this function is a vector "v" with 4 components if "D < 0", and 5 otherwise.
The correspond respectively to
"*" "v[1]" : the class number
"*" "v[2]" : a vector giving the structure of the class group as a product of cyclic
groups;
"*" "v[3]" : a vector giving generators of those cyclic groups (as binary quadratic
forms).
"*" "v[4]" : (omitted if "D < 0") the regulator, computed to an accuracy which is the
maximum of an internal accuracy determined by the program and the current default (note
that once the regulator is known to a small accuracy it is trivial to compute it to very
high accuracy, see the tutorial).
"*" "v[5]" : a measure of the correctness of the result. If it is close to 1, the result
is correct (under GRH). If it is close to a larger integer, this shows that the class
number is off by a factor equal to this integer, and you must start again with a larger
value for "c_1" or a different random seed. In this case, a warning message is printed.
The library syntax is quadclassunit0"(D,flag,tech)". Also available are
"buchimag(D,c_1,c_2)" and "buchreal(D,flag,c_1,c_2)".
quaddisc"(x)"
discriminant of the quadratic field "Q( sqrt {x})", where "x belongs to Q".
The library syntax is quaddisc"(x)".
quadhilbert"(D,{flag = 0})"
relative equation defining the Hilbert class field of the quadratic field of discriminant
"D". If "flag" is non-zero and "D < 0", outputs "[form,root(form)]" (to be used for
constructing subfields). If "flag" is non-zero and "D > 0", try hard to get the best
modulus. Uses complex multiplication in the imaginary case and Stark units in the real
case.
The library syntax is quadhilbert"(D,flag,prec)".
quadgen"(x)"
creates the quadratic number "omega = (a+ sqrt {x})/2" where "a = 0" if "x = 0 mod 4", "a
= 1" if "x = 1 mod 4", so that "(1,omega)" is an integral basis for the quadratic order of
discriminant "x". "x" must be an integer congruent to 0 or 1 modulo 4.
The library syntax is quadgen"(x)".
quadpoly"(D,{v = x})"
creates the ``canonical'' quadratic polynomial (in the variable "v") corresponding to the
discriminant "D", i.e. the minimal polynomial of quadgen(x). "D" must be an integer
congruent to 0 or 1 modulo 4.
The library syntax is quadpoly0"(x,v)".
quadray"(D,f,{flag = 0})"
relative equation for the ray class field of conductor "f" for the quadratic field of
discriminant "D" (which can also be a "bnf"), using analytic methods.
For "D < 0", uses the "sigma" function. "flag" has the following meaning: if it's an odd
integer, outputs instead the vector of "[ideal, corresponding root]". It can also be a
two-component vector "[lambda,flag]", where flag is as above and "lambda" is the technical
element of "bnf" necessary for Schertz's method. In that case, returns 0 if "lambda" is
not suitable.
For "D > 0", uses Stark's conjecture. If "flag" is non-zero, try hard to get the best
modulus. The function may fail with the following message
"Cannot find a suitable modulus in FindModulus"
See "bnrstark" for more details about the real case.
The library syntax is quadray"(D,f,flag)".
quadregulator"(x)"
regulator of the quadratic field of positive discriminant "x". Returns an error if "x" is
not a discriminant (fundamental or not) or if "x" is a square. See also "quadclassunit" if
"x" is large.
The library syntax is regula"(x,prec)".
quadunit"(x)"
fundamental unit of the real quadratic field "Q( sqrt x)" where "x" is the positive
discriminant of the field. If "x" is not a fundamental discriminant, this probably gives
the fundamental unit of the corresponding order. "x" must be of type integer, and the
result is a quadratic number.
The library syntax is fundunit"(x)".
removeprimes"({x = []})"
removes the primes listed in "x" from the prime number table. In particular
"removeprimes(addprimes)" empties the extra prime table. "x" can also be a single integer.
List the current extra primes if "x" is omitted.
The library syntax is removeprimes"(x)".
sigma"(x,{k = 1})"
sum of the "k^{th}" powers of the positive divisors of "|x|". "x" must be of type integer.
The library syntax is sumdiv"(x)" ( = "sigma(x)") or "gsumdivk(x,k)" ( = "sigma(x,k)"),
where "k" is a C long integer.
sqrtint"(x)"
integer square root of "x", which must be of PARI type integer. The result is non-negative
and rounded towards zero. A negative "x" is allowed, and the result in that case is
"I*sqrtint(-x)".
The library syntax is racine"(x)".
znlog"(x,g)"
"g" must be a primitive root mod a prime "p", and the result is the discrete log of "x" in
the multiplicative group "(Z/pZ)^*". This function using a simple-minded
baby-step/giant-step approach and requires "O( sqrt {p})" storage, hence it cannot be used
for "p" greater than about "10^{13}".
The library syntax is znlog"(x,g)".
znorder"(x)"
"x" must be an integer mod "n", and the result is the order of "x" in the multiplicative
group "(Z/nZ)^*". Returns an error if "x" is not invertible.
The library syntax is order"(x)".
znprimroot"(x)"
returns a primitive root of "x", where "x" is a prime power.
The library syntax is gener"(x)".
znstar"(n)"
gives the structure of the multiplicative group "(Z/nZ)^*" as a 3-component row vector
"v", where "v[1] = phi(n)" is the order of that group, "v[2]" is a "k"-component row-
vector "d" of integers "d[i]" such that "d[i] > 1" and "d[i] | d[i-1]" for "i >= 2" and
"(Z/nZ)^* ~ prod_{i = 1}^k(Z/d[i]Z)", and "v[3]" is a "k"-component row vector giving
generators of the image of the cyclic groups "Z/d[i]Z".
The library syntax is znstar"(n)".
Functions related to elliptic curves
We have implemented a number of functions which are useful for number theorists working on
elliptic curves. We always use Tate's notations. The functions assume that the curve is
given by a general Weierstrass model
" y^2+a_1xy+a_3y = x^3+a_2x^2+a_4x+a_6, "
where a priori the "a_i" can be of any scalar type. This curve can be considered as a
five-component vector "E = [a1,a2,a3,a4,a6]". Points on "E" are represented as two-
component vectors "[x,y]", except for the point at infinity, i.e. the identity element of
the group law, represented by the one-component vector "[0]".
It is useful to have at one's disposal more information. This is given by the function
"ellinit" (see there), which usually gives a 19 component vector (which we will call a
long vector in this section). If a specific flag is added, a vector with only 13 component
will be output (which we will call a medium vector). A medium vector just gives the first
13 components of the long vector corresponding to the same curve, but is of course faster
to compute. The following member functions are available to deal with the output of
"ellinit":
"a1"--"a6", "b2"--"b8", "c4"--"c6" : coefficients of the elliptic curve.
"area" : volume of the complex lattice defining "E".
"disc" : discriminant of the curve.
"j" : "j"-invariant of the curve.
"omega" : "[omega_1,omega_2]", periods forming a basis of the complex lattice defining
"E" ("omega_1" is the
real period, and "omega_2/omega_1" belongs to Poincare's half-plane).
"eta" : quasi-periods "[eta_1, eta_2]", such that "eta_1omega_2-eta_2omega_1 = iPi".
"roots" : roots of the associated Weierstrass equation.
"tate" : "[u^2,u,v]" in the notation of Tate.
"w" : Mestre's "w" (this is technical).
Their use is best described by an example: assume that "E" was output by "ellinit", then
typing "E.disc" will retrieve the curve's discriminant. The member functions "area", "eta"
and "omega" are only available for curves over Q. Conversely, "tate" and "w" are only
available for curves defined over "Q_p".
Some functions, in particular those relative to height computations (see "ellheight")
require also that the curve be in minimal Weierstrass form. This is achieved by the
function "ellglobalred".
All functions related to elliptic curves share the prefix "ell", and the precise curve we
are interested in is always the first argument, in either one of the three formats
discussed above, unless otherwise specified. For instance, in functions which do not use
the extra information given by long vectors, the curve can be given either as a five-
component vector, or by one of the longer vectors computed by "ellinit".
elladd"(E,z1,z2)"
sum of the points "z1" and "z2" on the elliptic curve corresponding to the vector "E".
The library syntax is addell"(E,z1,z2)".
ellak"(E,n)"
computes the coefficient "a_n" of the "L"-function of the elliptic curve "E", i.e. in
principle coefficients of a newform of weight 2 assuming Taniyama-Weil conjecture (which
is now known to hold in full generality thanks to the work of Breuil, Conrad, Diamond,
Taylor and Wiles). "E" must be a medium or long vector of the type given by "ellinit". For
this function to work for every "n" and not just those prime to the conductor, "E" must be
a minimal Weierstrass equation. If this is not the case, use the function "ellglobalred"
first before using "ellak".
The library syntax is akell"(E,n)".
ellan"(E,n)"
computes the vector of the first "n" "a_k" corresponding to the elliptic curve "E". All
comments in "ellak" description remain valid.
The library syntax is anell"(E,n)", where "n" is a C integer.
ellap"(E,p,{flag = 0})"
computes the "a_p" corresponding to the elliptic curve "E" and the prime number "p". These
are defined by the equation "#E(F_p) = p+1 - a_p", where "#E(F_p)" stands for the number
of points of the curve "E" over the finite field "F_p". When "flag" is 0, this uses the
baby-step giant-step method and a trick due to Mestre. This runs in time "O(p^{1/4})" and
requires "O(p^{1/4})" storage, hence becomes unreasonable when "p" has about 30 digits.
If "flag" is 1, computes the "a_p" as a sum of Legendre symbols. This is slower than the
previous method as soon as "p" is greater than 100, say.
No checking is done that "p" is indeed prime. "E" must be a medium or long vector of the
type given by "ellinit", defined over Q, "F_p" or "Q_p". "E" must be given by a
Weierstrass equation minimal at "p".
The library syntax is ellap0"(E,p,flag)". Also available are "apell(E,p)", corresponding
to "flag = 0", and "apell2(E,p)" ("flag = 1").
ellbil"(E,z1,z2)"
if "z1" and "z2" are points on the elliptic curve "E", this function computes the value of
the canonical bilinear form on "z1", "z2":
" ellheight(E,z1+z2) - ellheight(E,z1) - ellheight(E,z2) "
where "+" denotes of course addition on "E". In addition, "z1" or "z2" (but not both) can
be vectors or matrices. Note that this is equal to twice some normalizations. "E" is
assumed to be integral, given by a minimal model.
The library syntax is bilhell"(E,z1,z2,prec)".
ellchangecurve"(E,v)"
changes the data for the elliptic curve "E" by changing the coordinates using the vector
"v = [u,r,s,t]", i.e. if "x'" and "y'" are the new coordinates, then "x = u^2x'+r", "y =
u^3y'+su^2x'+t". The vector "E" must be a medium or long vector of the type given by
"ellinit".
The library syntax is coordch"(E,v)".
ellchangepoint"(x,v)"
changes the coordinates of the point or vector of points "x" using the vector "v =
[u,r,s,t]", i.e. if "x'" and "y'" are the new coordinates, then "x = u^2x'+r", "y =
u^3y'+su^2x'+t" (see also "ellchangecurve").
The library syntax is pointch"(x,v)".
elleisnum"(E,k,{flag = 0})"
"E" being an elliptic curve as output by "ellinit" (or, alternatively, given by a
2-component vector "[omega_1,omega_2]"), and "k" being an even positive integer, computes
the numerical value of the Eisenstein series of weight "k" at "E". When flag is non-zero
and "k = 4" or 6, returns "g_2" or "g_3" with the correct normalization.
The library syntax is elleisnum"(E,k,flag)".
elleta"(om)"
returns the two-component row vector "[eta_1,eta_2]" of quasi-periods associated to "om =
[omega_1, omega_2]"
The library syntax is elleta"(om, prec)"
ellglobalred"(E)"
calculates the arithmetic conductor, the global minimal model of "E" and the global
Tamagawa number "c". Here "E" is an elliptic curve given by a medium or long vector of the
type given by "ellinit", and is supposed to have all its coefficients "a_i" in Q. The
result is a 3 component vector "[N,v,c]". "N" is the arithmetic conductor of the curve,
"v" is itself a vector "[u,r,s,t]" with rational components. It gives a coordinate change
for "E" over Q such that the resulting model has integral coefficients, is everywhere
minimal, "a_1" is 0 or 1, "a_2" is 0, 1 or "-1" and "a_3" is 0 or 1. Such a model is
unique, and the vector "v" is unique if we specify that "u" is positive. To get the new
model, simply type "ellchangecurve(E,v)". Finally "c" is the product of the local Tamagawa
numbers "c_p", a quantity which enters in the Birch and Swinnerton-Dyer conjecture.
The library syntax is globalreduction"(E)".
ellheight"(E,z,{flag = 0})"
global Neron-Tate height of the point "z" on the elliptic curve "E". The vector "E" must
be a long vector of the type given by "ellinit", with "flag = 1". If "flag = 0", this
computation is done using sigma and theta-functions and a trick due to J. Silverman. If
"flag = 1", use Tate's "4^n" algorithm, which is much slower. "E" is assumed to be
integral, given by a minimal model.
The library syntax is ellheight0"(E,z,flag,prec)". The Archimedean contribution alone is
given by the library function "hell(E,z,prec)". Also available are "ghell(E,z,prec)"
("flag = 0") and "ghell2(E,z,prec)" ("flag = 1").
ellheightmatrix"(E,x)"
"x" being a vector of points, this function outputs the Gram matrix of "x" with respect to
the Neron-Tate height, in other words, the "(i,j)" component of the matrix is equal to
"ellbil(E,x[i],x[j])". The rank of this matrix, at least in some approximate sense, gives
the rank of the set of points, and if "x" is a basis of the Mordell-Weil group of "E", its
determinant is equal to the regulator of "E". Note that this matrix should be divided by 2
to be in accordance with certain normalizations. "E" is assumed to be integral, given by a
minimal model.
The library syntax is mathell"(E,x,prec)".
ellinit"(E,{flag = 0})"
computes some fixed data concerning the elliptic curve given by the five-component vector
"E", which will be essential for most further computations on the curve. The result is a
19-component vector E (called a long vector in this section), shortened to 13 components
(medium vector) if "flag = 1". Both contain the following information in the first 13
components:
" a_1,a_2,a_3,a_4,a_6,b_2,b_4,b_6,b_8,c_4,c_6,Delta,j."
In particular, the discriminant is "E[12]" (or "E.disc"), and the "j"-invariant is "E[13]"
(or "E.j").
The other six components are only present if "flag" is 0 (or omitted!). Their content
depends on whether the curve is defined over R or not:
"*" When "E" is defined over R, "E[14]" ("E.roots") is a vector whose three components
contain the roots of the associated Weierstrass equation. If the roots are all real, then
they are ordered by decreasing value. If only one is real, it is the first component of
"E[14]".
"E[15]" ("E.omega[1]") is the real period of "E" (integral of "dx/(2y+a_1x+a_3)" over the
connected component of the identity element of the real points of the curve), and "E[16]"
("E.omega[2]") is a complex period. In other words, "omega_1 = E[15]" and "omega_2 =
E[16]" form a basis of the complex lattice defining "E" ("E.omega"), with "tau =
(omega_2)/(omega_1)" having positive imaginary part.
"E[17]" and "E[18]" are the corresponding values "eta_1" and "eta_2" such that
"eta_1omega_2-eta_2omega_1 = iPi", and both can be retrieved by typing "E.eta" (as a row
vector whose components are the "eta_i").
Finally, "E[19]" ("E.area") is the volume of the complex lattice defining "E".
"*" When "E" is defined over "Q_p", the "p"-adic valuation of "j" must be negative. Then
"E[14]" ("E.roots") is the vector with a single component equal to the "p"-adic root of
the associated Weierstrass equation corresponding to "-1" under the Tate parametrization.
"E[15]" is equal to the square of the "u"-value, in the notation of Tate.
"E[16]" is the "u"-value itself, if it belongs to "Q_p", otherwise zero.
"E[17]" is the value of Tate's "q" for the curve "E".
"E.tate" will yield the three-component vector "[u^2,u,q]".
"E[18]" ("E.w") is the value of Mestre's "w" (this is technical), and "E[19]" is
arbitrarily set equal to zero.
For all other base fields or rings, the last six components are arbitrarily set equal to
zero (see also the description of member functions related to elliptic curves at the
beginning of this section).
The library syntax is ellinit0"(E,flag,prec)". Also available are "initell(E,prec)" ("flag
= 0") and "smallinitell(E,prec)" ("flag = 1").
ellisoncurve"(E,z)"
gives 1 (i.e. true) if the point "z" is on the elliptic curve "E", 0 otherwise. If "E" or
"z" have imprecise coefficients, an attempt is made to take this into account, i.e. an
imprecise equality is checked, not a precise one.
The library syntax is oncurve"(E,z)", and the result is a "long".
ellj"(x)"
elliptic "j"-invariant. "x" must be a complex number with positive imaginary part, or
convertible into a power series or a "p"-adic number with positive valuation.
The library syntax is jell"(x,prec)".
elllocalred"(E,p)"
calculates the Kodaira type of the local fiber of the elliptic curve "E" at the prime "p".
"E" must be given by a medium or long vector of the type given by "ellinit", and is
assumed to have all its coefficients "a_i" in Z. The result is a 4-component vector
"[f,kod,v,c]". Here "f" is the exponent of "p" in the arithmetic conductor of "E", and
"kod" is the Kodaira type which is coded as follows:
1 means good reduction (type I"_0"), 2, 3 and 4 mean types II, III and IV respectively,
"4+nu" with "nu > 0" means type I"_nu"; finally the opposite values "-1", "-2", etc. refer
to the starred types I"_0^*", II"^*", etc. The third component "v" is itself a vector
"[u,r,s,t]" giving the coordinate changes done during the local reduction. Normally, this
has no use if "u" is 1, that is, if the given equation was already minimal. Finally, the
last component "c" is the local Tamagawa number "c_p".
The library syntax is localreduction"(E,p)".
elllseries"(E,s,{A = 1})"
"E" being a medium or long vector given by "ellinit", this computes the value of the
L-series of "E" at "s". It is assumed that "E" is a minimal model over Z and that the
curve is a modular elliptic curve. The optional parameter "A" is a cutoff point for the
integral, which must be chosen close to 1 for best speed. The result must be independent
of "A", so this allows some internal checking of the function.
Note that if the conductor of the curve is large, say greater than "10^{12}", this
function will take an unreasonable amount of time since it uses an "O(N^{1/2})" algorithm.
The library syntax is lseriesell"(E,s,A,prec)" where "prec" is a "long" and an omitted "A"
is coded as "NULL".
ellorder"(E,z)"
gives the order of the point "z" on the elliptic curve "E" if it is a torsion point, zero
otherwise. In the present version 2.2.0, this is implemented only for elliptic curves
defined over Q.
The library syntax is orderell"(E,z)".
ellordinate"(E,x)"
gives a 0, 1 or 2-component vector containing the "y"-coordinates of the points of the
curve "E" having "x" as "x"-coordinate.
The library syntax is ordell"(E,x)".
ellpointtoz"(E,z)"
if "E" is an elliptic curve with coefficients in R, this computes a complex number "t"
(modulo the lattice defining "E") corresponding to the point "z", i.e. such that, in the
standard Weierstrass model, " wp (t) = z[1], wp '(t) = z[2]". In other words, this is the
inverse function of "ellztopoint".
If "E" has coefficients in "Q_p", then either Tate's "u" is in "Q_p", in which case the
output is a "p"-adic number "t" corresponding to the point "z" under the Tate
parametrization, or only its square is, in which case the output is "t+1/t". "E" must be a
long vector output by "ellinit".
The library syntax is zell"(E,z,prec)".
ellpow"(E,z,n)"
computes "n" times the point "z" for the group law on the elliptic curve "E". Here, "n"
can be in Z, or "n" can be a complex quadratic integer if the curve "E" has complex
multiplication by "n" (if not, an error message is issued).
The library syntax is powell"(E,z,n)".
ellrootno"(E,{p = 1})"
"E" being a medium or long vector given by "ellinit", this computes the local (if "p ! =
1") or global (if "p = 1") root number of the L-series of the elliptic curve "E". Note
that the global root number is the sign of the functional equation and conjecturally is
the parity of the rank of the Mordell-Weil group. The equation for "E" must have
coefficients in Q but need not be minimal.
The library syntax is ellrootno"(E,p)" and the result (equal to "+-1") is a "long".
ellsigma"(E,z,{flag = 0})"
value of the Weierstrass "sigma" function of the lattice associated to "E" as given by
"ellinit" (alternatively, "E" can be given as a lattice "[omega_1,omega_2]").
If "flag = 1", computes an (arbitrary) determination of " log (sigma(z))".
If "flag = 2,3", same using the product expansion instead of theta series. The library
syntax is ellsigma"(E,z,flag)"
ellsub"(E,z1,z2)"
difference of the points "z1" and "z2" on the elliptic curve corresponding to the vector
"E".
The library syntax is subell"(E,z1,z2)".
elltaniyama"(E)"
computes the modular parametrization of the elliptic curve "E", where "E" is given in the
(long or medium) format output by "ellinit", in the form of a two-component vector "[u,v]"
of power series, given to the current default series precision. This vector is
characterized by the following two properties. First the point "(x,y) = (u,v)" satisfies
the equation of the elliptic curve. Second, the differential "du/(2v+a_1u+a_3)" is equal
to "f(z)dz", a differential form on "H/Gamma_0(N)" where "N" is the conductor of the
curve. The variable used in the power series for "u" and "v" is "x", which is implicitly
understood to be equal to " exp (2iPi z)". It is assumed that the curve is a strong Weil
curve, and the Manin constant is equal to 1. The equation of the curve "E" must be minimal
(use "ellglobalred" to get a minimal equation).
The library syntax is taniyama"(E)", and the precision of the result is determined by the
global variable "precdl".
elltors"(E,{flag = 0})"
if "E" is an elliptic curve defined over Q, outputs the torsion subgroup of "E" as a
3-component vector "[t,v1,v2]", where "t" is the order of the torsion group, "v1" gives
the structure of the torsion group as a product of cyclic groups (sorted by decreasing
order), and "v2" gives generators for these cyclic groups. "E" must be a long vector as
output by "ellinit".
? E = ellinit([0,0,0,-1,0]);
? elltors(E)
%1 = [4, [2, 2], [[0, 0], [1, 0]]]
Here, the torsion subgroup is isomorphic to "Z/2Z x Z/2Z", with generators "[0,0]" and
"[1,0]".
If "flag = 0", use Doud's algorithm : bound torsion by computing "#E(F_p)" for small
primes of good reduction, then look for torsion points using Weierstrass parametrization
(and Mazur's classification).
If "flag = 1", use Lutz--Nagell (much slower), "E" is allowed to be a medium vector.
The library syntax is elltors0"(E,flag)".
ellwp"(E,{z = x},{flag = 0})"
Computes the value at "z" of the Weierstrass " wp " function attached to the elliptic
curve "E" as given by "ellinit" (alternatively, "E" can be given as a lattice
"[omega_1,omega_2]").
If "z" is omitted or is a simple variable, computes the power series expansion in "z"
(starting "z^{-2}+O(z^2)"). The number of terms to an even power in the expansion is the
default serieslength in GP, and the second argument (C long integer) in library mode.
Optional flag is (for now) only taken into account when "z" is numeric, and means 0:
compute only " wp (z)", 1: compute "[ wp (z), wp '(z)]".
The library syntax is ellwp0"(E,z,flag,prec,precdl)". Also available is
weipell"(E,precdl)" for the power series (in "x = polx[0]").
ellzeta"(E,z)"
value of the Weierstrass "zeta" function of the lattice associated to "E" as given by
"ellinit" (alternatively, "E" can be given as a lattice "[omega_1,omega_2]").
The library syntax is ellzeta"(E,z)".
ellztopoint"(E,z)"
"E" being a long vector, computes the coordinates "[x,y]" on the curve "E" corresponding
to the complex number "z". Hence this is the inverse function of "ellpointtoz". In other
words, if the curve is put in Weierstrass form, "[x,y]" represents the Weierstrass " wp
"-function and its derivative. If "z" is in the lattice defining "E" over C, the result
is the point at infinity "[0]".
The library syntax is pointell"(E,z,prec)".
Functions related to general number fields
In this section can be found functions which are used almost exclusively for working in
general number fields. Other less specific functions can be found in the next section on
polynomials. Functions related to quadratic number fields can be found in the section
"Label se:arithmetic" (Arithmetic functions).
We shall use the following conventions:
"*" "nf" denotes a number field, i.e. a 9-component vector in the format output by
"nfinit". This contains the basic arithmetic data associated to the number field:
signature, maximal order, discriminant, etc.
"*" "bnf" denotes a big number field, i.e. a 10-component vector in the format output by
"bnfinit". This contains "nf" and the deeper invariants of the field: units, class groups,
as well as a lot of technical data necessary for some complex fonctions like
"bnfisprincipal".
"*" "bnr" denotes a big ``ray number field'', i.e. some data structure output by
"bnrinit", even more complicated than "bnf", corresponding to the ray class group
structure of the field, for some modulus.
"*" "rnf" denotes a relative number field (see below).
"*" "ideal" can mean any of the following:
-- a Z-basis, in Hermite normal form (HNF) or not. In this case "x" is a square matrix.
-- an idele, i.e. a 2-component vector, the first being an ideal given as a Z--basis,
the second being a "r_1+r_2"-component row vector giving the complex logarithmic
Archimedean information.
-- a "Z_K"-generating system for an ideal.
-- a column vector "x" expressing an element of the number field on the integral basis,
in which case the ideal is treated as being the principal idele (or ideal) generated by
"x".
-- a prime ideal, i.e. a 5-component vector in the format output by "idealprimedec".
-- a polmod "x", i.e. an algebraic integer, in which case the ideal is treated as being
the principal idele generated by "x".
-- an integer or a rational number, also treated as a principal idele.
"*" a {character} on the Abelian group "\bigoplus (Z/N_iZ) g_i" is given by a row vector
"chi = [a_1,...,a_n]" such that "chi(prod g_i^{n_i}) = exp(2iPisum a_i n_i / N_i)".
Warnings:
1) An element in "nf" can be expressed either as a polmod or as a vector of components on
the integral basis "nf.zk". It is absolutely essential that all such vectors be column
vectors.
2) When giving an ideal by a "Z_K" generating system to a function expecting an ideal, it
must be ensured that the function understands that it is a "Z_K"-generating system and not
a Z-generating system. When the number of generators is strictly less than the degree of
the field, there is no ambiguity and the program assumes that one is giving a
"Z_K"-generating set. When the number of generators is greater than or equal to the
degree of the field, however, the program assumes on the contrary that you are giving a
Z-generating set. If this is not the case, you must absolutely change it into a
Z-generating set, the simplest manner being to use "idealhnf(nf,x)".
Concerning relative extensions, some additional definitions are necessary.
"*" A {relative matrix} will be a matrix whose entries are elements of a (given) number
field "nf", always expressed as column vectors on the integral basis "nf.zk". Hence it is
a matrix of vectors.
"*" An ideal list will be a row vector of (fractional) ideals of the number field "nf".
"*" A pseudo-matrix will be a pair "(A,I)" where "A" is a relative matrix and "I" an ideal
list whose length is the same as the number of columns of "A". This pair will be
represented by a 2-component row vector.
"*" The module generated by a pseudo-matrix "(A,I)" is the sum "sum_i{a}_jA_j" where the
"{a}_j" are the ideals of "I" and "A_j" is the "j"-th column of "A".
"*" A pseudo-matrix "(A,I)" is a pseudo-basis of the module it generates if "A" is a
square matrix with non-zero determinant and all the ideals of "I" are non-zero. We say
that it is in Hermite Normal Form (HNF) if it is upper triangular and all the elements of
the diagonal are equal to 1.
"*" The determinant of a pseudo-basis "(A,I)" is the ideal equal to the product of the
determinant of "A" by all the ideals of "I". The determinant of a pseudo-matrix is the
determinant of any pseudo-basis of the module it generates.
Finally, when defining a relative extension, the base field should be defined by a
variable having a lower priority (i.e. a higher number) than the variable defining the
extension. For example, under GP you can use the variable name "y" (or "t") to define the
base field, and the variable name "x" to define the relative extension.
Now a last set of definitions concerning the way big ray number fields (or bnr) are input,
using class field theory. These are defined by a triple "a1", "a2", "a3", where the
defining set "[a1,a2,a3]" can have any of the following forms: "[bnr]", "[bnr,subgroup]",
"[bnf,module]", "[bnf,module,subgroup]", where:
"*" "bnf" is as output by "bnfclassunit" or "bnfinit", where units are mandatory unless
the ideal is trivial; bnr by "bnrclass" (with "flag > 0") or "bnrinit". This is the ground
field.
"*" module is either an ideal in any form (see above) or a two-component row vector
containing an ideal and an "r_1"-component row vector of flags indicating which real
Archimedean embeddings to take in the module.
"*" subgroup is the HNF matrix of a subgroup of the ray class group of the ground field
for the modulus module. This is input as a square matrix expressing generators of a
subgroup of the ray class group "bnr.clgp" on the given generators.
The corresponding bnr is then the subfield of the ray class field of the ground field for
the given modulus, associated to the given subgroup.
All the functions which are specific to relative extensions, number fields, big number
fields, big number rays, share the prefix "rnf", "nf", "bnf", "bnr" respectively. They are
meant to take as first argument a number field of that precise type, respectively output
by "rnfinit", "nfinit", "bnfinit", and "bnrinit".
However, and even though it may not be specified in the descriptions of the functions
below, it is permissible, if the function expects a "nf", to use a "bnf" instead (which
contains much more information). The program will make the effort of converting to what it
needs. On the other hand, if the program requires a big number field, the program will not
launch "bnfinit" for you, which can be a costly operation. Instead, it will give you a
specific error message.
The data types corresponding to the structures described above are rather complicated.
Thus, as we already have seen it with elliptic curves, GP provides you with some ``member
functions'' to retrieve the data you need from these structures (once they have been
initialized of course). The relevant types of number fields are indicated between
parentheses:
"bnf" (bnr, bnf ) : big number field.
"clgp" (bnr, bnf ) : classgroup. This one admits the following three subclasses:
"cyc" : cyclic decomposition (SNF).
"gen" : generators.
"no" : number of elements.
"diff" (bnr, bnf, nf ) : the different ideal.
"codiff" (bnr, bnf, nf ) : the codifferent (inverse of the different in the ideal
group).
"disc" (bnr, bnf, nf ) : discriminant.
"fu" (bnr, bnf, nf ) : fundamental units.
"futu" (bnr, bnf ) : "[u,w]", "u" is a vector of fundamental units, "w" generates the
torsion.
"nf" (bnr, bnf, nf ) : number field.
"reg" (bnr, bnf, ) : regulator.
"roots" (bnr, bnf, nf ) : roots of the polnomial generating the field.
"sign" (bnr, bnf, nf ) : "[r_1,r_2]" the signature of the field. This means that the
field has "r_1" real embeddings, "2r_2" complex ones.
"t2" (bnr, bnf, nf ) : the T2 matrix (see "nfinit").
"tu" (bnr, bnf, ) : a generator for the torsion units.
"tufu" (bnr, bnf, ) : as "futu", but outputs "[w,u]".
"zk" (bnr, bnf, nf ) : integral basis, i.e. a Z-basis of the maximal order.
"zkst" (bnr ) : structure of "(Z_K/m)^*" (can be extracted also from an
idealstar).
For instance, assume that "bnf = bnfinit(pol)", for some polynomial. Then "bnf.clgp"
retrieves the class group, and "bnf.clgp.no" the class number. If we had set "bnf =
nfinit(pol)", both would have output an error message. All these functions are completely
recursive, thus for instance "bnr.bnf.nf.zk" will yield the maximal order of bnr (which
you could get directly with a simple "bnr.zk" of course).
The following functions, starting with "buch" in library mode, and with "bnf" under GP,
are implementations of the sub-exponential algorithms for finding class and unit groups
under GRH, due to Hafner-McCurley, Buchmann and Cohen-Diaz-Olivier.
The general call to the functions concerning class groups of general number fields
(i.e. excluding "quadclassunit") involves a polynomial "P" and a technical vector
"tech = [c,c2,nrel,borne,nrpid,minsfb],"
where the parameters are to be understood as follows:
"P" is the defining polynomial for the number field, which must be in "Z[X]", irreducible
and, preferably, monic. In fact, if you supply a non-monic polynomial at this point, GP
will issue a warning, then transform your polynomial so that it becomes monic. Instead of
the normal result, say "res", you then get a vector "[res,Mod(a,Q)]", where "Mod(a,Q) =
Mod(X,P)" gives the change of variables.
The numbers "c" and "c2" are positive real numbers which control the execution time and
the stack size. To get maximum speed, set "c2 = c". To get a rigorous result (under GRH)
you must take "c2 = 12" (or "c2 = 6" in the quadratic case, but then you should use the
much faster function "quadclassunit"). Reasonable values for "c" are between 0.1 and 2.
(The defaults are "c = c2 = 0.3").
"nrel" is the number of initial extra relations requested in computing the relation
matrix. Reasonable values are between 5 and 20. (The default is 5).
"borne" is a multiplicative coefficient of the Minkowski bound which controls the search
for small norm relations. If this parameter is set equal to 0, the program does not search
for small norm relations. Otherwise reasonable values are between 0.5 and 2.0. (The
default is 1.0).
"nrpid" is the maximal number of small norm relations associated to each ideal in the
factor base. Irrelevant when "borne = 0". Otherwise, reasonable values are between 4 and
20. (The default is 4).
"minsfb" is the minimal number of elements in the ``sub-factorbase''. If the program does
not seem to succeed in finding a full rank matrix (which you can see in GP by typing "\g
2"), increase this number. Reasonable values are between 2 and 5. (The default is 3).
Remarks.
Apart from the polynomial "P", you don't need to supply any of the technical parameters
(under the library you still need to send at least an empty vector, "cgetg(1,t_VEC)").
However, should you choose to set some of them, they must be given in the requested order.
For example, if you want to specify a given value of "nrel", you must give some values as
well for "c" and "c2", and provide a vector "[c,c2,nrel]".
Note also that you can use an "nf" instead of "P", which avoids recomputing the integral
basis and analogous quantities.
bnfcertify"(bnf)"
"bnf" being a big number field as output by "bnfinit" or "bnfclassunit", checks whether
the result is correct, i.e. whether it is possible to remove the assumption of the
Generalized Riemann Hypothesis. If it is correct, the answer is 1. If not, the program
may output some error message, but more probably will loop indefinitely. In no occasion
can the program give a wrong answer (barring bugs of course): if the program answers 1,
the answer is certified.
The library syntax is certifybuchall"(bnf)", and the result is a C long.
bnfclassunit"(P,{flag = 0},{tech = []})"
Buchmann's sub-exponential algorithm for computing the class group, the regulator and a
system of fundamental units of the general algebraic number field "K" defined by the
irreducible polynomial "P" with integer coefficients.
The result of this function is a vector "v" with 10 components (it is not a "bnf", you
need "bnfinit" for that), which for ease of presentation is in fact output as a one column
matrix. First we describe the default behaviour ("flag = 0"):
"v[1]" is equal to the polynomial "P". Note that for optimum performance, "P" should have
gone through "polred" or "nfinit(x,2)".
"v[2]" is the 2-component vector "[r1,r2]", where "r1" and "r2" are as usual the number of
real and half the number of complex embeddings of the number field "K".
"v[3]" is the 2-component vector containing the field discriminant and the index.
"v[4]" is an integral basis in Hermite normal form.
"v[5]" ("v.clgp") is a 3-component vector containing the class number ("v.clgp.no"), the
structure of the class group as a product of cyclic groups of order "n_i" ("v.clgp.cyc"),
and the corresponding generators of the class group of respective orders "n_i"
("v.clgp.gen").
"v[6]" ("v.reg") is the regulator computed to an accuracy which is the maximum of an
internally determined accuracy and of the default.
"v[7]" is a measure of the correctness of the result. If it is close to 1, the results are
correct (under GRH). If it is close to a larger integer, this shows that the product of
the class number by the regulator is off by a factor equal to this integer, and you must
start again with a larger value for "c" or a different random seed, i.e. use the function
"setrand". (Since the computation involves a random process, starting again with exactly
the same parameters may give the correct result.) In this case a warning message is
printed.
"v[8]" ("v.tu") a vector with 2 components, the first being the number "w" of roots of
unity in "K" and the second a primitive "w"-th root of unity expressed as a polynomial.
"v[9]" ("v.fu") is a system of fundamental units also expressed as polynomials.
"v[10]" gives a measure of the correctness of the computations of the fundamental units
(not of the regulator), expressed as a number of bits. If this number is greater than 20,
say, everything is OK. If "v[10] <= 0", then we have lost all accuracy in computing the
units (usually an error message will be printed and the units not given). In the
intermediate cases, one must proceed with caution (for example by increasing the current
precision).
If "flag = 1", and the precision happens to be insufficient for obtaining the fundamental
units exactly, the internal precision is doubled and the computation redone, until the
exact results are obtained. The user should be warned that this can take a very long time
when the coefficients of the fundamental units on the integral basis are very large, for
example in the case of large real quadratic fields. In that case, there are alternate
methods for representing algebraic numbers which are not implemented in PARI.
If "flag = 2", the fundamental units and roots of unity are not computed. Hence the
result has only 7 components, the first seven ones.
"tech" is a technical vector (empty by default) containing "c", "c2", nrel, borne, nbpid,
minsfb, in this order (see the beginning of the section or the keyword "bnf"). You can
supply any number of these provided you give an actual value to each of them (the ``empty
arg'' trick won't work here). Careful use of these parameters may speed up your
computations considerably.
The library syntax is bnfclassunit0"(P,flag,tech,prec)".
bnfclgp"(P,{tech = []})"
as "bnfclassunit", but only outputs "v[5]", i.e. the class group.
The library syntax is bnfclassgrouponly"(P,tech,prec)", where tech is as described under
"bnfclassunit".
bnfdecodemodule"(nf,m)"
if "m" is a module as output in the first component of an extension given by
"bnrdisclist", outputs the true module.
The library syntax is decodemodule"(nf,m)".
bnfinit"(P,{flag = 0},{tech = []})"
essentially identical to "bnfclassunit" except that the output contains a lot of technical
data, and should not be printed out explicitly in general. The result of "bnfinit" is used
in programs such as "bnfisprincipal", "bnfisunit" or "bnfnarrow". The result is a
10-component vector "bnf".
"*" The first 6 and last 2 components are technical and in principle are not used by the
casual user. However, for the sake of completeness, their description is as follows. We
use the notations explained in the book by H. Cohen, A Course in Computational Algebraic
Number Theory, Graduate Texts in Maths 138, Springer-Verlag, 1993, Section 6.5, and
subsection 6.5.5 in particular.
"bnf[1]" contains the matrix "W", i.e. the matrix in Hermite normal form giving relations
for the class group on prime ideal generators "(p_i)_{1 <= i <= r}".
"bnf[2]" contains the matrix "B", i.e. the matrix containing the expressions of the prime
ideal factorbase in terms of the "p_i". It is an "r x c" matrix.
"bnf[3]" contains the complex logarithmic embeddings of the system of fundamental units
which has been found. It is an "(r_1+r_2) x (r_1+r_2-1)" matrix.
"bnf[4]" contains the matrix "M''_C" of Archimedean components of the relations of the
matrix "(W|B)".
"bnf[5]" contains the prime factor base, i.e. the list of prime ideals used in finding the
relations.
"bnf[6]" contains the permutation of the prime factor base which was necessary to reduce
the relation matrix to the form explained in subsection 6.5.5 of GTM 138 (i.e. with a big
"c x c" identity matrix on the lower right). Note that in the above mentioned book, the
need to permute the rows of the relation matrices which occur was not emphasized.
"bnf[9]" is a 3-element row vector used in "bnfisprincipal" only and obtained as follows.
Let "D = U W V" obtained by applying the Smith normal form algorithm to the matrix "W" ( =
"bnf[1]") and let "U_r" be the reduction of "U" modulo "D". The first elements of the
factorbase are given (in terms of "bnf.gen") by the columns of "U_r", with archimedian
component "g_a"; let also "GD_a" be the archimedian components of the generators of the
(principal) ideals defined by the "bnf.gen[i]^bnf.cyc[i]". Then "bnf[9] = [U_r, g_a,
GD_a]".
Finally, "bnf[10]" is by default unused and set equal to 0. This field is used to store
further information about the field as it becomes available (which is rarely needed, hence
would be too expensive to compute during the initial "bnfinit" call). For instance, the
generators of the principal ideals "bnf.gen[i]^bnf.cyc[i]" (during a call to
"bnrisprincipal"), or those corresponding to the relations in "W" and "B" (when the "bnf"
internal precision needs to be increased).
"*" The less technical components are as follows:
"bnf[7]" or "bnf.nf" is equal to the number field data "nf" as would be given by "nfinit".
"bnf[8]" is a vector containing the last 6 components of "bnfclassunit[,1]", i.e. the
classgroup "bnf.clgp", the regulator "bnf.reg", the general ``check'' number which should
be close to 1, the number of roots of unity and a generator "bnf.tu", the fundamental
units "bnf.fu", and finally the check on their computation. If the precision becomes
insufficient, GP outputs a warning ("fundamental units too large, not given") and does not
strive to compute the units by default ("flag = 0").
When "flag = 1", GP insists on finding the fundamental units exactly, the internal
precision being doubled and the computation redone, until the exact results are obtained.
The user should be warned that this can take a very long time when the coefficients of the
fundamental units on the integral basis are very large.
When "flag = 2", on the contrary, it is initially agreed that GP will not compute units.
When "flag = 3", computes a very small version of "bnfinit", a ``small big number field''
(or sbnf for short) which contains enough information to recover the full "bnf" vector
very rapidly, but which is much smaller and hence easy to store and print. It is supposed
to be used in conjunction with "bnfmake". The output is a 12 component vector "v", as
follows. Let "bnf" be the result of a full "bnfinit", complete with units. Then "v[1]" is
the polynomial "P", "v[2]" is the number of real embeddings "r_1", "v[3]" is the field
discriminant, "v[4]" is the integral basis, "v[5]" is the list of roots as in the sixth
component of "nfinit", "v[6]" is the matrix "MD" of "nfinit" giving a Z-basis of the
different, "v[7]" is the matrix "W = bnf[1]", "v[8]" is the matrix "matalpha = bnf[2]",
"v[9]" is the prime ideal factor base "bnf[5]" coded in a compact way, and ordered
according to the permutation "bnf[6]", "v[10]" is the 2-component vector giving the number
of roots of unity and a generator, expressed on the integral basis, "v[11]" is the list of
fundamental units, expressed on the integral basis, "v[12]" is a vector containing the
algebraic numbers alpha corresponding to the columns of the matrix "matalpha", expressed
on the integral basis.
Note that all the components are exact (integral or rational), except for the roots in
"v[5]". In practice, this is the only component which a user is allowed to modify, by
recomputing the roots to a higher accuracy if desired. Note also that the member functions
will not work on sbnf, you have to use "bnfmake" explicitly first.
The library syntax is bnfinit0"(P,flag,tech,prec)".
bnfisintnorm"(bnf,x)"
computes a complete system of solutions (modulo units of positive norm) of the absolute
norm equation "Norm(a) = x", where "a" is an integer in "bnf". If "bnf" has not been
certified, the correctness of the result depends on the validity of GRH.
The library syntax is bnfisintnorm"(bnf,x)".
bnfisnorm"(bnf,x,{flag = 1})"
tries to tell whether the rational number "x" is the norm of some element y in "bnf".
Returns a vector "[a,b]" where "x = Norm(a)*b". Looks for a solution which is an "S"-unit,
with "S" a certain set of prime ideals containing (among others) all primes dividing "x".
If "bnf" is known to be Galois, set "flag = 0" (in this case, "x" is a norm iff "b = 1").
If "flag" is non zero the program adds to "S" the following prime ideals, depending on the
sign of "flag". If "flag > 0", the ideals of norm less than "flag". And if "flag < 0" the
ideals dividing "flag".
If you are willing to assume GRH, the answer is guaranteed (i.e. "x" is a norm iff "b =
1"), if "S" contains all primes less than "12 log (disc(Bnf))^2", where "Bnf" is the
Galois closure of "bnf".
The library syntax is bnfisnorm"(bnf,x,flag,prec)", where "flag" and "prec" are "long"s.
bnfissunit"(bnf,sfu,x)"
"bnf" being output by "bnfinit", sfu by "bnfsunit", gives the column vector of exponents
of "x" on the fundamental "S"-units and the roots of unity. If "x" is not a unit, outputs
an empty vector.
The library syntax is bnfissunit"(bnf,sfu,x)".
bnfisprincipal"(bnf,x,{flag = 1})"
"bnf" being the number field data output by "bnfinit", and "x" being either a Z-basis of
an ideal in the number field (not necessarily in HNF) or a prime ideal in the format
output by the function "idealprimedec", this function tests whether the ideal is principal
or not. The result is more complete than a simple true/false answer: it gives a row vector
"[v_1,v_2,check]", where
"v_1" is the vector of components "c_i" of the class of the ideal "x" in the class group,
expressed on the generators "g_i" given by "bnfinit" (specifically "bnf.clgp.gen" which is
the same as "bnf[8][1][3]"). The "c_i" are chosen so that "0 <= c_i < n_i" where "n_i" is
the order of "g_i" (the vector of "n_i" being "bnf.clgp.cyc", that is "bnf[8][1][2]").
"v_2" gives on the integral basis the components of "alpha" such that "x =
alphaprod_ig_i^{c_i}". In particular, "x" is principal if and only if "v_1" is equal to
the zero vector, and if this the case "x = alphaZ_K" where "alpha" is given by "v_2". Note
that if "alpha" is too large to be given, a warning message will be printed and "v_2" will
be set equal to the empty vector.
Finally the third component check is analogous to the last component of "bnfclassunit": it
gives a check on the accuracy of the result, in bits. check should be at least 10, and
preferably much more. In any case, the result is checked for correctness.
If "flag = 0", outputs only "v_1", which is much easier to compute.
If "flag = 2", does as if "flag" were 0, but doubles the precision until a result is
obtained.
If "flag = 3", as in the default behaviour ("flag = 1"), but doubles the precision until a
result is obtained.
The user is warned that these two last setting may induce very lengthy computations.
The library syntax is isprincipalall"(bnf,x,flag)".
bnfisunit"(bnf,x)"
"bnf" being the number field data output by "bnfinit" and "x" being an algebraic number
(type integer, rational or polmod), this outputs the decomposition of "x" on the
fundamental units and the roots of unity if "x" is a unit, the empty vector otherwise.
More precisely, if "u_1",...,"u_r" are the fundamental units, and "zeta" is the generator
of the group of roots of unity (found by "bnfclassunit" or "bnfinit"), the output is a
vector "[x_1,...,x_r,x_{r+1}]" such that "x = u_1^{x_1}...u_r^{x_r}.zeta^{x_{r+1}}". The
"x_i" are integers for "i <= r" and is an integer modulo the order of "zeta" for "i =
r+1".
The library syntax is isunit"(bnf,x)".
bnfmake"(sbnf)"
sbnf being a ``small "bnf"'' as output by "bnfinit""(x,3)", computes the complete
"bnfinit" information. The result is not identical to what "bnfinit" would yield, but is
functionally identical. The execution time is very small compared to a complete "bnfinit".
Note that if the default precision in GP (or "prec" in library mode) is greater than the
precision of the roots "sbnf[5]", these are recomputed so as to get a result with greater
accuracy.
Note that the member functions are not available for sbnf, you have to use "bnfmake"
explicitly first.
The library syntax is makebigbnf"(sbnf,prec)", where "prec" is a C long integer.
bnfnarrow"(bnf)"
"bnf" being a big number field as output by "bnfinit", computes the narrow class group of
"bnf". The output is a 3-component row vector "v" analogous to the corresponding class
group component "bnf.clgp" ("bnf[8][1]"): the first component is the narrow class number
"v.no", the second component is a vector containing the SNF cyclic components "v.cyc" of
the narrow class group, and the third is a vector giving the generators of the
corresponding "v.gen" cyclic groups. Note that this function is a special case of
"bnrclass".
The library syntax is buchnarrow"(bnf)".
bnfsignunit"(bnf)"
"bnf" being a big number field output by "bnfinit", this computes an "r_1 x (r_1+r_2-1)"
matrix having "+-1" components, giving the signs of the real embeddings of the fundamental
units.
The library syntax is signunits"(bnf)".
bnfreg"(bnf)"
"bnf" being a big number field output by "bnfinit", computes its regulator.
The library syntax is regulator"(bnf,tech,prec)", where tech is as in "bnfclassunit".
bnfsunit"(bnf,S)"
computes the fundamental "S"-units of the number field "bnf" (output by "bnfinit"), where
"S" is a list of prime ideals (output by "idealprimedec"). The output is a vector "v" with
6 components.
"v[1]" gives a minimal system of (integral) generators of the "S"-unit group modulo the
unit group.
"v[2]" contains technical data needed by "bnfissunit".
"v[3]" is an empty vector (used to give the logarithmic embeddings of the generators in
"v[1]" in version 2.0.16).
"v[4]" is the "S"-regulator (this is the product of the regulator, the determinant of
"v[2]" and the natural logarithms of the norms of the ideals in "S").
"v[5]" gives the "S"-class group structure, in the usual format (a row vector whose three
components give in order the "S"-class number, the cyclic components and the generators).
"v[6]" is a copy of "S".
The library syntax is bnfsunit"(bnf,S,prec)".
bnfunit"(bnf)"
"bnf" being a big number field as output by "bnfinit", outputs a two-component row vector
giving in the first component the vector of fundamental units of the number field, and in
the second component the number of bit of accuracy which remained in the computation
(which is always correct, otherwise an error message is printed). This function is mainly
for people who used the wrong flag in "bnfinit" and would like to skip part of a lengthy
"bnfinit" computation.
The library syntax is buchfu"(bnf)".
bnrL1"(bnr,subgroup,{flag = 0})"
bnr being the number field data which is output by "bnrinit(,,1)" and subgroup being a
square matrix defining a congruence subgroup of the ray class group corresponding to bnr
(or 0 for the trivial congruence subgroup), returns for each character "chi" of the ray
class group which is trivial on this subgroup, the value at "s = 1" (or "s = 0") of the
abelian "L"-function associated to "chi". For the value at "s = 0", the function returns
in fact for each character "chi" a vector "[r_chi , c_chi]" where "r_chi" is the order of
"L(s, chi)" at "s = 0" and "c_chi" the first non-zero term in the expansion of "L(s, chi)"
at "s = 0"; in other words
"L(s, chi) = c_chi.s^{r_chi} + O(s^{r_chi + 1})"
near 0. flag is optional, default value is 0; its binary digits mean 1: compute at "s = 1"
if set to 1 or "s = 0" if set to 0, 2: compute the primitive "L"-functions associated to
"chi" if set to 0 or the "L"-function with Euler factors at prime ideals dividing the
modulus of bnr removed if set to 1 (this is the so-called "L_S(s, chi)" function where "S"
is the set of infinite places of the number field together with the finite prime ideals
dividing the modulus of bnr, see the example below), 3: returns also the character.
Example:
bnf = bnfinit(x^2 - 229);
bnr = bnrinit(bnf,1,1);
bnrL1(bnr, 0)
returns the order and the first non-zero term of the abelian "L"-functions "L(s, chi)" at
"s = 0" where "chi" runs through the characters of the class group of "Q( sqrt {229})".
Then
bnr2 = bnrinit(bnf,2,1);
bnrL1(bnr2,0,2)
returns the order and the first non-zero terms of the abelian "L"-functions "L_S(s, chi)"
at "s = 0" where "chi" runs through the characters of the class group of "Q( sqrt {229})"
and "S" is the set of infinite places of "Q( sqrt {229})" together with the finite prime 2
(note that the ray class group modulo 2 is in fact the class group, so "bnrL1(bnr2,0)"
returns exactly the same answer as "bnrL1(bnr,0)"!).
The library syntax is bnrL1"(bnr,subgroup,flag,prec)"
bnrclass"(bnf,ideal,{flag = 0})"
"bnf" being a big number field as output by "bnfinit" (the units are mandatory unless the
ideal is trivial), and ideal being either an ideal in any form or a two-component row
vector containing an ideal and an "r_1"-component row vector of flags indicating which
real Archimedean embeddings to take in the module, computes the ray class group of the
number field for the module ideal, as a 3-component vector as all other finite Abelian
groups (cardinality, vector of cyclic components, corresponding generators).
If "flag = 2", the output is different. It is a 6-component vector "w". "w[1]" is "bnf".
"w[2]" is the result of applying "idealstar(bnf,I,2)". "w[3]", "w[4]" and "w[6]" are
technical components used only by the function "bnrisprincipal". "w[5]" is the structure
of the ray class group as would have been output with "flag = 0".
If "flag = 1", as above, except that the generators of the ray class group are not
computed, which saves time.
The library syntax is bnrclass0"(bnf,ideal,flag,prec)".
bnrclassno"(bnf,I)"
"bnf" being a big number field as output by "bnfinit" (units are mandatory unless the
ideal is trivial), and "I" being either an ideal in any form or a two-component row vector
containing an ideal and an "r_1"-component row vector of flags indicating which real
Archimedean embeddings to take in the modulus, computes the ray class number of the number
field for the modulus "I". This is faster than "bnrclass" and should be used if only the
ray class number is desired.
The library syntax is rayclassno"(bnf,I)".
bnrclassnolist"(bnf,list)"
"bnf" being a big number field as output by "bnfinit" (units are mandatory unless the
ideal is trivial), and list being a list of modules as output by "ideallist" of
"ideallistarch", outputs the list of the class numbers of the corresponding ray class
groups.
The library syntax is rayclassnolist"(bnf,list)".
bnrconductor"(a_1,{a_2},{a_3}, {flag = 0})"
conductor of the subfield of a ray class field as defined by "[a_1,a_2,a_3]" (see "bnr" at
the beginning of this section).
The library syntax is bnrconductor"(a_1,a_2,a_3,flag,prec)", where an omitted argument
among the "a_i" is input as "gzero", and "flag" is a C long.
bnrconductorofchar"(bnr,chi)"
bnr being a big ray number field as output by "bnrclass", and chi being a row vector
representing a character as expressed on the generators of the ray class group, gives the
conductor of this character as a modulus.
The library syntax is bnrconductorofchar"(bnr,chi,prec)" where "prec" is a "long".
bnrdisc"(a1,{a2},{a3},{flag = 0})"
"a1", "a2", "a3" defining a big ray number field "L" over a groud field "K" (see "bnr" at
the beginning of this section for the meaning of "a1", "a2", "a3"), outputs a 3-component
row vector "[N,R_1,D]", where "N" is the (absolute) degree of "L", "R_1" the number of
real places of "L", and "D" the discriminant of "L/Q", including sign (if "flag = 0").
If "flag = 1", as above but outputs relative data. "N" is now the degree of "L/K", "R_1"
is the number of real places of "K" unramified in "L" (so that the number of real places
of "L" is equal to "R_1" times the relative degree "N"), and "D" is the relative
discriminant ideal of "L/K".
If "flag = 2", does as in case 0, except that if the modulus is not the exact conductor
corresponding to the "L", no data is computed and the result is 0 ("gzero").
If "flag = 3", as case 2, outputting relative data.
The library syntax is bnrdisc0"(a1,a2,a3,flag,prec)".
bnrdisclist"(bnf,bound,{arch},{flag = 0})"
"bnf" being a big number field as output by "bnfinit" (the units are mandatory), computes
a list of discriminants of Abelian extensions of the number field by increasing modulus
norm up to bound bound, where the ramified Archimedean places are given by arch
(unramified at infinity if arch is void or omitted). If flag is non-zero, give arch all
the possible values. (See "bnr" at the beginning of this section for the meaning of "a1",
"a2", "a3".)
The alternative syntax "bnrdisclist(bnf,list)" is supported, where list is as output by
"ideallist" or "ideallistarch" (with units).
The output format is as follows. The output "v" is a row vector of row vectors, allowing
the bound to be greater than "2^{16}" for 32-bit machines, and "v[i][j]" is understood to
be in fact "V[2^{15}(i-1)+j]" of a unique big vector "V" (note that "2^{15}" is hardwired
and can be increased in the source code only on 64-bit machines and higher).
Such a component "V[k]" is itself a vector "W" (maybe of length 0) whose components
correspond to each possible ideal of norm "k". Each component "W[i]" corresponds to an
Abelian extension "L" of "bnf" whose modulus is an ideal of norm "k" and no Archimedean
components (hence the extension is unramified at infinity). The extension "W[i]" is
represented by a 4-component row vector "[m,d,r,D]" with the following meaning. "m" is the
prime ideal factorization of the modulus, "d = [L:Q]" is the absolute degree of "L", "r"
is the number of real places of "L", and "D" is the factorization of the absolute
discriminant. Each prime ideal "pr = [p,alpha,e,f,beta]" in the prime factorization "m" is
coded as "p.n^2+(f-1).n+(j-1)", where "n" is the degree of the base field and "j" is such
that
"pr = idealprimedec(nf,p)[j]".
"m" can be decoded using "bnfdecodemodule".
The library syntax is bnrdisclist0"(a1,a2,a3,bound,arch,flag)".
bnrinit"(bnf,ideal,{flag = 0})"
"bnf" is as output by "bnfinit", ideal is a valid ideal (or a module), initializes data
linked to the ray class group structure corresponding to this module. This is the same as
"bnrclass(bnf,ideal,flag+1)".
The library syntax is bnrinit0"(bnf,ideal,flag,prec)".
bnrisconductor"(a1,{a2},{a3})"
"a1", "a2", "a3" represent an extension of the base field, given by class field theory for
some modulus encoded in the parameters. Outputs 1 if this modulus is the conductor, and 0
otherwise. This is slightly faster than "bnrconductor".
The library syntax is bnrisconductor"(a1,a2,a3)" and the result is a "long".
bnrisprincipal"(bnr,x,{flag = 1})"
bnr being the number field data which is output by "bnrinit""(,,1)" and "x" being an ideal
in any form, outputs the components of "x" on the ray class group generators in a way
similar to "bnfisprincipal". That is a 3-component vector "v" where "v[1]" is the vector
of components of "x" on the ray class group generators, "v[2]" gives on the integral basis
an element "alpha" such that "x = alphaprod_ig_i^{x_i}". Finally "v[3]" indicates the
number of bits of accuracy left in the result. In any case the result is checked for
correctness, but "v[3]" is included to see if it is necessary to increase the accuracy in
other computations.
If "flag = 0", outputs only "v_1". In that case, bnr need not contain the ray class group
generators, i.e. it may be created with "bnrinit""(,,0)"
The library syntax is isprincipalrayall"(bnr,x,flag)".
bnrrootnumber"(bnr,chi,{flag = 0})"
if "chi = chi" is a (not necessarily primitive) character over bnr, let "L(s,chi) =
sum_{id} chi(id) N(id)^{-s}" be the associated Artin L-function. Returns the so-called
Artin root number, i.e. the complex number "W(chi)" of modulus 1 such that
"Lambda(1-s,chi) = W(chi) Lambda(s,\overline{chi})"
where "Lambda(s,chi) = A(chi)^{s/2}gamma_chi(s) L(s,chi)" is the enlarged L-function
associated to "L".
The generators of the ray class group are needed, and you can set "flag = 1" if the
character is known to be primitive. Example:
bnf = bnfinit(x^2 - 145);
bnr = bnrinit(bnf,7,1);
bnrrootnumber(bnr, [5])
returns the root number of the character "chi" of "Cl_7(Q( sqrt {145}))" such that "chi(g)
= zeta^5", where "g" is the generator of the ray-class field and "zeta = e^{2iPi/N}" where
"N" is the order of "g" ("N = 12" as "bnr.cyc" readily tells us).
The library syntax is bnrrootnumber"(bnf,chi,flag)"
bnrstark"{(bnr,subgroup,{flag = 0})}"
bnr being as output by "bnrinit(,,1)", finds a relative equation for the class field
corresponding to the modulus in bnr and the given congruence subgroup using Stark units
(set "subgroup = 0" if you want the whole ray class group). The main variable of bnr must
not be "x", and the ground field and the class field must be totally real and not
isomorphic to Q (over the rationnals, use "polsubcyclo" or "galoissubcyclo"). flag is
optional and may be set to 0 to obtain a reduced relative polynomial, 1 to be satisfied
with any relative polynomial, 2 to obtain an absolute polynomial and 3 to obtain the
irreducible relative polynomial of the Stark unit, 0 being default. Example:
bnf = bnfinit(y^2 - 3);
bnr = bnrinit(bnf, 5, 1);
bnrstark(bnr, 0)
returns the ray class field of "Q( sqrt {3})" modulo 5.
Remark. The result of the computation depends on the choice of a modulus verifying special
conditions. By default the function will try few moduli, choosing the one giving the
smallest result. In some cases where the result is however very large, you can tell the
function to try more moduli by adding 4 to the value of flag. Whether this flag is set or
not, the function may fail in some extreme cases, returning the error message
"Cannot find a suitable modulus in FindModule".
In this case, the corresponding congruence group is a product of cyclic groups and, for
the time being, the class field has to be obtained by splitting this group into its cyclic
components.
The library syntax is bnrstark"(bnr,subgroup,flag)".
dirzetak"(nf,b)"
gives as a vector the first "b" coefficients of the Dedekind zeta function of the number
field "nf" considered as a Dirichlet series.
The library syntax is dirzetak"(nf,b)".
factornf"(x,t)"
factorization of the univariate polynomial "x" over the number field defined by the
(univariate) polynomial "t". "x" may have coefficients in Q or in the number field. The
main variable of "t" must be of lower priority than that of "x" (in other words the
variable number of "t" must be greater than that of "x"). However if the coefficients of
the number field occur explicitly (as polmods) as coefficients of "x", the variable of
these polmods must be the same as the main variable of "t". For example
? factornf(x^2 + Mod(y, y^2+1), y^2+1);
? factornf(x^2 + 1, y^2+1); \\ these two are OK
? factornf(x^2 + Mod(z,z^2+1), y^2+1)
*** incorrect type in gmulsg
The library syntax is polfnf"(x,t)".
galoisfixedfield"(gal,perm,{fl = 0},{v = y}))"
gal being be a Galois field as output by "galoisinit" and perm an element of "gal.group"
or a vector of such elements, computes the fixed field of gal by the automorphism defined
by the permutations perm of the roots "gal.roots". "P" is guaranteed to be squarefree
modulo "gal.p".
If no flags or "flag = 0", output format is the same as for "nfsubfield", returning
"[P,x]" such that "P" is a polynomial defining the fixed field, and "x" is a root of "P"
expressed as a polmod in "gal.pol".
If "flag = 1" return only the polynomial "P".
If "flag = 2" return "[P,x,F]" where "P" and "x" are as above and "F" is the factorization
of "gal.pol" over the field defined by "P", where variable "v" ("y" by default) stands for
a root of "P". The priority of "v" must be less than the priority of the variable of
"gal.pol".
Example:
G = galoisinit(x^4+1);
galoisfixedfield(G,G.group[2],2)
[x^2 + 2, Mod(x^3 + x, x^4 + 1), [x^2 - y*x - 1, x^2 + y*x - 1]]
computes the factorization "x^4+1 = (x^2- sqrt {-2}x-1)(x^2+ sqrt {-2}x-1)"
The library syntax is galoisfixedfield"(gal,perm,p)".
galoisinit"(pol,{den})"
computes the Galois group and all neccessary information for computing the fixed fields of
the Galois extension "K/Q" where "K" is the number field defined by "pol" (monic
irreducible polynomial in "Z[X]" or a number field as output by "nfinit"). The extension
"K/Q" must be Galois with Galois group ``weakly'' super-solvable (see "nfgaloisconj")
Warning: The interface of this function is experimental, so the described output can be
subject to important changes in the near future. However the function itself should work
as described. For any remarks about this interface, please mail
"allomber AT math.fr".
The output is an 8-component vector gal.
"gal[1]" contains the polynomial pol ("gal.pol").
"gal[2]" is a three--components vector "[p,e,q]" where "p" is a prime number ("gal.p")
such that pol totally split modulo "p" , "e" is an integer and "q = p^e" ("gal.mod") is
the modulus of the roots in "gal.roots".
"gal[3]" is a vector "L" containing the "p"-adic roots of pol as integers implicitly
modulo "gal.mod". ("gal.roots").
"gal[4]" is the inverse of the Van der Monde matrix of the "p"-adic roots of pol,
multiplied by "gal[5]".
"gal[5]" is a multiple of the least common denominator of the automorphisms expressed as
polynomial in a root of pol.
"gal[6]" is the Galois group "G" expressed as a vector of permutations of "L"
("gal.group").
"gal[7]" is a generating subset "S = [s_1,...,s_g]" of "G" expressed as a vector of
permutations of "L" ("gal.gen").
"gal[8]" contains the relative orders "[o_1,...,o_g]" of the generators of "S"
("gal.orders").
Let "H" be the maximal normal supersolvable subgroup of "G", we have the following
properties:
"*" if "G/H ~ A_4" then "[o_1,...,o_g]" ends by "[2,2,3]".
"*" if "G/H ~ S_4" then "[o_1,...,o_g]" ends by "[2,2,3,2]".
"*" else "G" is super-solvable.
"*" for "1 <= i <= g" the subgroup of "G" generated by "[s_1,...,s_g]" is normal, with
the exception of "i = g-2" in the second case and of "i = g-3" in the third.
"*" the relative order "o_i" of "s_i" is its order in the quotient group
"G/<s_1,...,s_{i-1}>", with the same exceptions.
"*" for any "x belongs to G" there exists a unique family "[e_1,...,e_g]" such that (no
exceptions):
-- for "1 <= i <= g" we have "0 <= e_i < o_i"
-- "x = g_1^{e_1}g_2^{e_2}...g_n^{e_n}"
If present "den" must be a suitable value for "gal[5]".
The library syntax is galoisinit"(gal,den)".
galoispermtopol"(gal,perm)"
gal being a galois field as output by "galoisinit" and perm a element of "gal.group",
return the polynomial defining the Galois automorphism, as output by "nfgaloisconj",
associated with the permutation perm of the roots "gal.roots". perm can also be a vector
or matrix, in this case, "galoispermtopol" is applied to all components recursively.
Note that
G = galoisinit(pol);
galoispermtopol(G, G[6])~
is equivalent to "nfgaloisconj(pol)", if degree of pol is greater or equal to 2.
The library syntax is galoispermtopol"(gal,perm)".
galoissubcyclo"(n,H,{Z},{v})"
compute a polynomial defining the subfield of "Q(zeta_n)" fixed by the subgroup H of
"Z/nZ". The subgroup H can be given by a generator, a set of generators given by a vector
or a HNF matrix. If present "Z" must be znstar(n), and is currently only used when H is a
HNF matrix. If v is given, the polynomial is given in the variable v.
The following function can be used to compute all subfields of "Q(zeta_n)" (of order less
than "d", if "d" is set):
subcyclo(n, d = -1)=
{
local(Z,G,S);
if (d < 0, d = n);
Z = znstar(n);
G = matdiagonal(Z[2]);
S = [];
forsubgroup(H = G, d,
S = concat(S, galoissubcyclo(n, mathnf(concat(G,H)),Z));
);
S
}
The library syntax is galoissubcyclo"(n,H,Z,v)" where n is a C long integer.
idealadd"(nf,x,y)"
sum of the two ideals "x" and "y" in the number field "nf". When "x" and "y" are given by
Z-bases, this does not depend on "nf" and can be used to compute the sum of any two
Z-modules. The result is given in HNF.
The library syntax is idealadd"(nf,x,y)".
idealaddtoone"(nf,x,{y})"
"x" and "y" being two co-prime integral ideals (given in any form), this gives a two-
component row vector "[a,b]" such that "a belongs to x", "b belongs to y" and "a+b = 1".
The alternative syntax "idealaddtoone(nf,v)", is supported, where "v" is a "k"-component
vector of ideals (given in any form) which sum to "Z_K". This outputs a "k"-component
vector "e" such that "e[i] belongs to x[i]" for "1 <= i <= k" and "sum_{1 <= i <= k}e[i] =
1".
The library syntax is idealaddtoone0"(nf,x,y)", where an omitted "y" is coded as "NULL".
idealappr"(nf,x,{flag = 0})"
if "x" is a fractional ideal (given in any form), gives an element "alpha" in "nf" such
that for all prime ideals "p" such that the valuation of "x" at "p" is non-zero, we have
"v_{p}(alpha) = v_{p}(x)", and. "v_{p}(alpha) >= 0" for all other "{p}".
If "flag" is non-zero, "x" must be given as a prime ideal factorization, as output by
"idealfactor", but possibly with zero or negative exponents. This yields an element
"alpha" such that for all prime ideals "p" occurring in "x", "v_{p}(alpha)" is equal to
the exponent of "p" in "x", and for all other prime ideals, "v_{p}(alpha) >= 0". This
generalizes "idealappr(nf,x,0)" since zero exponents are allowed. Note that the algorithm
used is slightly different, so that "idealapp(nf,idealfactor(nf,x))" may not be the same
as "idealappr(nf,x,1)".
The library syntax is idealappr0"(nf,x,flag)".
idealchinese"(nf,x,y)"
"x" being a prime ideal factorization (i.e. a 2 by 2 matrix whose first column contain
prime ideals, and the second column integral exponents), "y" a vector of elements in "nf"
indexed by the ideals in "x", computes an element "b" such that
"v_p(b - y_p) >= v_p(x)" for all prime ideals in "x" and "v_p(b) >= 0" for all other "p".
The library syntax is idealchinese"(nf,x,y)".
idealcoprime"(nf,x,y)"
given two integral ideals "x" and "y" in the number field "nf", finds a "beta" in the
field, expressed on the integral basis "nf[7]", such that "beta.y" is an integral ideal
coprime to "x".
The library syntax is idealcoprime"(nf,x)".
idealdiv"(nf,x,y,{flag = 0})"
quotient "x.y^{-1}" of the two ideals "x" and "y" in the number field "nf". The result is
given in HNF.
If "flag" is non-zero, the quotient "x.y^{-1}" is assumed to be an integral ideal. This
can be much faster when the norm of the quotient is small even though the norms of "x" and
"y" are large.
The library syntax is idealdiv0"(nf,x,y,flag)". Also available are "idealdiv(nf,x,y)"
("flag = 0") and "idealdivexact(nf,x,y)" ("flag = 1").
idealfactor"(nf,x)"
factors into prime ideal powers the ideal "x" in the number field "nf". The output format
is similar to the "factor" function, and the prime ideals are represented in the form
output by the "idealprimedec" function, i.e. as 5-element vectors.
The library syntax is idealfactor"(nf,x)".
idealhnf"(nf,a,{b})"
gives the Hermite normal form matrix of the ideal "a". The ideal can be given in any form
whatsoever (typically by an algebraic number if it is principal, by a "Z_K"-system of
generators, as a prime ideal as given by "idealprimedec", or by a Z-basis).
If "b" is not omitted, assume the ideal given was "aZ_K+bZ_K", where "a" and "b" are
elements of "K" given either as vectors on the integral basis "nf[7]" or as algebraic
numbers.
The library syntax is idealhnf0"(nf,a,b)" where an omitted "b" is coded as "NULL". Also
available is "idealhermite(nf,a)" ("b" omitted).
idealintersect"(nf,x,y)"
intersection of the two ideals "x" and "y" in the number field "nf". When "x" and "y" are
given by Z-bases, this does not depend on "nf" and can be used to compute the intersection
of any two Z-modules. The result is given in HNF.
The library syntax is idealintersect"(nf,x,y)".
idealinv"(nf,x)"
inverse of the ideal "x" in the number field "nf". The result is the Hermite normal form
of the inverse of the ideal, together with the opposite of the Archimedean information if
it is given.
The library syntax is idealinv"(nf,x)".
ideallist"(nf,bound,{flag = 4})"
computes the list of all ideals of norm less or equal to bound in the number field nf. The
result is a row vector with exactly bound components. Each component is itself a row
vector containing the information about ideals of a given norm, in no specific order. This
information can be either the HNF of the ideal or the "idealstar" with possibly some
additional information.
If "flag" is present, its binary digits are toggles meaning
1: give also the generators in the "idealstar".
2: output "[L,U]", where "L" is as before and "U" is a vector of "zinternallog"s of the
units.
4: give only the ideals and not the "idealstar" or the "ideallog" of the units.
The library syntax is ideallist0"(nf,bound,flag)", where bound must be a C long integer.
Also available is "ideallist(nf,bound)", corresponding to the case "flag = 0".
ideallistarch"(nf,list,{arch = []},{flag = 0})"
vector of vectors of all "idealstarinit" (see "idealstar") of all modules in list, with
Archimedean part arch added (void if omitted). list is a vector of big ideals, as output
by "ideallist""(..., flag)" for instance. "flag" is optional; its binary digits are
toggles meaning: 1: give generators as well, 2: list format is "[L,U]" (see "ideallist").
The library syntax is ideallistarch0"(nf,list,arch,flag)", where an omitted arch is coded
as "NULL".
ideallog"(nf,x,bid)"
"nf" being a number field, bid being a ``big ideal'' as output by "idealstar" and "x"
being a non-necessarily integral element of nf which must have valuation equal to 0 at all
prime ideals dividing "I = bid[1]", computes the ``discrete logarithm'' of "x" on the
generators given in "bid[2]". In other words, if "g_i" are these generators, of orders
"d_i" respectively, the result is a column vector of integers "(x_i)" such that "0 <= x_i
< d_i" and
"x = prod_ig_i^{x_i} (mod ^*I) ."
Note that when "I" is a module, this implies also sign conditions on the embeddings.
The library syntax is zideallog"(nf,x,bid)".
idealmin"(nf,x,{vdir})"
computes a minimum of the ideal "x" in the direction vdir in the number field nf.
The library syntax is minideal"(nf,x,vdir,prec)", where an omitted vdir is coded as
"NULL".
idealmul"(nf,x,y,{flag = 0})"
ideal multiplication of the ideals "x" and "y" in the number field nf. The result is a
generating set for the ideal product with at most "n" elements, and is in Hermite normal
form if either "x" or "y" is in HNF or is a prime ideal as output by "idealprimedec", and
this is given together with the sum of the Archimedean information in "x" and "y" if both
are given.
If "flag" is non-zero, reduce the result using "idealred".
The library syntax is idealmul"(nf,x,y)" ("flag = 0") or "idealmulred(nf,x,y,prec)" ("flag
! = 0"), where as usual, "prec" is a C long integer representing the precision.
idealnorm"(nf,x)"
computes the norm of the ideal "x" in the number field "nf".
The library syntax is idealnorm"(nf, x)".
idealpow"(nf,x,k,{flag = 0})"
computes the "k"-th power of the ideal "x" in the number field "nf". "k" can be positive,
negative or zero. The result is NOT reduced, it is really the "k"-th ideal power, and is
given in HNF.
If "flag" is non-zero, reduce the result using "idealred". Note however that this is NOT
the same as as "idealpow(nf,x,k)" followed by reduction, since the reduction is performed
throughout the powering process.
The library syntax corresponding to "flag = 0" is "idealpow(nf,x,k)". If "k" is a "long",
you can use "idealpows(nf,x,k)". Corresponding to "flag = 1" is
"idealpowred(nf,vp,k,prec)", where "prec" is a "long".
idealprimedec"(nf,p)"
computes the prime ideal decomposition of the prime number "p" in the number field "nf".
"p" must be a (positive) prime number. Note that the fact that "p" is prime is not
checked, so if a non-prime number "p" is given it may lead to unpredictable results.
The result is a vector of 5-component vectors, each representing one of the prime ideals
above "p" in the number field "nf". The representation "vp = [p,a,e,f,b]" of a prime ideal
means the following. The prime ideal is equal to "pZ_K+alphaZ_K" where "Z_K" is the ring
of integers of the field and "alpha = sum_i a_iomega_i" where the "omega_i" form the
integral basis "nf.zk", "e" is the ramification index, "f" is the residual index, and "b"
is an "n"-component column vector representing a "beta belongs to Z_K" such that "vp^{-1}
= Z_K+beta/pZ_K" which will be useful for computing valuations, but which the user can
ignore. The number "alpha" is guaranteed to have a valuation equal to 1 at the prime ideal
(this is automatic if "e > 1").
The library syntax is idealprimedec"(nf,p)".
idealprincipal"(nf,x)"
creates the principal ideal generated by the algebraic number "x" (which must be of type
integer, rational or polmod) in the number field "nf". The result is a one-column matrix.
The library syntax is principalideal"(nf,x)".
idealred"(nf,I,{vdir = 0})"
LLL reduction of the ideal "I" in the number field nf, along the direction vdir. If vdir
is present, it must be an "r1+r2"-component vector ("r1" and "r2" number of real and
complex places of nf as usual).
This function finds a ``small'' "a" in "I" (it is an LLL pseudo-minimum along direction
vdir). The result is the Hermite normal form of the LLL-reduced ideal "r I/a", where "r"
is a rational number such that the resulting ideal is integral and primitive. This is
often, but not always, a reduced ideal in the sense of Buchmann. If "I" is an idele, the
logarithmic embeddings of "a" are subtracted to the Archimedean part.
More often than not, a principal ideal will yield the identity matrix. This is a quick and
dirty way to check if ideals are principal without computing a full "bnf" structure, but
it's not a necessary condition; hence, a non-trivial result doesn't prove the ideal is
non-trivial in the class group.
Note that this is not the same as the LLL reduction of the lattice "I" since ideal
operations are involved.
The library syntax is ideallllred"(nf,x,vdir,prec)", where an omitted vdir is coded as
"NULL".
idealstar"(nf,I,{flag = 1})"
nf being a number field, and "I" either and ideal in any form, or a row vector whose first
component is an ideal and whose second component is a row vector of "r_1" 0 or 1, outputs
necessary data for computing in the group "(Z_K/I)^*".
If "flag = 2", the result is a 5-component vector "w". "w[1]" is the ideal or module "I"
itself. "w[2]" is the structure of the group. The other components are difficult to
describe and are used only in conjunction with the function "ideallog".
If "flag = 1" (default), as "flag = 2", but do not compute explicit generators for the
cyclic components, which saves time.
If "flag = 0", computes the structure of "(Z_K/I)^*" as a 3-component vector "v". "v[1]"
is the order, "v[2]" is the vector of SNF cyclic components and "v[3]" the corresponding
generators. When the row vector is explicitly included, the non-zero elements of this
vector are considered as real embeddings of nf in the order given by "polroots", i.e. in
nf[6] ("nf.roots"), and then "I" is a module with components at infinity.
To solve discrete logarithms (using "ideallog"), you have to choose "flag = 2".
The library syntax is idealstar0"(nf,I,flag)".
idealtwoelt"(nf,x,{a})"
computes a two-element representation of the ideal "x" in the number field "nf", using a
straightforward (exponential time) search. "x" can be an ideal in any form, (including
perhaps an Archimedean part, which is ignored) and the result is a row vector "[a,alpha]"
with two components such that "x = aZ_K+alphaZ_K" and "a belongs to Z", where "a" is the
one passed as argument if any. If "x" is given by at least two generators, "a" is chosen
to be the positive generator of "x cap Z".
Note that when an explicit "a" is given, we use an asymptotically faster method, however
in practice it is usually slower.
The library syntax is ideal_two_elt0"(nf,x,a)", where an omitted "a" is entered as "NULL".
idealval"(nf,x,vp)"
gives the valuation of the ideal "x" at the prime ideal vp in the number field "nf", where
vp must be a 5-component vector as given by "idealprimedec".
The library syntax is idealval"(nf,x,vp)", and the result is a "long" integer.
ideleprincipal"(nf,x)"
creates the principal idele generated by the algebraic number "x" (which must be of type
integer, rational or polmod) in the number field "nf". The result is a two-component
vector, the first being a one-column matrix representing the corresponding principal
ideal, and the second being the vector with "r_1+r_2" components giving the complex
logarithmic embedding of "x".
The library syntax is principalidele"(nf,x)".
matalgtobasis"(nf,x)"
"nf" being a number field in "nfinit" format, and "x" a matrix whose coefficients are
expressed as polmods in "nf", transforms this matrix into a matrix whose coefficients are
expressed on the integral basis of "nf". This is the same as applying "nfalgtobasis" to
each entry, but it would be dangerous to use the same name.
The library syntax is matalgtobasis"(nf,x)".
matbasistoalg"(nf,x)"
"nf" being a number field in "nfinit" format, and "x" a matrix whose coefficients are
expressed as column vectors on the integral basis of "nf", transforms this matrix into a
matrix whose coefficients are algebraic numbers expressed as polmods. This is the same as
applying "nfbasistoalg" to each entry, but it would be dangerous to use the same name.
The library syntax is matbasistoalg"(nf,x)".
modreverse"(a)"
"a" being a polmod A(X) modulo T(X), finds the ``reverse polmod'' B(X) modulo Q(X), where
"Q" is the minimal polynomial of "a", which must be equal to the degree of "T", and such
that if "theta" is a root of "T" then "theta = B(alpha)" for a certain root "alpha" of
"Q".
This is very useful when one changes the generating element in algebraic extensions.
The library syntax is polmodrecip"(x)".
newtonpoly"(x,p)"
gives the vector of the slopes of the Newton polygon of the polynomial "x" with respect to
the prime number "p". The "n" components of the vector are in decreasing order, where "n"
is equal to the degree of "x". Vertical slopes occur iff the constant coefficient of "x"
is zero and are denoted by "VERYBIGINT", the biggest single precision integer
representable on the machine ("2^{31}-1" (resp. "2^{63}-1") on 32-bit (resp. 64-bit)
machines), see "Label se:valuation".
The library syntax is newtonpoly"(x,p)".
nfalgtobasis"(nf,x)"
this is the inverse function of "nfbasistoalg". Given an object "x" whose entries are
expressed as algebraic numbers in the number field "nf", transforms it so that the entries
are expressed as a column vector on the integral basis "nf.zk".
The library syntax is algtobasis"(nf,x)".
nfbasis"(x,{flag = 0},{p})"
integral basis of the number field defined by the irreducible, preferably monic,
polynomial "x", using a modified version of the round 4 algorithm by default. The binary
digits of "flag" have the following meaning:
1: assume that no square of a prime greater than the default "primelimit" divides the
discriminant of "x", i.e. that the index of "x" has only small prime divisors.
2: use round 2 algorithm. For small degrees and coefficient size, this is sometimes a
little faster. (This program is the translation into C of a program written by David Ford
in Algeb.)
Thus for instance, if "flag = 3", this uses the round 2 algorithm and outputs an order
which will be maximal at all the small primes.
If "p" is present, we assume (without checking!) that it is the two-column matrix of the
factorization of the discriminant of the polynomial "x". Note that it does not have to be
a complete factorization. This is especially useful if only a local integral basis for
some small set of places is desired: only factors with exponents greater or equal to 2
will be considered.
The library syntax is nfbasis0"(x,flag,p)". An extended version is "nfbasis(x,&d,flag,p)",
where "d" will receive the discriminant of the number field (not of the polynomial "x"),
and an omitted "p" should be input as "gzero". Also available are "base(x,&d)" ("flag =
0"), "base2(x,&d)" ("flag = 2") and "factoredbase(x,p,&d)".
nfbasistoalg"(nf,x)"
this is the inverse function of "nfalgtobasis". Given an object "x" whose entries are
expressed on the integral basis "nf.zk", transforms it into an object whose entries are
algebraic numbers (i.e. polmods).
The library syntax is basistoalg"(nf,x)".
nfdetint"(nf,x)"
given a pseudo-matrix "x", computes a non-zero ideal contained in (i.e. multiple of) the
determinant of "x". This is particularly useful in conjunction with "nfhnfmod".
The library syntax is nfdetint"(nf,x)".
nfdisc"(x,{flag = 0},{p})"
field discriminant of the number field defined by the integral, preferably monic,
irreducible polynomial "x". "flag" and "p" are exactly as in "nfbasis". That is, "p"
provides the matrix of a partial factorization of the discriminant of "x", and binary
digits of "flag" are as follows:
1: assume that no square of a prime greater than "primelimit" divides the discriminant.
2: use the round 2 algorithm, instead of the default round 4. This should be slower
except maybe for polynomials of small degree and coefficients.
The library syntax is nfdiscf0"(x,flag,p)" where, to omit "p", you should input "gzero".
You can also use "discf(x)" ("flag = 0").
nfeltdiv"(nf,x,y)"
given two elements "x" and "y" in nf, computes their quotient "x/y" in the number field
"nf".
The library syntax is element_div"(nf,x,y)".
nfeltdiveuc"(nf,x,y)"
given two elements "x" and "y" in nf, computes an algebraic integer "q" in the number
field "nf" such that the components of "x-qy" are reasonably small. In fact, this is
functionally identical to "round(nfeltdiv(nf,x,y))".
The library syntax is nfdiveuc"(nf,x,y)".
nfeltdivmodpr"(nf,x,y,pr)"
given two elements "x" and "y" in nf and pr a prime ideal in "modpr" format (see
"nfmodprinit"), computes their quotient "x / y" modulo the prime ideal pr.
The library syntax is element_divmodpr"(nf,x,y,pr)".
nfeltdivrem"(nf,x,y)"
given two elements "x" and "y" in nf, gives a two-element row vector "[q,r]" such that "x
= qy+r", "q" is an algebraic integer in "nf", and the components of "r" are reasonably
small.
The library syntax is nfdivres"(nf,x,y)".
nfeltmod"(nf,x,y)"
given two elements "x" and "y" in nf, computes an element "r" of "nf" of the form "r =
x-qy" with "q" and algebraic integer, and such that "r" is small. This is functionally
identical to
"x - nfeltmul(nf,round(nfeltdiv(nf,x,y)),y)."
The library syntax is nfmod"(nf,x,y)".
nfeltmul"(nf,x,y)"
given two elements "x" and "y" in nf, computes their product "x*y" in the number field
"nf".
The library syntax is element_mul"(nf,x,y)".
nfeltmulmodpr"(nf,x,y,pr)"
given two elements "x" and "y" in nf and pr a prime ideal in "modpr" format (see
"nfmodprinit"), computes their product "x*y" modulo the prime ideal pr.
The library syntax is element_mulmodpr"(nf,x,y,pr)".
nfeltpow"(nf,x,k)"
given an element "x" in nf, and a positive or negative integer "k", computes "x^k" in the
number field "nf".
The library syntax is element_pow"(nf,x,k)".
nfeltpowmodpr"(nf,x,k,pr)"
given an element "x" in nf, an integer "k" and a prime ideal pr in "modpr" format (see
"nfmodprinit"), computes "x^k" modulo the prime ideal pr.
The library syntax is element_powmodpr"(nf,x,k,pr)".
nfeltreduce"(nf,x,ideal)"
given an ideal in Hermite normal form and an element "x" of the number field "nf", finds
an element "r" in "nf" such that "x-r" belongs to the ideal and "r" is small.
The library syntax is element_reduce"(nf,x,ideal)".
nfeltreducemodpr"(nf,x,pr)"
given an element "x" of the number field "nf" and a prime ideal pr in "modpr" format
compute a canonical representative for the class of "x" modulo pr.
The library syntax is nfreducemodpr2"(nf,x,pr)".
nfeltval"(nf,x,pr)"
given an element "x" in nf and a prime ideal pr in the format output by "idealprimedec",
computes their the valuation at pr of the element "x". The same result could be obtained
using "idealval(nf,x,pr)" (since "x" would then be converted to a principal ideal), but it
would be less efficient.
The library syntax is element_val"(nf,x,pr)", and the result is a "long".
nffactor"(nf,x)"
factorization of the univariate polynomial "x" over the number field "nf" given by
"nfinit". "x" has coefficients in "nf" (i.e. either scalar, polmod, polynomial or column
vector). The main variable of "nf" must be of lower priority than that of "x" (in other
words, the variable number of "nf" must be greater than that of "x"). However if the
polynomial defining the number field occurs explicitly in the coefficients of "x" (as
modulus of a "t_POLMOD"), its main variable must be the same as the main variable of "x".
For example,
? nf = nfinit(y^2 + 1);
? nffactor(nf, x^2 + y); \\ OK
? nffactor(nf, x^2 + Mod(y, y^2+1)); \\ OK
? nffactor(nf, x^2 + Mod(z, z^2+1)); \\ WRONG
The library syntax is nffactor"(nf,x)".
nffactormod"(nf,x,pr)"
factorization of the univariate polynomial "x" modulo the prime ideal pr in the number
field "nf". "x" can have coefficients in the number field (scalar, polmod, polynomial,
column vector) or modulo the prime ideal (integermod modulo the rational prime under pr,
polmod or polynomial with integermod coefficients, column vector of integermod). The prime
ideal pr must be in the format output by "idealprimedec". The main variable of "nf" must
be of lower priority than that of "x" (in other words the variable number of "nf" must be
greater than that of "x"). However if the coefficients of the number field occur
explicitly (as polmods) as coefficients of "x", the variable of these polmods must be the
same as the main variable of "t" (see "nffactor").
The library syntax is nffactormod"(nf,x,pr)".
nfgaloisapply"(nf,aut,x)"
"nf" being a number field as output by "nfinit", and aut being a Galois automorphism of
"nf" expressed either as a polynomial or a polmod (such automorphisms being found using
for example one of the variants of "nfgaloisconj"), computes the action of the
automorphism aut on the object "x" in the number field. "x" can be an element (scalar,
polmod, polynomial or column vector) of the number field, an ideal (either given by
"Z_K"-generators or by a Z-basis), a prime ideal (given as a 5-element row vector) or an
idele (given as a 2-element row vector). Because of possible confusion with elements and
ideals, other vector or matrix arguments are forbidden.
The library syntax is galoisapply"(nf,aut,x)".
nfgaloisconj"(nf,{flag = 0},{d})"
"nf" being a number field as output by "nfinit", computes the conjugates of a root "r" of
the non-constant polynomial "x = nf[1]" expressed as polynomials in "r". This can be used
even if the number field "nf" is not Galois since some conjugates may lie in the field. As
a note to old-timers of PARI, starting with version 2.0.17 this function works much better
than in earlier versions.
"nf" can simply be a polynomial if "flag ! = 1".
If no flags or "flag = 0", if "nf" is a number field use a combination of flag 4 and 1 and
the result is always complete, else use a combination of flag 4 and 2 and the result is
subject to the restriction of "flag = 2", but a warning is issued when it is not proven
complete.
If "flag = 1", use "nfroots" (require a number field).
If "flag = 2", use complex approximations to the roots and an integral LLL. The result is
not guaranteed to be complete: some conjugates may be missing (no warning issued),
especially so if the corresponding polynomial has a huge index. In that case, increasing
the default precision may help.
If "flag = 4", use Allombert's algorithm and permutation testing. If the field is Galois
with ``weakly'' super solvable Galois group, return the complete list of automorphisms,
else only the identity element. If present, "d" is assumed to be a multiple of the least
common denominator of the conjugates expressed as polynomial in a root of pol.
A group G is ``weakly'' super solvable if it contains a super solvable normal subgroup "H"
such that "G = H" , or "G/H ~ A_4" , or "G/H ~ S_4". Abelian and nilpotent groups are
``weakly'' super solvable. In practice, almost all groups of small order are ``weakly''
super solvable, the exceptions having order 36(1 exception), 48(2), 56(1), 60(1), 72(5),
75(1), 80(1), 96(10) and " >= 108".
Hence "flag = 4" permits to quickly check whether a polynomial of order strictly less than
36 is Galois or not. This method is much faster than "nfroots" and can be applied to
polynomials of degree larger than 50.
The library syntax is galoisconj0"(nf,flag,d,prec)". Also available are "galoisconj(nf)"
for "flag = 0", "galoisconj2(nf,n,prec)" for "flag = 2" where "n" is a bound on the number
of conjugates, and "galoisconj4(nf,d)" corresponding to "flag = 4".
nfhilbert"(nf,a,b,{pr})"
if pr is omitted, compute the global Hilbert symbol "(a,b)" in "nf", that is 1 if "x^2 - a
y^2 - b z^2" has a non trivial solution "(x,y,z)" in "nf", and "-1" otherwise. Otherwise
compute the local symbol modulo the prime ideal pr (as output by "idealprimedec").
The library syntax is nfhilbert"(nf,a,b,pr)", where an omitted pr is coded as "NULL".
nfhnf"(nf,x)"
given a pseudo-matrix "(A,I)", finds a pseudo-basis in Hermite normal form of the module
it generates.
The library syntax is nfhermite"(nf,x)".
nfhnfmod"(nf,x,detx)"
given a pseudo-matrix "(A,I)" and an ideal detx which is contained in (read integral
multiple of) the determinant of "(A,I)", finds a pseudo-basis in Hermite normal form of
the module generated by "(A,I)". This avoids coefficient explosion. detx can be computed
using the function "nfdetint".
The library syntax is nfhermitemod"(nf,x,detx)".
nfinit"(pol,{flag = 0})"
pol being a non-constant, preferably monic, irreducible polynomial in "Z[X]", initializes
a number field structure ("nf") associated to the field "K" defined by pol. As such, it's
a technical object passed as the first argument to most "nf"xxx functions, but it contains
some information which may be directly useful. Access to this information via member
functions is prefered since the specific data organization specified below may change in
the future. Currently, "nf" is a row vector with 9 components:
"nf[1]" contains the polynomial pol ("nf.pol").
"nf[2]" contains "[r1,r2]" ("nf.sign"), the number of real and complex places of "K".
"nf[3]" contains the discriminant d(K) ("nf.disc") of "K".
"nf[4]" contains the index of "nf[1]", i.e. "[Z_K : Z[theta]]", where "theta" is any root
of "nf[1]".
"nf[5]" is a vector containing 7 matrices "M", "MC", "T2", "T", "MD", "TI", "MDI" useful
for certain computations in the number field "K".
"*" "M" is the "(r1+r2) x n" matrix whose columns represent the numerical values of the
conjugates of the elements of the integral basis.
"*" "MC" is essentially the conjugate of the transpose of "M", except that the last "r2"
columns are also multiplied by 2.
"*" "T2" is an "n x n" matrix equal to the real part of the product "MC.M" (which is a
real positive definite symmetric matrix), the so-called "T_2"-matrix ("nf.t2").
"*" "T" is the "n x n" matrix whose coefficients are "Tr(omega_iomega_j)" where the
"omega_i" are the elements of the integral basis. Note that "T = \overline{MC}.M" and in
particular that "T = T_2" if the field is totally real (in practice "T_2" will have real
approximate entries and "T" will have integer entries). Note also that " det (T)" is equal
to the discriminant of the field "K".
"*" The columns of "MD" ("nf.diff") express a Z-basis of the different of "K" on the
integral basis.
"*" "TI" is equal to "d(K)T^{-1}", which has integral coefficients. Note that,
understood as as ideal, the matrix "T^{-1}" generates the codifferent ideal.
"*" Finally, "MDI" is a two-element representation (for faster ideal product) of d(K)
times the codifferent ideal ("nf.disc*nf.codiff", which is an integral ideal). "MDI" is
only used in "idealinv".
"nf[6]" is the vector containing the "r1+r2" roots ("nf.roots") of "nf[1]" corresponding
to the "r1+r2" embeddings of the number field into C (the first "r1" components are real,
the next "r2" have positive imaginary part).
"nf[7]" is an integral basis in Hermite normal form for "Z_K" ("nf.zk") expressed on the
powers of "theta".
"nf[8]" is the "n x n" integral matrix expressing the power basis in terms of the integral
basis, and finally
"nf[9]" is the "n x n^2" matrix giving the multiplication table of the integral basis.
If a non monic polynomial is input, "nfinit" will transform it into a monic one, then
reduce it (see "flag = 3"). It is allowed, though not very useful given the existence of
nfnewprec, to input a "nf" or a "bnf" instead of a polynomial.
The special input format "[x,B]" is also accepted where "x" is a polynomial as above and
"B" is the integer basis, as computed by "nfbasis". This can be useful since "nfinit" uses
the round 4 algorithm by default, which can be very slow in pathological cases where round
2 ("nfbasis(x,2)") would succeed very quickly.
If "flag = 2": pol is changed into another polynomial "P" defining the same number field,
which is as simple as can easily be found using the "polred" algorithm, and all the
subsequent computations are done using this new polynomial. In particular, the first
component of the result is the modified polynomial.
If "flag = 3", does a "polred" as in case 2, but outputs "[nf,Mod(a,P)]", where "nf" is as
before and "Mod(a,P) = Mod(x,pol)" gives the change of variables. This is implicit when
pol is not monic: first a linear change of variables is performed, to get a monic
polynomial, then a "polred" reduction.
If "flag = 4", as 2 but uses a partial "polred".
If "flag = 5", as 3 using a partial "polred".
The library syntax is nfinit0"(x,flag,prec)".
nfisideal"(nf,x)"
returns 1 if "x" is an ideal in the number field "nf", 0 otherwise.
The library syntax is isideal"(x)".
nfisincl"(x,y)"
tests whether the number field "K" defined by the polynomial "x" is conjugate to a
subfield of the field "L" defined by "y" (where "x" and "y" must be in "Q[X]"). If they
are not, the output is the number 0. If they are, the output is a vector of polynomials,
each polynomial "a" representing an embedding of "K" into "L", i.e. being such that "y | x
o a".
If "y" is a number field (nf), a much faster algorithm is used (factoring "x" over "y"
using "nffactor"). Before version 2.0.14, this wasn't guaranteed to return all the
embeddings, hence was triggered by a special flag. This is no more the case.
The library syntax is nfisincl"(x,y,flag)".
nfisisom"(x,y)"
as "nfisincl", but tests for isomorphism. If either "x" or "y" is a number field, a much
faster algorithm will be used.
The library syntax is nfisisom"(x,y,flag)".
nfnewprec"(nf)"
transforms the number field "nf" into the corresponding data using current (usually
larger) precision. This function works as expected if "nf" is in fact a "bnf" (update
"bnf" to current precision) but may be quite slow (many generators of principal ideals
have to be computed).
The library syntax is nfnewprec"(nf,prec)".
nfkermodpr"(nf,a,pr)"
kernel of the matrix "a" in "Z_K/pr", where pr is in modpr format (see "nfmodprinit").
The library syntax is nfkermodpr"(nf,a,pr)".
nfmodprinit"(nf,pr)"
transforms the prime ideal pr into "modpr" format necessary for all operations modulo pr
in the number field nf. Returns a two-component vector "[P,a]", where "P" is the Hermite
normal form of pr, and "a" is an integral element congruent to 1 modulo pr, and congruent
to 0 modulo "p / pr^e". Here "p = Z cap pr" and "e" is the absolute ramification index.
The library syntax is nfmodprinit"(nf,pr)".
nfsubfields"(nf,{d = 0})"
finds all subfields of degree "d" of the number field "nf" (all subfields if "d" is null
or omitted). The result is a vector of subfields, each being given by "[g,h]", where "g"
is an absolute equation and "h" expresses one of the roots of "g" in terms of the root "x"
of the polynomial defining "nf". This is a crude implementation by M. Olivier of an
algorithm due to J. Klueners.
The library syntax is subfields"(nf,d)".
nfroots"(nf,x)"
roots of the polynomial "x" in the number field "nf" given by "nfinit" without
multiplicity. "x" has coefficients in the number field (scalar, polmod, polynomial, column
vector). The main variable of "nf" must be of lower priority than that of "x" (in other
words the variable number of "nf" must be greater than that of "x"). However if the
coefficients of the number field occur explicitly (as polmods) as coefficients of "x", the
variable of these polmods must be the same as the main variable of "t" (see "nffactor").
The library syntax is nfroots"(nf,x)".
nfrootsof1"(nf)"
computes the number of roots of unity "w" and a primitive "w"-th root of unity (expressed
on the integral basis) belonging to the number field "nf". The result is a two-component
vector "[w,z]" where "z" is a column vector expressing a primitive "w"-th root of unity on
the integral basis "nf.zk".
The library syntax is rootsof1"(nf)".
nfsnf"(nf,x)"
given a torsion module "x" as a 3-component row vector "[A,I,J]" where "A" is a square
invertible "n x n" matrix, "I" and "J" are two ideal lists, outputs an ideal list
"d_1,...,d_n" which is the Smith normal form of "x". In other words, "x" is isomorphic to
"Z_K/d_1 oplus ... oplus Z_K/d_n" and "d_i" divides "d_{i-1}" for "i >= 2". The link
between "x" and "[A,I,J]" is as follows: if "e_i" is the canonical basis of "K^n", "I =
[b_1,...,b_n]" and "J = [a_1,...,a_n]", then "x" is isomorphic to
" (b_1e_1 oplus ... oplus b_ne_n) / (a_1A_1 oplus ... oplus a_nA_n)
, "
where the "A_j" are the columns of the matrix "A". Note that every finitely generated
torsion module can be given in this way, and even with "b_i = Z_K" for all "i".
The library syntax is nfsmith"(nf,x)".
nfsolvemodpr"(nf,a,b,pr)"
solution of "a.x = b" in "Z_K/pr", where "a" is a matrix and "b" a column vector, and
where pr is in modpr format (see "nfmodprinit").
The library syntax is nfsolvemodpr"(nf,a,b,pr)".
polcompositum"(x,y,{flag = 0})"
"x" and "y" being polynomials in "Z[X]" in the same variable, outputs a vector giving the
list of all possible composita of the number fields defined by "x" and "y", if "x" and "y"
are irreducible, or of the corresponding etale algebras, if they are only squarefree.
Returns an error if one of the polynomials is not squarefree. When one of the polynomials
is irreducible (say "x"), it is often much faster to use "nffactor(nfinit(x), y)" then
"rnfequation".
If "flag = 1", outputs a vector of 4-component vectors "[z,a,b,k]", where "z" ranges
through the list of all possible compositums as above, and "a" (resp. "b") expresses the
root of "x" (resp. "y") as a polmod in a root of "z", and "k" is a small integer k such
that "a+kb" is the chosen root of "z".
The compositum will quite often be defined by a complicated polynomial, which it is
advisable to reduce before further work. Here is a simple example involving the field
"Q(zeta_5, 5^{1/5})":
? z = polcompositum(x^5 - 5, polcyclo(5), 1)[1];
? pol = z[1] \\ pol defines the compositum
%2 = x^20 + 5*x^19 + 15*x^18 + 35*x^17 + 70*x^16 + 141*x^15 + 260*x^14 \
+ 355*x^13 + 95*x^12 - 1460*x^11 - 3279*x^10 - 3660*x^9 - 2005*x^8 \
+ 705*x^7 + 9210*x^6 + 13506*x^5 + 7145*x^4 - 2740*x^3 + 1040*x^2 \
- 320*x + 256
? a = z[2]; a^5 - 5 \\ a is a fifth root of 5
%3 = 0
? z = polredabs(pol, 1); \\ look for a simpler polynomial
? pol = z[1]
%5 = x^20 + 25*x^10 + 5
? a = subst(a.pol, x, z[2]) \\ a in the new coordinates
%6 = Mod(-5/22*x^19 + 1/22*x^14 - 123/22*x^9 + 9/11*x^4, x^20 + 25*x^10 + 5)
The library syntax is polcompositum0"(x,y,flag)".
polgalois"(x)"
Galois group of the non-constant polynomial "x belongs to Q[X]". In the present version
2.2.0, "x" must be irreducible and the degree of "x" must be less than or equal to 7. On
certain versions for which the data file of Galois resolvents has been installed
(available in the Unix distribution as a separate package), degrees 8, 9, 10 and 11 are
also implemented.
The output is a 3-component vector "[n,s,k]" with the following meaning: "n" is the
cardinality of the group, "s" is its signature ("s = 1" if the group is a subgroup of the
alternating group "A_n", "s = -1" otherwise), and "k" is the number of the group
corresponding to a given pair "(n,s)" ("k = 1" except in 2 cases). Specifically, the
groups are coded as follows, using standard notations (see GTM 138, quoted at the
beginning of this section; see also ``The transitive groups of degree up to eleven'', by
G. Butler and J. McKay in Communications in Algebra, vol. 11, 1983, pp. 863--911):
In degree 1: "S_1 = [1,-1,1]".
In degree 2: "S_2 = [2,-1,1]".
In degree 3: "A_3 = C_3 = [3,1,1]", "S_3 = [6,-1,1]".
In degree 4: "C_4 = [4,-1,1]", "V_4 = [4,1,1]", "D_4 = [8,-1,1]", "A_4 = [12,1,1]", "S_4 =
[24,-1,1]".
In degree 5: "C_5 = [5,1,1]", "D_5 = [10,1,1]", "M_{20} = [20,-1,1]", "A_5 = [60,1,1]",
"S_5 = [120,-1,1]".
In degree 6: "C_6 = [6,-1,1]", "S_3 = [6,-1,2]", "D_6 = [12,-1,1]", "A_4 = [12,1,1]",
"G_{18} = [18,-1,1]", "S_4^ -= [24,-1,1]", "A_4 x C_2 = [24,-1,2]", "S_4^ += [24,1,1]",
"G_{36}^ -= [36,-1,1]", "G_{36}^ += [36,1,1]", "S_4 x C_2 = [48,-1,1]", "A_5 = PSL_2(5) =
[60,1,1]", "G_{72} = [72,-1,1]", "S_5 = PGL_2(5) = [120,-1,1]", "A_6 = [360,1,1]", "S_6 =
[720,-1,1]".
In degree 7: "C_7 = [7,1,1]", "D_7 = [14,-1,1]", "M_{21} = [21,1,1]", "M_{42} =
[42,-1,1]", "PSL_2(7) = PSL_3(2) = [168,1,1]", "A_7 = [2520,1,1]", "S_7 = [5040,-1,1]".
The method used is that of resolvent polynomials and is sensitive to the current
precision. The precision is updated internally but, in very rare cases, a wrong result may
be returned if the initial precision was not sufficient.
The library syntax is galois"(x,prec)".
polred"(x,{flag = 0},{p})"
finds polynomials with reasonably small coefficients defining subfields of the number
field defined by "x". One of the polynomials always defines Q (hence is equal to "x-1"),
and another always defines the same number field as "x" if "x" is irreducible. All "x"
accepted by "nfinit" are also allowed here (e.g. non-monic polynomials, "nf", "bnf",
"[x,Z_K_basis]").
The following binary digits of "flag" are significant:
1: does a partial reduction only. This means that only a suborder of the maximal order may
be used.
2: gives also elements. The result is a two-column matrix, the first column giving the
elements defining these subfields, the second giving the corresponding minimal
polynomials.
If "p" is given, it is assumed that it is the two-column matrix of the factorization of
the discriminant of the polynomial "x".
The library syntax is polred0"(x,flag,p,prec)", where an omitted "p" is coded by "gzero".
Also available are "polred(x,prec)" and "factoredpolred(x,p,prec)", both corresponding to
"flag = 0".
polredabs"(x,{flag = 0})"
finds one of the polynomial defining the same number field as the one defined by "x", and
such that the sum of the squares of the modulus of the roots (i.e. the "T_2"-norm) is
minimal. All "x" accepted by "nfinit" are also allowed here (e.g. non-monic polynomials,
"nf", "bnf", "[x,Z_K_basis]").
The binary digits of "flag" mean
1: outputs a two-component row vector "[P,a]", where "P" is the default output and "a" is
an element expressed on a root of the polynomial "P", whose minimal polynomial is equal to
"x".
4: gives all polynomials of minimal "T_2" norm (of the two polynomials P(x) and "P(-x)",
only one is given).
The library syntax is polredabs0"(x,flag,prec)".
polredord"(x)"
finds polynomials with reasonably small coefficients and of the same degree as that of "x"
defining suborders of the order defined by "x". One of the polynomials always defines Q
(hence is equal to "(x-1)^n", where "n" is the degree), and another always defines the
same order as "x" if "x" is irreducible.
The library syntax is ordred"(x)".
poltschirnhaus"(x)"
applies a random Tschirnhausen transformation to the polynomial "x", which is assumed to
be non-constant and separable, so as to obtain a new equation for the etale algebra
defined by "x". This is for instance useful when computing resolvents, hence is used by
the "polgalois" function.
The library syntax is tschirnhaus"(x)".
rnfalgtobasis"(rnf,x)"
"rnf" being a relative number field extension "L/K" as output by "rnfinit" and "x" being
an element of "L" expressed as a polynomial or polmod with polmod coefficients, expresses
"x" on the relative integral basis.
The library syntax is rnfalgtobasis"(rnf,x)".
rnfbasis"(bnf,x)"
given a big number field "bnf" as output by "bnfinit", and either a polynomial "x" with
coefficients in "bnf" defining a relative extension "L" of "bnf", or a pseudo-basis "x" of
such an extension, gives either a true "bnf"-basis of "L" if it exists, or an
"n+1"-element generating set of "L" if not, where "n" is the rank of "L" over "bnf".
The library syntax is rnfbasis"(bnf,x)".
rnfbasistoalg"(rnf,x)"
"rnf" being a relative number field extension "L/K" as output by "rnfinit" and "x" being
an element of "L" expressed on the relative integral basis, computes the representation of
"x" as a polmod with polmods coefficients.
The library syntax is rnfbasistoalg"(rnf,x)".
rnfcharpoly"(nf,T,a,{v = x})"
characteristic polynomial of "a" over "nf", where "a" belongs to the algebra defined by
"T" over "nf", i.e. "nf[X]/(T)". Returns a polynomial in variable "v" ("x" by default).
The library syntax is rnfcharpoly"(nf,T,a,v)", where "v" is a variable number.
rnfconductor"(bnf,pol)"
"bnf" being a big number field as output by "bnfinit", and pol a relative polynomial
defining an Abelian extension, computes the class field theory conductor of this Abelian
extension. The result is a 3-component vector "[conductor,rayclgp,subgroup]", where
conductor is the conductor of the extension given as a 2-component row vector "[f_0,f_ oo
]", rayclgp is the full ray class group corresponding to the conductor given as a
3-component vector [h,cyc,gen] as usual for a group, and subgroup is a matrix in HNF
defining the subgroup of the ray class group on the given generators gen.
The library syntax is rnfconductor"(rnf,pol,prec)".
rnfdedekind"(nf,pol,pr)"
given a number field "nf" as output by "nfinit" and a polynomial pol with coefficients in
"nf" defining a relative extension "L" of "nf", evaluates the relative Dedekind criterion
over the order defined by a root of pol for the prime ideal pr and outputs a 3-component
vector as the result. The first component is a flag equal to 1 if the enlarged order could
be proven to be pr-maximal and to 0 otherwise (it may be maximal in the latter case if pr
is ramified in "L"), the second component is a pseudo-basis of the enlarged order and the
third component is the valuation at pr of the order discriminant.
The library syntax is rnfdedekind"(nf,pol,pr)".
rnfdet"(nf,M)"
given a pseudomatrix "M" over the maximal order of "nf", computes its pseudodeterminant.
The library syntax is rnfdet"(nf,M)".
rnfdisc"(nf,pol)"
given a number field "nf" as output by "nfinit" and a polynomial pol with coefficients in
"nf" defining a relative extension "L" of "nf", computes the relative discriminant of "L".
This is a two-element row vector "[D,d]", where "D" is the relative ideal discriminant and
"d" is the relative discriminant considered as an element of "nf^*/{nf^*}^2". The main
variable of "nf" must be of lower priority than that of pol.
Note: As usual, "nf" can be a "bnf" as output by "nfinit".
The library syntax is rnfdiscf"(bnf,pol)".
rnfeltabstorel"(rnf,x)"
"rnf" being a relative number field extension "L/K" as output by "rnfinit" and "x" being
an element of "L" expressed as a polynomial modulo the absolute equation "rnf[11][1]",
computes "x" as an element of the relative extension "L/K" as a polmod with polmod
coefficients.
The library syntax is rnfelementabstorel"(rnf,x)".
rnfeltdown"(rnf,x)"
"rnf" being a relative number field extension "L/K" as output by "rnfinit" and "x" being
an element of "L" expressed as a polynomial or polmod with polmod coefficients, computes
"x" as an element of "K" as a polmod, assuming "x" is in "K" (otherwise an error will
occur). If "x" is given on the relative integral basis, apply "rnfbasistoalg" first,
otherwise PARI will believe you are dealing with a vector.
The library syntax is rnfelementdown"(rnf,x)".
rnfeltreltoabs"(rnf,x)"
"rnf" being a relative number field extension "L/K" as output by "rnfinit" and "x" being
an element of "L" expressed as a polynomial or polmod with polmod coefficients, computes
"x" as an element of the absolute extension "L/Q" as a polynomial modulo the absolute
equation "rnf[11][1]". If "x" is given on the relative integral basis, apply
"rnfbasistoalg" first, otherwise PARI will believe you are dealing with a vector.
The library syntax is rnfelementreltoabs"(rnf,x)".
rnfeltup"(rnf,x)"
"rnf" being a relative number field extension "L/K" as output by "rnfinit" and "x" being
an element of "K" expressed as a polynomial or polmod, computes "x" as an element of the
absolute extension "L/Q" as a polynomial modulo the absolute equation "rnf[11][1]". Note
that it is unnecessary to compute "x" as an element of the relative extension "L/K" (its
expression would be identical to itself). If "x" is given on the integral basis of "K",
apply "nfbasistoalg" first, otherwise PARI will believe you are dealing with a vector.
The library syntax is rnfelementup"(rnf,x)".
rnfequation"(nf,pol,{flag = 0})"
given a number field "nf" as output by "nfinit" (or simply a polynomial) and a polynomial
pol with coefficients in "nf" defining a relative extension "L" of "nf", computes the
absolute equation of "L" over Q.
If "flag" is non-zero, outputs a 3-component row vector "[z,a,k]", where "z" is the
absolute equation of "L" over Q, as in the default behaviour, "a" expresses as an element
of "L" a root "alpha" of the polynomial defining the base field "nf", and "k" is a small
integer such that "theta = beta+kalpha" where "theta" is a root of "z" and "beta" a root
of "pol".
The main variable of "nf" must be of lower priority than that of pol. Note that for
efficiency, this does not check whether the relative equation is irreducible over "nf",
but only if it is squarefree. If it is reducible but squarefree, the result will be the
absolute equation of the etale algebra defined by pol. If pol is not squarefree, an error
message will be issued.
The library syntax is rnfequation0"(nf,pol,flag)".
rnfhnfbasis"(bnf,x)"
given a big number field "bnf" as output by "bnfinit", and either a polynomial "x" with
coefficients in "bnf" defining a relative extension "L" of "bnf", or a pseudo-basis "x" of
such an extension, gives either a true "bnf"-basis of "L" in upper triangular Hermite
normal form, if it exists, zero otherwise.
The library syntax is rnfhermitebasis"(nf,x)".
rnfidealabstorel"(rnf,x)"
"rnf" being a relative number field extension "L/K" as output by "rnfinit" and "x" being
an ideal of the absolute extension "L/Q" given in HNF (if it is not, apply "idealhnf"
first), computes the relative pseudomatrix in HNF giving the ideal "x" considered as an
ideal of the relative extension "L/K".
The library syntax is rnfidealabstorel"(rnf,x)".
rnfidealdown"(rnf,x)"
"rnf" being a relative number field extension "L/K" as output by "rnfinit" and "x" being
an ideal of the absolute extension "L/Q" given in HNF (if it is not, apply "idealhnf"
first), gives the ideal of "K" below "x", i.e. the intersection of "x" with "K". Note
that, if "x" is given as a relative ideal (i.e. a pseudomatrix in HNF), then it is not
necessary to use this function since the result is simply the first ideal of the ideal
list of the pseudomatrix.
The library syntax is rnfidealdown"(rnf,x)".
rnfidealhnf"(rnf,x)"
"rnf" being a relative number field extension "L/K" as output by "rnfinit" and "x" being a
relative ideal (which can be, as in the absolute case, of many different types, including
of course elements), computes as a 2-component row vector the relative Hermite normal form
of "x", the first component being the HNF matrix (with entries on the integral basis), and
the second component the ideals.
The library syntax is rnfidealhermite"(rnf,x)".
rnfidealmul"(rnf,x,y)"
"rnf" being a relative number field extension "L/K" as output by "rnfinit" and "x" and "y"
being ideals of the relative extension "L/K" given by pseudo-matrices, outputs the ideal
product, again as a relative ideal.
The library syntax is rnfidealmul"(rnf,x,y)".
rnfidealnormabs"(rnf,x)"
"rnf" being a relative number field extension "L/K" as output by "rnfinit" and "x" being a
relative ideal (which can be, as in the absolute case, of many different types, including
of course elements), computes the norm of the ideal "x" considered as an ideal of the
absolute extension "L/Q". This is identical to "idealnorm(rnfidealnormrel(rnf,x))", only
faster.
The library syntax is rnfidealnormabs"(rnf,x)".
rnfidealnormrel"(rnf,x)"
"rnf" being a relative number field extension "L/K" as output by "rnfinit" and "x" being a
relative ideal (which can be, as in the absolute case, of many different types, including
of course elements), computes the relative norm of "x" as a ideal of "K" in HNF.
The library syntax is rnfidealnormrel"(rnf,x)".
rnfidealreltoabs"(rnf,x)"
"rnf" being a relative number field extension "L/K" as output by "rnfinit" and "x" being a
relative ideal (which can be, as in the absolute case, of many different types, including
of course elements), computes the HNF matrix of the ideal "x" considered as an ideal of
the absolute extension "L/Q".
The library syntax is rnfidealreltoabs"(rnf,x)".
rnfidealtwoelt"(rnf,x)"
"rnf" being a relative number field extension "L/K" as output by "rnfinit" and "x" being
an ideal of the relative extension "L/K" given by a pseudo-matrix, gives a vector of two
generators of "x" over "Z_L" expressed as polmods with polmod coefficients.
The library syntax is rnfidealtwoelement"(rnf,x)".
rnfidealup"(rnf,x)"
"rnf" being a relative number field extension "L/K" as output by "rnfinit" and "x" being
an ideal of "K", gives the ideal "xZ_L" as an absolute ideal of "L/Q" (the relative ideal
representation is trivial: the matrix is the identity matrix, and the ideal list starts
with "x", all the other ideals being "Z_K").
The library syntax is rnfidealup"(rnf,x)".
rnfinit"(nf,pol)"
"nf" being a number field in "nfinit" format considered as base field, and pol a
polynomial defining a relative extension over "nf", this computes all the necessary data
to work in the relative extension. The main variable of pol must be of higher priority
(i.e. lower number) than that of "nf", and the coefficients of pol must be in "nf".
The result is an 11-component row vector as follows (most of the components are
technical), the numbering being very close to that of "nfinit". In the following
description, we let "K" be the base field defined by "nf", "m" the degree of the base
field, "n" the relative degree, "L" the large field (of relative degree "n" or absolute
degree "nm"), "r_1" and "r_2" the number of real and complex places of "K".
"rnf[1]" contains the relative polynomial pol.
"rnf[2]" is a row vector with "r_1+r_2" entries, entry "j" being a 2-component row vector
"[r_{j,1},r_{j,2}]" where "r_{j,1}" and "r_{j,2}" are the number of real and complex
places of "L" above the "j"-th place of "K" so that "r_{j,1} = 0" and "r_{j,2} = n" if "j"
is a complex place, while if "j" is a real place we have "r_{j,1}+2r_{j,2} = n".
"rnf[3]" is a two-component row vector "[d(L/K),s]" where "d(L/K)" is the relative ideal
discriminant of "L/K" and "s" is the discriminant of "L/K" viewed as an element of
"K^*/(K^*)^2", in other words it is the output of "rnfdisc".
"rnf[4]" is the ideal index "f", i.e. such that "d(pol)Z_K = f^2d(L/K)".
"rnf[5]" is a vector vm with 7 entries useful for certain computations in the relative
extension "L/K". "vm[1]" is a vector of "r_1+r_2" matrices, the "j"-th matrix being an
"(r_{1,j}+r_{2,j}) x n" matrix "M_j" representing the numerical values of the conjugates
of the "j"-th embedding of the elements of the integral basis, where "r_{i,j}" is as in
"rnf[2]". "vm[2]" is a vector of "r_1+r_2" matrices, the "j"-th matrix "MC_j" being
essentially the conjugate of the matrix "M_j" except that the last "r_{2,j}" columns are
also multiplied by 2. "vm[3]" is a vector of "r_1+r_2" matrices "T2_j", where "T2_j" is
an "n x n" matrix equal to the real part of the product "MC_j.M_j" (which is a real
positive definite matrix). "vm[4]" is the "n x n" matrix "T" whose entries are the
relative traces of "omega_iomega_j" expressed as polmods in "nf", where the "omega_i" are
the elements of the relative integral basis. Note that the "j"-th embedding of "T" is
equal to "\overline{MC_j}.M_j", and in particular will be equal to "T2_j" if "r_{2,j} =
0". Note also that the relative ideal discriminant of "L/K" is equal to " det (T)" times
the square of the product of the ideals in the relative pseudo-basis (in "rnf[7][2]"). The
last 3 entries "vm[5]", "vm[6]" and "vm[7]" are linked to the different as in "nfinit",
but have not yet been implemented.
"rnf[6]" is a row vector with "r_1+r_2" entries, the "j"-th entry being the row vector
with "r_{1,j}+r_{2,j}" entries of the roots of the "j"-th embedding of the relative
polynomial pol.
"rnf[7]" is a two-component row vector, where the first component is the relative integral
pseudo basis expressed as polynomials (in the variable of "pol") with polmod coefficients
in "nf", and the second component is the ideal list of the pseudobasis in HNF.
"rnf[8]" is the inverse matrix of the integral basis matrix, with coefficients polmods in
"nf".
"rnf[9]" may be the multiplication table of the integral basis, but is not implemented at
present.
"rnf[10]" is "nf".
"rnf[11]" is a vector vabs with 5 entries describing the absolute extension "L/Q".
"vabs[1]" is an absolute equation. "vabs[2]" expresses the generator "alpha" of the
number field "nf" as a polynomial modulo the absolute equation "vabs[1]". "vabs[3]" is a
small integer "k" such that, if "beta" is an abstract root of pol and "alpha" the
generator of "nf", the generator whose root is vabs will be "beta + k alpha". Note that
one must be very careful if "k ! = 0" when dealing simultaneously with absolute and
relative quantities since the generator chosen for the absolute extension is not the same
as for the relative one. If this happens, one can of course go on working, but we strongly
advise to change the relative polynomial so that its root will be "beta + k alpha".
Typically, the GP instruction would be
"pol = subst(pol, x, x - k*Mod(y,nf.pol))"
Finally, "vabs[4]" is the absolute integral basis of "L" expressed in HNF (hence as would
be output by "nfinit(vabs[1])"), and "vabs[5]" the inverse matrix of the integral basis,
allowing to go from polmod to integral basis representation.
The library syntax is rnfinitalg"(nf,pol,prec)".
rnfisfree"(bnf,x)"
given a big number field "bnf" as output by "bnfinit", and either a polynomial "x" with
coefficients in "bnf" defining a relative extension "L" of "bnf", or a pseudo-basis "x" of
such an extension, returns true (1) if "L/bnf" is free, false (0) if not.
The library syntax is rnfisfree"(bnf,x)", and the result is a "long".
rnfisnorm"(bnf,ext,el,{flag = 1})"
similar to "bnfisnorm" but in the relative case. This tries to decide whether the element
el in bnf is the norm of some "y" in ext. "bnf" is as output by "bnfinit".
"ext" is a relative extension which has to be a row vector whose components are:
"ext[1]": a relative equation of the number field ext over bnf. As usual, the priority of
the variable of the polynomial defining the ground field bnf (say "y") must be lower than
the main variable of "ext[1]", say "x".
"ext[2]": the generator "y" of the base field as a polynomial in "x" (as given by
"rnfequation" with "flag = 1").
"ext[3]": is the "bnfinit" of the absolute extension "ext/Q".
This returns a vector "[a,b]", where "el = Norm(a)*b". It looks for a solution which is an
"S"-integer, with "S" a list of places (of bnf) containing the ramified primes, the
generators of the class group of ext, as well as those primes dividing el. If "ext/bnf" is
known to be Galois, set "flag = 0" (here el is a norm iff "b = 1"). If "flag" is non zero
add to "S" all the places above the primes which: divide "flag" if "flag < 0", or are less
than "flag" if "flag > 0". The answer is guaranteed (i.e. el is a norm iff "b = 1") under
GRH, if "S" contains all primes less than "12 log ^2|disc(Ext)|", where Ext is the normal
closure of "ext / bnf". Example:
bnf = bnfinit(y^3 + y^2 - 2*y - 1);
p = x^2 + Mod(y^2 + 2*y + 1, bnf.pol);
rnf = rnfequation(bnf,p,1);
ext = [p, rnf[2], bnfinit(rnf[1])];
rnfisnorm(bnf,ext,17, 1)
checks whether 17 is a norm in the Galois extension "Q(beta) / Q(alpha)", where "alpha^3 +
alpha^2 - 2alpha - 1 = 0" and "beta^2 + alpha^2 + 2*alpha + 1 = 0" (it is).
The library syntax is rnfisnorm"(bnf,ext,x,flag,prec)".
rnfkummer"(bnr,subgroup,{deg = 0})"
bnr being as output by "bnrinit", finds a relative equation for the class field
corresponding to the module in bnr and the given congruence subgroup. If deg is positive,
outputs the list of all relative equations of degree deg contained in the ray class field
defined by bnr.
(THIS PROGRAM IS STILL IN DEVELOPMENT STAGE)
The library syntax is rnfkummer"(bnr,subgroup,deg,prec)", where deg is a "long".
rnflllgram"(nf,pol,order)"
given a polynomial pol with coefficients in nf and an order order as output by
"rnfpseudobasis" or similar, gives "[[neworder],U]", where neworder is a reduced order and
"U" is the unimodular transformation matrix.
The library syntax is rnflllgram"(nf,pol,order,prec)".
rnfnormgroup"(bnr,pol)"
bnr being a big ray class field as output by "bnrinit" and pol a relative polynomial
defining an Abelian extension, computes the norm group (alias Artin or Takagi group)
corresponding to the Abelian extension of "bnf = bnr[1]" defined by pol, where the module
corresponding to bnr is assumed to be a multiple of the conductor (i.e. polrel defines a
subextension of bnr). The result is the HNF defining the norm group on the given
generators of "bnr[5][3]". Note that neither the fact that pol defines an Abelian
extension nor the fact that the module is a multiple of the conductor is checked. The
result is undefined if the assumption is not correct.
The library syntax is rnfnormgroup"(bnr,pol)".
rnfpolred"(nf,pol)"
relative version of "polred". Given a monic polynomial pol with coefficients in "nf",
finds a list of relative polynomials defining some subfields, hopefully simpler and
containing the original field. In the present version 2.2.0, this is slower than
"rnfpolredabs".
The library syntax is rnfpolred"(nf,pol,prec)".
rnfpolredabs"(nf,pol,{flag = 0})"
relative version of "polredabs". Given a monic polynomial pol with coefficients in "nf",
finds a simpler relative polynomial defining the same field. If "flag = 1", returns
"[P,a]" where "P" is the default output and "a" is an element expressed on a root of "P"
whose characteristic polynomial is pol, if "flag = 2", returns an absolute polynomial
(same as
"rnfequation(nf,rnfpolredabs(nf,pol))"
but faster).
Remark. In the present implementation, this is both faster and much more efficient than
"rnfpolred", the difference being more dramatic than in the absolute case. This is because
the implementation of "rnfpolred" is based on (a partial implementation of) an incomplete
reduction theory of lattices over number fields (i.e. the function "rnflllgram") which
deserves to be improved.
The library syntax is rnfpolredabs"(nf,pol,flag,prec)".
rnfpseudobasis"(nf,pol)"
given a number field "nf" as output by "nfinit" and a polynomial pol with coefficients in
"nf" defining a relative extension "L" of "nf", computes a pseudo-basis "(A,I)" and the
relative discriminant of "L". This is output as a four-element row vector "[A,I,D,d]",
where "D" is the relative ideal discriminant and "d" is the relative discriminant
considered as an element of "nf^*/{nf^*}^2".
Note: As usual, "nf" can be a "bnf" as output by "bnfinit".
The library syntax is rnfpseudobasis"(nf,pol)".
rnfsteinitz"(nf,x)"
given a number field "nf" as output by "nfinit" and either a polynomial "x" with
coefficients in "nf" defining a relative extension "L" of "nf", or a pseudo-basis "x" of
such an extension as output for example by "rnfpseudobasis", computes another pseudo-basis
"(A,I)" (not in HNF in general) such that all the ideals of "I" except perhaps the last
one are equal to the ring of integers of "nf", and outputs the four-component row vector
"[A,I,D,d]" as in "rnfpseudobasis". The name of this function comes from the fact that the
ideal class of the last ideal of "I" (which is well defined) is called the Steinitz class
of the module "Z_L".
Note: "nf" can be a "bnf" as output by "bnfinit".
The library syntax is rnfsteinitz"(nf,x)".
subgrouplist"(bnr,{bound},{flag = 0})"
bnr being as output by "bnrinit" or a list of cyclic components of a finite Abelian group
"G", outputs the list of subgroups of "G" (of index bounded by bound, if not omitted).
Subgroups are given as HNF left divisors of the SNF matrix corresponding to "G". If "flag
= 0" (default) and bnr is as output by "bnrinit", gives only the subgroups whose modulus
is the conductor.
The library syntax is subgrouplist0"(bnr,bound,flag,prec)", where bound, "flag" and "prec"
are long integers.
zetak"(znf,x,{flag = 0})"
znf being a number field initialized by "zetakinit" (not by "nfinit"), computes the value
of the Dedekind zeta function of the number field at the complex number "x". If "flag = 1"
computes Dedekind "Lambda" function instead (i.e. the product of the Dedekind zeta
function by its gamma and exponential factors).
The accuracy of the result depends in an essential way on the accuracy of both the
"zetakinit" program and the current accuracy, but even so the result may be off by up to 5
or 10 decimal digits.
The library syntax is glambdak"(znf,x,prec)" or "gzetak(znf,x,prec)".
zetakinit"(x)"
computes a number of initialization data concerning the number field defined by the
polynomial "x" so as to be able to compute the Dedekind zeta and lambda functions
(respectively zetak(x) and "zetak(x,1)"). This function calls in particular the "bnfinit"
program. The result is a 9-component vector "v" whose components are very technical and
cannot really be used by the user except through the "zetak" function. The only component
which can be used if it has not been computed already is "v[1][4]" which is the result of
the "bnfinit" call.
This function is very inefficient and should be rewritten. It needs to computes millions
of coefficients of the corresponding Dirichlet series if the precision is big. Unless the
discriminant is small it will not be able to handle more than 9 digits of relative
precision (e.g "zetakinit(x^8 - 2)" needs 440MB of memory at default precision).
The library syntax is initzeta"(x)".
Polynomials and power series
We group here all functions which are specific to polynomials or power series. Many other
functions which can be applied on these objects are described in the other sections. Also,
some of the functions described here can be applied to other types.
O"(a""^""b)"
"p"-adic (if "a" is an integer greater or equal to 2) or power series zero (in all other
cases), with precision given by "b".
The library syntax is ggrandocp"(a,b)", where "b" is a "long".
deriv"(x,{v})"
derivative of "x" with respect to the main variable if "v" is omitted, and with respect to
"v" otherwise. "x" can be any type except polmod. The derivative of a scalar type is zero,
and the derivative of a vector or matrix is done componentwise. One can use "x'" as a
shortcut if the derivative is with respect to the main variable of "x".
The library syntax is deriv"(x,v)", where "v" is a "long", and an omitted "v" is coded as
"-1".
eval"(x)"
replaces in "x" the formal variables by the values that have been assigned to them after
the creation of "x". This is mainly useful in GP, and not in library mode. Do not confuse
this with substitution (see "subst"). Applying this function to a character string yields
the output from the corresponding GP command, as if directly input from the keyboard (see
"Label se:strings").
The library syntax is geval"(x)". The more basic functions "poleval(q,x)", "qfeval(q,x)",
and "hqfeval(q,x)" evaluate "q" at "x", where "q" is respectively assumed to be a
polynomial, a quadratic form (a symmetric matrix), or an Hermitian form (an Hermitian
complex matrix).
factorpadic"(pol,p,r,{flag = 0})"
"p"-adic factorization of the polynomial pol to precision "r", the result being a two-
column matrix as in "factor". The factors are normalized so that their leading coefficient
is a power of "p". "r" must be strictly larger than the "p"-adic valuation of the
discriminant of pol for the result to make any sense. The method used is a modified
version of the round 4 algorithm of Zassenhaus.
If "flag = 1", use an algorithm due to Buchmann and Lenstra, which is usually less
efficient.
The library syntax is factorpadic4"(pol,p,r)", where "r" is a "long" integer.
intformal"(x,{v})"
formal integration of "x" with respect to the main variable if "v" is omitted, with
respect to the variable "v" otherwise. Since PARI does not know about ``abstract''
logarithms (they are immediately evaluated, if only to a power series), logarithmic terms
in the result will yield an error. "x" can be of any type. When "x" is a rational
function, it is assumed that the base ring is an integral domain of characteristic zero.
The library syntax is integ"(x,v)", where "v" is a "long" and an omitted "v" is coded as
"-1".
padicappr"(pol,a)"
vector of "p"-adic roots of the polynomial "pol" congruent to the "p"-adic number "a"
modulo "p" (or modulo 4 if "p = 2"), and with the same "p"-adic precision as "a". The
number "a" can be an ordinary "p"-adic number (type "t_PADIC", i.e. an element of "Q_p")
or can be an element of a finite extension of "Q_p", in which case it is of type
"t_POLMOD", where at least one of the coefficients of the polmod is a "p"-adic number. In
this case, the result is the vector of roots belonging to the same extension of "Q_p" as
"a".
The library syntax is apprgen9"(pol,a)", but if "a" is known to be simply a "p"-adic
number (type "t_PADIC"), the syntax "apprgen(pol,a)" can be used.
polcoeff"(x,s,{v})"
coefficient of degree "s" of the polynomial "x", with respect to the main variable if "v"
is omitted, with respect to "v" otherwise.
The library syntax is polcoeff0"(x,s,v)", where "v" is a "long" and an omitted "v" is
coded as "-1". Also available is truecoeff"(x,v)".
poldegree"(x,{v})"
degree of the polynomial "x" in the main variable if "v" is omitted, in the variable "v"
otherwise. This is to be understood as follows. When "x" is a polynomial or a rational
function, it gives the degree of "x", the degree of 0 being "-1" by convention. When "x"
is a non-zero scalar, it gives 0, and when "x" is a zero scalar, it gives "-1". Return an
error otherwise.
The library syntax is poldegree"(x,v)", where "v" and the result are "long"s (and an
omitted "v" is coded as "-1"). Also available is degree"(x)", which is equivalent to
"poldegree(x,-1)".
polcyclo"(n,{v = x})"
"n"-th cyclotomic polynomial, in variable "v" ("x" by default). The integer "n" must be
positive.
The library syntax is cyclo"(n,v)", where "n" and "v" are "long" integers ("v" is a
variable number, usually obtained through "varn").
poldisc"(pol,{v})"
discriminant of the polynomial pol in the main variable is "v" is omitted, in "v"
otherwise. The algorithm used is the subresultant algorithm.
The library syntax is poldisc0"(x,v)". Also available is discsr"(x)", equivalent to
"poldisc0(x,-1)".
poldiscreduced"(f)"
reduced discriminant vector of the (integral, monic) polynomial "f". This is the vector of
elementary divisors of "Z[alpha]/f'(alpha)Z[alpha]", where "alpha" is a root of the
polynomial "f". The components of the result are all positive, and their product is equal
to the absolute value of the discriminant of "f".
The library syntax is reduceddiscsmith"(x)".
polhensellift"(x, y, p, e)"
given a vector "y" of polynomials that are pairwise relatively prime modulo the prime "p",
and whose product is congruent to "x" modulo "p", lift the elements of "y" to polynomials
whose product is congruent to "x" modulo "p^e".
The library syntax is polhensellift"(x,y,p,e)" where "e" must be a "long".
polinterpolate"(xa,{ya},{v = x},{&e})"
given the data vectors "xa" and "ya" of the same length "n" ("xa" containing the
"x"-coordinates, and "ya" the corresponding "y"-coordinates), this function finds the
interpolating polynomial passing through these points and evaluates it at "v". If "ya" is
omitted, return the polynomial interpolating the "(i,xa[i])". If present, "e" will contain
an error estimate on the returned value.
The library syntax is polint"(xa,ya,v,&e)", where "e" will contain an error estimate on
the returned value.
polisirreducible"(pol)"
pol being a polynomial (univariate in the present version 2.2.0), returns 1 if pol is non-
constant and irreducible, 0 otherwise. Irreducibility is checked over the smallest base
field over which pol seems to be defined.
The library syntax is gisirreducible"(pol)".
pollead"(x,{v})"
leading coefficient of the polynomial or power series "x". This is computed with respect
to the main variable of "x" if "v" is omitted, with respect to the variable "v" otherwise.
The library syntax is pollead"(x,v)", where "v" is a "long" and an omitted "v" is coded as
"-1". Also available is leadingcoeff"(x)".
pollegendre"(n,{v = x})"
creates the "n^{th}" Legendre polynomial, in variable "v".
The library syntax is legendre"(n)", where "x" is a "long".
polrecip"(pol)"
reciprocal polynomial of pol, i.e. the coefficients are in reverse order. pol must be a
polynomial.
The library syntax is polrecip"(x)".
polresultant"(x,y,{v},{flag = 0})"
resultant of the two polynomials "x" and "y" with exact entries, with respect to the main
variables of "x" and "y" if "v" is omitted, with respect to the variable "v" otherwise.
The algorithm used is the subresultant algorithm by default.
If "flag = 1", uses the determinant of Sylvester's matrix instead (here "x" and "y" may
have non-exact coefficients).
If "flag = 2", uses Ducos's modified subresultant algorithm. It should be much faster than
the default if the coefficient ring is complicated (e.g multivariate polynomials or huge
coefficients), and slightly slower otherwise.
The library syntax is polresultant0"(x,y,v,flag)", where "v" is a "long" and an omitted
"v" is coded as "-1". Also available are "subres(x,y)" ("flag = 0") and "resultant2(x,y)"
("flag = 1").
polroots"(pol,{flag = 0})"
complex roots of the polynomial pol, given as a column vector where each root is repeated
according to its multiplicity. The precision is given as for transcendental functions:
under GP it is kept in the variable "realprecision" and is transparent to the user, but it
must be explicitly given as a second argument in library mode.
The algorithm used is a modification of A. Schoenhage's remarkable root-finding algorithm,
due to and implemented by X. Gourdon. Barring bugs, it is guaranteed to converge and to
give the roots to the required accuracy.
If "flag = 1", use a variant of the Newton-Raphson method, which is not guaranteed to
converge, but is rather fast. If you get the messages ``too many iterations in roots'' or
``INTERNAL ERROR: incorrect result in roots'', use the default function (i.e. no flag or
"flag = 0"). This used to be the default root-finding function in PARI until version
1.39.06.
The library syntax is roots"(pol,prec)" or "rootsold(pol,prec)".
polrootsmod"(pol,p,{flag = 0})"
row vector of roots modulo "p" of the polynomial pol. The particular non-prime value "p =
4" is accepted, mainly for 2-adic computations. Multiple roots are not repeated.
If "p < 100", you may try setting "flag = 1", which uses a naive search. In this case,
multiple roots are repeated with their order of multiplicity.
The library syntax is rootmod"(pol,p)" ("flag = 0") or "rootmod2(pol,p)" ("flag = 1").
polrootspadic"(pol,p,r)"
row vector of "p"-adic roots of the polynomial pol with "p"-adic precision equal to "r".
Multiple roots are not repeated. "p" is assumed to be a prime.
The library syntax is rootpadic"(pol,p,r)", where "r" is a "long".
polsturm"(pol,{a},{b})"
number of real roots of the real polynomial pol in the interval "]a,b]", using Sturm's
algorithm. "a" (resp. "b") is taken to be "- oo " (resp. "+ oo ") if omitted.
The library syntax is sturmpart"(pol,a,b)". Use "NULL" to omit an argument. sturm"(pol)"
is equivalent to sturmpart"(pol,NULL,NULL)". The result is a "long".
polsubcyclo"(n,d,{v = x})"
gives a polynomial (in variable "v") defining the sub-Abelian extension of degree "d" of
the cyclotomic field "Q(zeta_n)", where "d | phi(n)". "(Z/nZ)^*" has to be cyclic (i.e. "n
= 2", 4, "p^k" or "2p^k" for an odd prime "p"). The function "galoissubcyclo" covers the
general case.
The library syntax is subcyclo"(n,d,v)", where "v" is a variable number.
polsylvestermatrix"(x,y)"
forms the Sylvester matrix corresponding to the two polynomials "x" and "y", where the
coefficients of the polynomials are put in the columns of the matrix (which is the natural
direction for solving equations afterwards). The use of this matrix can be essential when
dealing with polynomials with inexact entries, since polynomial Euclidean division doesn't
make much sense in this case.
The library syntax is sylvestermatrix"(x,y)".
polsym"(x,n)"
creates the vector of the symmetric powers of the roots of the polynomial "x" up to power
"n", using Newton's formula.
The library syntax is polsym"(x)".
poltchebi"(n,{v = x})"
creates the "n^{th}" Chebyshev polynomial, in variable "v".
The library syntax is tchebi"(n,v)", where "n" and "v" are "long" integers ("v" is a
variable number).
polzagier"(n,m)"
creates Zagier's polynomial "P_{n,m}" used in the functions "sumalt" and "sumpos" (with
"flag = 1"). The exact definition can be found in a forthcoming paper. One must have "m <=
n".
The library syntax is polzagreel"(n,m,prec)" if the result is only wanted as a polynomial
with real coefficients to the precision "prec", or "polzag(n,m)" if the result is wanted
exactly, where "n" and "m" are "long"s.
serconvol"(x,y)"
convolution (or Hadamard product) of the two power series "x" and "y"; in other words if
"x = sum a_k*X^k" and "y = sum b_k*X^k" then "serconvol(x,y) = sum a_k*b_k*X^k".
The library syntax is convol"(x,y)".
serlaplace"(x)"
"x" must be a power series with only non-negative exponents. If "x = sum (a_k/k!)*X^k"
then the result is "sum a_k*X^k".
The library syntax is laplace"(x)".
serreverse"(x)"
reverse power series (i.e. "x^{-1}", not "1/x") of "x". "x" must be a power series whose
valuation is exactly equal to one.
The library syntax is recip"(x)".
subst"(x,y,z)"
replace the simple variable "y" by the argument "z" in the ``polynomial'' expression "x".
Every type is allowed for "x", but if it is not a genuine polynomial (or power series, or
rational function), the substitution will be done as if the scalar components were
polynomials of degree one. In particular, beware that:
? subst(1, x, [1,2; 3,4])
%1 =
[1 0]
[0 1]
? subst(1, x, Mat([0,1]))
*** forbidden substitution by a non square matrix
If "x" is a power series, "z" must be either a polynomial, a power series, or a rational
function. "y" must be a simple variable name.
The library syntax is gsubst"(x,v,z)", where "v" is the number of the variable "y".
taylor"(x,y)"
Taylor expansion around 0 of "x" with respect to the simple variable "y". "x" can be of
any reasonable type, for example a rational function. The number of terms of the expansion
is transparent to the user under GP, but must be given as a second argument in library
mode.
The library syntax is tayl"(x,y,n)", where the "long" integer "n" is the desired number of
terms in the expansion.
thue"(tnf,a,{sol})"
solves the equation "P(x,y) = a" in integers "x" and "y", where tnf was created with
thueinit(P). sol, if present, contains the solutions of "Norm(x) = a" modulo units of
positive norm in the number field defined by "P" (as computed by "bnfisintnorm"). If tnf
was computed without assuming GRH ("flag = 1" in "thueinit"), the result is unconditional.
For instance, here's how to solve the Thue equation "x^{13} - 5y^{13} = - 4":
? tnf = thueinit(x^13 - 5);
? thue(tnf, -4)
%1 = [[1, 1]]
Hence, assuming GRH, the only solution is "x = 1", "y = 1".
The library syntax is thue"(tnf,a,sol)", where an omitted sol is coded as "NULL".
thueinit"(P,{flag = 0})"
initializes the tnf corresponding to "P". It is meant to be used in conjunction with
"thue" to solve Thue equations "P(x,y) = a", where "a" is an integer. If "flag" is non-
zero, certify the result unconditionnaly, Otherwise, assume GRH, this being much faster of
course.
The library syntax is thueinit"(P,flag,prec)".
Vectors, matrices, linear algebra and sets
Note that most linear algebra functions operating on subspaces defined by generating sets
(such as "mathnf", "qflll", etc.) take matrices as arguments. As usual, the generating
vectors are taken to be the columns of the given matrix.
algdep"(x,k,{flag = 0})"
"x" being real, complex, or "p"-adic, finds a polynomial of degree at most "k" with
integer coefficients having "x" as approximate root. Note that the polynomial which is
obtained is not necessarily the ``correct'' one (it's not even guaranteed to be
irreducible!). One can check the closeness either by a polynomial evaluation or
substitution, or by computing the roots of the polynomial given by algdep.
If "x" is padic, "flag" is meaningless and the algorithm LLL-reduces the ``dual lattice''
corresponding to the powers of "x".
Otherwise, if "flag" is zero, the algorithm used is a variant of the LLL algorithm due to
Hastad, Lagarias and Schnorr (STACS 1986). If the precision is too low, the routine may
enter an infinite loop.
If "flag" is non-zero, use a standard LLL. "flag" then indicates a precision, which should
be between 0.5 and 1.0 times the number of decimal digits to which "x" was computed.
The library syntax is algdep0"(x,k,flag,prec)", where "k" and "flag" are "long"s. Also
available is "algdep(x,k,prec)" ("flag = 0").
charpoly"(A,{v = x},{flag = 0})"
characteristic polynomial of "A" with respect to the variable "v", i.e. determinant of
"v*I-A" if "A" is a square matrix, determinant of the map ``multiplication by "A"'' if "A"
is a scalar, in particular a polmod (e.g. "charpoly(I,x) = x^2+1"). Note that in the
latter case, the minimal polynomial can be obtained as
minpoly(A)=
{
local(y);
y = charpoly(A);
y / gcd(y,y')
}
The value of "flag" is only significant for matrices.
If "flag = 0", the method used is essentially the same as for computing the adjoint
matrix, i.e. computing the traces of the powers of "A".
If "flag = 1", uses Lagrange interpolation which is almost always slower.
If "flag = 2", uses the Hessenberg form. This is faster than the default when the
coefficients are integermod a prime or real numbers, but is usually slower in other base
rings.
The library syntax is charpoly0"(A,v,flag)", where "v" is the variable number. Also
available are the functions "caract(A,v)" ("flag = 1"), "carhess(A,v)" ("flag = 2"), and
"caradj(A,v,pt)" where, in this last case, pt is a "GEN*" which, if not equal to "NULL",
will receive the address of the adjoint matrix of "A" (see "matadjoint"), so both can be
obtained at once.
concat"(x,{y})"
concatenation of "x" and "y". If "x" or "y" is not a vector or matrix, it is considered as
a one-dimensional vector. All types are allowed for "x" and "y", but the sizes must be
compatible. Note that matrices are concatenated horizontally, i.e. the number of rows
stays the same. Using transpositions, it is easy to concatenate them vertically.
To concatenate vectors sideways (i.e. to obtain a two-row or two-column matrix), first
transform the vector into a one-row or one-column matrix using the function "Mat".
Concatenating a row vector to a matrix having the same number of columns will add the row
to the matrix (top row if the vector is "x", i.e. comes first, and bottom row otherwise).
The empty matrix "[;]" is considered to have a number of rows compatible with any
operation, in particular concatenation. (Note that this is definitely not the case for
empty vectors "[ ]" or "[ ]~".)
If "y" is omitted, "x" has to be a row vector or a list, in which case its elements are
concatenated, from left to right, using the above rules.
? concat([1,2], [3,4])
%1 = [1, 2, 3, 4]
? a = [[1,2]~, [3,4]~]; concat(a)
%2 = [1, 2, 3, 4]~
? a[1] = Mat(a[1]); concat(a)
%3 =
[1 3]
[2 4]
? concat([1,2; 3,4], [5,6]~)
%4 =
[1 2 5]
[3 4 6]
? concat([%, [7,8]~, [1,2,3,4]])
%5 =
[1 2 5 7]
[3 4 6 8]
[1 2 3 4]
The library syntax is concat"(x,y)".
lindep"(x,{flag = 0})"
"x" being a vector with real or complex coefficients, finds a small integral linear
combination among these coefficients.
If "flag = 0", uses a variant of the LLL algorithm due to Hastad, Lagarias and Schnorr
(STACS 1986).
If "flag > 0", uses the LLL algorithm. "flag" is a parameter which should be between one
half the number of decimal digits of precision and that number (see "algdep").
If "flag < 0", returns as soon as one relation has been found.
The library syntax is lindep0"(x,flag,prec)". Also available is "lindep(x,prec)" ("flag =
0").
listcreate"(n)"
creates an empty list of maximal length "n".
This function is useless in library mode.
listinsert"(list,x,n)"
inserts the object "x" at position "n" in list (which must be of type "t_LIST"). All the
remaining elements of list (from position "n+1" onwards) are shifted to the right. This
and "listput" are the only commands which enable you to increase a list's effective length
(as long as it remains under the maximal length specified at the time of the
"listcreate").
This function is useless in library mode.
listkill"(list)"
kill list. This deletes all elements from list and sets its effective length to 0. The
maximal length is not affected.
This function is useless in library mode.
listput"(list,x,{n})"
sets the "n"-th element of the list list (which must be of type "t_LIST") equal to "x". If
"n" is omitted, or greater than the list current effective length, just appends "x". This
and "listinsert" are the only commands which enable you to increase a list's effective
length (as long as it remains under the maximal length specified at the time of the
"listcreate").
If you want to put an element into an occupied cell, i.e. if you don't want to change the
effective length, you can consider the list as a vector and use the usual "list[n] = x"
construct.
This function is useless in library mode.
listsort"(list,{flag = 0})"
sorts list (which must be of type "t_LIST") in place. If "flag" is non-zero, suppresses
all repeated coefficients. This is much faster than the "vecsort" command since no copy
has to be made.
This function is useless in library mode.
matadjoint"(x)"
adjoint matrix of "x", i.e. the matrix "y" of cofactors of "x", satisfying "x*y = det
(x)*Id". "x" must be a (non-necessarily invertible) square matrix.
The library syntax is adj"(x)".
matcompanion"(x)"
the left companion matrix to the polynomial "x".
The library syntax is assmat"(x)".
matdet"(x,{flag = 0})"
determinant of "x". "x" must be a square matrix.
If "flag = 0", uses Gauss-Bareiss.
If "flag = 1", uses classical Gaussian elimination, which is better when the entries of
the matrix are reals or integers for example, but usually much worse for more complicated
entries like multivariate polynomials.
The library syntax is det"(x)" ("flag = 0") and "det2(x)" ("flag = 1").
matdetint"(x)"
"x" being an "m x n" matrix with integer coefficients, this function computes a multiple
of the determinant of the lattice generated by the columns of "x" if it is of rank "m",
and returns zero otherwise. This function can be useful in conjunction with the function
"mathnfmod" which needs to know such a multiple. Other ways to obtain this determinant
(assuming the rank is maximal) is "matdet(qflll(x,4)[2]*x)" or more simply
"matdet(mathnf(x))". Experiment to see which is faster for your applications.
The library syntax is detint"(x)".
matdiagonal"(x)"
"x" being a vector, creates the diagonal matrix whose diagonal entries are those of "x".
The library syntax is diagonal"(x)".
mateigen"(x)"
gives the eigenvectors of "x" as columns of a matrix.
The library syntax is eigen"(x)".
mathess"(x)"
Hessenberg form of the square matrix "x".
The library syntax is hess"(x)".
mathilbert"(x)"
"x" being a "long", creates the Hilbert matrix of order "x", i.e. the matrix whose
coefficient ("i","j") is "1/ (i+j-1)". matrix"
The library syntax is mathilbert"(x)".
mathnf"(x,{flag = 0})"
if "x" is a (not necessarily square) matrix of maximal rank, finds the upper triangular
Hermite normal form of "x". If the rank of "x" is equal to its number of rows, the result
is a square matrix. In general, the columns of the result form a basis of the lattice
spanned by the columns of "x".
If "flag = 0", uses the naive algorithm. If the Z-module generated by the columns is a
lattice, it is recommanded to use "mathnfmod(x, matdetint(x))" instead (much faster).
If "flag = 1", uses Batut's algorithm. Outputs a two-component row vector "[H,U]", where
"H" is the upper triangular Hermite normal form of "x" (i.e. the default result) and "U"
is the unimodular transformation matrix such that "xU = [0|H]". If the rank of "x" is
equal to its number of rows, "H" is a square matrix. In general, the columns of "H" form a
basis of the lattice spanned by the columns of "x".
If "flag = 2", uses Havas's algorithm. Outputs "[H,U,P]", such that "H" and "U" are as
before and "P" is a permutation of the rows such that "P" applied to "xU" gives "H". This
does not work very well in present version 2.2.0.
If "flag = 3", uses Batut's algorithm, and outputs "[H,U,P]" as in the previous case.
If "flag = 4", as in case 1 above, but uses LLL reduction along the way.
The library syntax is mathnf0"(x,flag)". Also available are "hnf(x)" ("flag = 0") and
"hnfall(x)" ("flag = 1"). To reduce huge (say "400 x 400" and more) relation matrices
(sparse with small entries), you can use the pair "hnfspec" / "hnfadd". Since this is
rather technical and the calling interface may change, they are not documented yet. Look
at the code in "basemath/alglin1.c".
mathnfmod"(x,d)"
if "x" is a (not necessarily square) matrix of maximal rank with integer entries, and "d"
is a multiple of the (non-zero) determinant of the lattice spanned by the columns of "x",
finds the upper triangular Hermite normal form of "x".
If the rank of "x" is equal to its number of rows, the result is a square matrix. In
general, the columns of the result form a basis of the lattice spanned by the columns of
"x". This is much faster than "mathnf" when "d" is known.
The library syntax is hnfmod"(x,d)".
mathnfmodid"(x,d)"
outputs the (upper triangular) Hermite normal form of "x" concatenated with "d" times the
identity matrix.
The library syntax is hnfmodid"(x,d)".
matid"(n)"
creates the "n x n" identity matrix.
The library syntax is idmat"(n)" where "n" is a "long".
Related functions are "gscalmat(x,n)", which creates "x" times the identity matrix ("x"
being a "GEN" and "n" a "long"), and "gscalsmat(x,n)" which is the same when "x" is a
"long".
matimage"(x,{flag = 0})"
gives a basis for the image of the matrix "x" as columns of a matrix. A priori the matrix
can have entries of any type. If "flag = 0", use standard Gauss pivot. If "flag = 1", use
"matsupplement".
The library syntax is matimage0"(x,flag)". Also available is "image(x)" ("flag = 0").
matimagecompl"(x)"
gives the vector of the column indices which are not extracted by the function "matimage".
Hence the number of components of matimagecompl(x) plus the number of columns of
matimage(x) is equal to the number of columns of the matrix "x".
The library syntax is imagecompl"(x)".
matindexrank"(x)"
"x" being a matrix of rank "r", gives two vectors "y" and "z" of length "r" giving a list
of rows and columns respectively (starting from 1) such that the extracted matrix obtained
from these two vectors using "vecextract(x,y,z)" is invertible.
The library syntax is indexrank"(x)".
matintersect"(x,y)"
"x" and "y" being two matrices with the same number of rows each of whose columns are
independent, finds a basis of the Q-vector space equal to the intersection of the spaces
spanned by the columns of "x" and "y" respectively. See also the function
"idealintersect", which does the same for free Z-modules.
The library syntax is intersect"(x,y)".
matinverseimage"(x,y)"
gives a column vector belonging to the inverse image of the column vector "y" by the
matrix "x" if one exists, the empty vector otherwise. To get the complete inverse image,
it suffices to add to the result any element of the kernel of "x" obtained for example by
"matker".
The library syntax is inverseimage"(x,y)".
matisdiagonal"(x)"
returns true (1) if "x" is a diagonal matrix, false (0) if not.
The library syntax is isdiagonal"(x)", and this returns a "long" integer.
matker"(x,{flag = 0})"
gives a basis for the kernel of the matrix "x" as columns of a matrix. A priori the matrix
can have entries of any type.
If "x" is known to have integral entries, set "flag = 1".
Note: The library function "ker_mod_p(x, p)", where "x" has integer entries and "p" is
prime, which is equivalent to but many orders of magnitude faster than
"matker(x*Mod(1,p))" and needs much less stack space. To use it under GP, type
"install(ker_mod_p, GG)" first.
The library syntax is matker0"(x,flag)". Also available are "ker(x)" ("flag = 0"),
"keri(x)" ("flag = 1") and "ker_mod_p(x,p)".
matkerint"(x,{flag = 0})"
gives an LLL-reduced Z-basis for the lattice equal to the kernel of the matrix "x" as
columns of the matrix "x" with integer entries (rational entries are not permitted).
If "flag = 0", uses a modified integer LLL algorithm.
If "flag = 1", uses "matrixqz(x,-2)". If LLL reduction of the final result is not desired,
you can save time using "matrixqz(matker(x),-2)" instead.
If "flag = 2", uses another modified LLL. In the present version 2.2.0, only independent
rows are allowed in this case.
The library syntax is matkerint0"(x,flag)". Also available is "kerint(x)" ("flag = 0").
matmuldiagonal"(x,d)"
product of the matrix "x" by the diagonal matrix whose diagonal entries are those of the
vector "d". Equivalent to, but much faster than "x*matdiagonal(d)".
The library syntax is matmuldiagonal"(x,d)".
matmultodiagonal"(x,y)"
product of the matrices "x" and "y" knowing that the result is a diagonal matrix. Much
faster than "x*y" in that case.
The library syntax is matmultodiagonal"(x,y)".
matpascal"(x,{q})"
creates as a matrix the lower triangular Pascal triangle of order "x+1" (i.e. with
binomial coefficients up to "x"). If "q" is given, compute the "q"-Pascal triangle
(i.e. using "q"-binomial coefficients).
The library syntax is matqpascal"(x,q)", where "x" is a "long" and "q = NULL" is used to
omit "q". Also available is matpascal{x}.
matrank"(x)"
rank of the matrix "x".
The library syntax is rank"(x)", and the result is a "long".
matrix"(m,n,{X},{Y},{expr = 0})"
creation of the "m x n" matrix whose coefficients are given by the expression expr. There
are two formal parameters in expr, the first one ("X") corresponding to the rows, the
second ("Y") to the columns, and "X" goes from 1 to "m", "Y" goes from 1 to "n". If one of
the last 3 parameters is omitted, fill the matrix with zeroes.
The library syntax is matrice"(GEN nlig,GEN ncol,entree *e1,entree *e2,char *expr)".
matrixqz"(x,p)"
"x" being an "m x n" matrix with "m >= n" with rational or integer entries, this function
has varying behaviour depending on the sign of "p":
If "p >= 0", "x" is assumed to be of maximal rank. This function returns a matrix having
only integral entries, having the same image as "x", such that the GCD of all its "n x n"
subdeterminants is equal to 1 when "p" is equal to 0, or not divisible by "p" otherwise.
Here "p" must be a prime number (when it is non-zero). However, if the function is used
when "p" has no small prime factors, it will either work or give the message ``impossible
inverse modulo'' and a non-trivial divisor of "p".
If "p = -1", this function returns a matrix whose columns form a basis of the lattice
equal to "Z^n" intersected with the lattice generated by the columns of "x".
If "p = -2", returns a matrix whose columns form a basis of the lattice equal to "Z^n"
intersected with the Q-vector space generated by the columns of "x".
The library syntax is matrixqz0"(x,p)".
matsize"(x)"
"x" being a vector or matrix, returns a row vector with two components, the first being
the number of rows (1 for a row vector), the second the number of columns (1 for a column
vector).
The library syntax is matsize"(x)".
matsnf"(X,{flag = 0})"
if "X" is a (singular or non-singular) square matrix outputs the vector of elementary
divisors of "X" (i.e. the diagonal of the Smith normal form of "X").
The binary digits of flag mean:
1 (complete output): if set, outputs "[U,V,D]", where "U" and "V" are two unimodular
matrices such that "UXV" is the diagonal matrix "D". Otherwise output only the diagonal of
"D".
2 (generic input): if set, allows polynomial entries. Otherwise, assume that "X" has
integer coefficients.
4 (cleanup): if set, cleans up the output. This means that elementary divisors equal to 1
will be deleted, i.e. outputs a shortened vector "D'" instead of "D". If complete output
was required, returns "[U',V',D']" so that "U'XV' = D'" holds. If this flag is set, "X" is
allowed to be of the form "D" or "[U,V,D]" as would normally be output with the cleanup
flag unset.
The library syntax is matsnf0"(X,flag)". Also available is "smith(X)" ("flag = 0").
matsolve"(x,y)"
"x" being an invertible matrix and "y" a column vector, finds the solution "u" of "x*u =
y", using Gaussian elimination. This has the same effect as, but is a bit faster, than
"x^{-1}*y".
The library syntax is gauss"(x,y)".
matsolvemod"(m,d,y,{flag = 0})"
"m" being any integral matrix, "d" a vector of positive integer moduli, and "y" an
integral column vector, gives a small integer solution to the system of congruences "sum_i
m_{i,j}x_j = y_i (mod d_i)" if one exists, otherwise returns zero. Shorthand notation: "y"
(resp. "d") can be given as a single integer, in which case all the "y_i" (resp. "d_i")
above are taken to be equal to "y" (resp. "d").
If "flag = 1", all solutions are returned in the form of a two-component row vector
"[x,u]", where "x" is a small integer solution to the system of congruences and "u" is a
matrix whose columns give a basis of the homogeneous system (so that all solutions can be
obtained by adding "x" to any linear combination of columns of "u"). If no solution
exists, returns zero.
The library syntax is matsolvemod0"(m,d,y,flag)". Also available are "gaussmodulo(m,d,y)"
("flag = 0") and "gaussmodulo2(m,d,y)" ("flag = 1").
matsupplement"(x)"
assuming that the columns of the matrix "x" are linearly independent (if they are not, an
error message is issued), finds a square invertible matrix whose first columns are the
columns of "x", i.e. supplement the columns of "x" to a basis of the whole space.
The library syntax is suppl"(x)".
mattranspose"(x)" or "x~"
transpose of "x". This has an effect only on vectors and matrices.
The library syntax is gtrans"(x)".
qfgaussred"(q)"
decomposition into squares of the quadratic form represented by the symmetric matrix "q".
The result is a matrix whose diagonal entries are the coefficients of the squares, and the
non-diagonal entries represent the bilinear forms. More precisely, if "(a_{ij})" denotes
the output, one has
" q(x) = sum_i a_{ii} (x_i + sum_{j > i} a_{ij} x_j)^2 "
The library syntax is sqred"(x)".
qfjacobi"(x)"
"x" being a real symmetric matrix, this gives a vector having two components: the first
one is the vector of eigenvalues of "x", the second is the corresponding orthogonal matrix
of eigenvectors of "x". The method used is Jacobi's method for symmetric matrices.
The library syntax is jacobi"(x)".
qflll"(x,{flag = 0})"
LLL algorithm applied to the columns of the (not necessarily square) matrix "x". The
columns of "x" must however be linearly independent, unless specified otherwise below. The
result is a transformation matrix "T" such that "x.T" is an LLL-reduced basis of the
lattice generated by the column vectors of "x".
If "flag = 0" (default), the computations are done with real numbers (i.e. not with
rational numbers) hence are fast but as presently programmed (version 2.2.0) are
numerically unstable.
If "flag = 1", it is assumed that the corresponding Gram matrix is integral. The
computation is done entirely with integers and the algorithm is both accurate and quite
fast. In this case, "x" needs not be of maximal rank, but if it is not, "T" will not be
square.
If "flag = 2", similar to case 1, except "x" should be an integer matrix whose columns are
linearly independent. The lattice generated by the columns of "x" is first partially
reduced before applying the LLL algorithm. [A basis is said to be partially reduced if
"|v_i +- v_j| >= |v_i|" for any two distinct basis vectors "v_i, v_j".]
This can be significantly faster than "flag = 1" when one row is huge compared to the
other rows.
If "flag = 3", all computations are done in rational numbers. This does not incur
numerical instability, but is extremely slow. This function is essentially superseded by
case 1, so will soon disappear.
If "flag = 4", "x" is assumed to have integral entries, but needs not be of maximal rank.
The result is a two-component vector of matrices : the columns of the first matrix
represent a basis of the integer kernel of "x" (not necessarily LLL-reduced) and the
second matrix is the transformation matrix "T" such that "x.T" is an LLL-reduced Z-basis
of the image of the matrix "x".
If "flag = 5", case as case 4, but "x" may have polynomial coefficients.
If "flag = 7", uses an older version of case 0 above.
If "flag = 8", same as case 0, where "x" may have polynomial coefficients.
If "flag = 9", variation on case 1, using content.
The library syntax is qflll0"(x,flag,prec)". Also available are "lll(x,prec)" ("flag =
0"), "lllint(x)" ("flag = 1"), and "lllkerim(x)" ("flag = 4").
qflllgram"(x,{flag = 0})"
same as "qflll" except that the matrix "x" which must now be a square symmetric real
matrix is the Gram matrix of the lattice vectors, and not the coordinates of the vectors
themselves. The result is again the transformation matrix "T" which gives (as columns) the
coefficients with respect to the initial basis vectors. The flags have more or less the
same meaning, but some are missing. In brief:
"flag = 0": numerically unstable in the present version 2.2.0.
"flag = 1": "x" has integer entries, the computations are all done in integers.
"flag = 4": "x" has integer entries, gives the kernel and reduced image.
"flag = 5": same as 4 for generic "x".
"flag = 7": an older version of case 0.
The library syntax is qflllgram0"(x,flag,prec)". Also available are "lllgram(x,prec)"
("flag = 0"), "lllgramint(x)" ("flag = 1"), and "lllgramkerim(x)" ("flag = 4").
qfminim"(x,b,m,{flag = 0})"
"x" being a square and symmetric matrix representing a positive definite quadratic form,
this function deals with the minimal vectors of "x", depending on "flag".
If "flag = 0" (default), seeks vectors of square norm less than or equal to "b" (for the
norm defined by "x"), and at most "2m" of these vectors. The result is a three-component
vector, the first component being the number of vectors, the second being the maximum norm
found, and the last vector is a matrix whose columns are the vectors found, only one being
given for each pair "+- v" (at most "m" such pairs).
If "flag = 1", ignores "m" and returns the first vector whose norm is less than "b".
In both these cases, "x" is assumed to have integral entries, and the function searches
for the minimal non-zero vectors whenever "b = 0".
If "flag = 2", "x" can have non integral real entries, but "b = 0" is now meaningless
(uses Fincke-Pohst algorithm).
The library syntax is qfminim0"(x,b,m,flag,prec)", also available are " minim(x,b,m)"
("flag = 0"), " minim2(x,b,m)" ("flag = 1"), and finally " fincke_pohst(x,b,m,prec)"
("flag = 2").
qfperfection"(x)"
"x" being a square and symmetric matrix with integer entries representing a positive
definite quadratic form, outputs the perfection rank of the form. That is, gives the rank
of the family of the "s" symmetric matrices "v_iv_i^t", where "s" is half the number of
minimal vectors and the "v_i" ("1 <= i <= s") are the minimal vectors.
As a side note to old-timers, this used to fail bluntly when "x" had more than 5000
minimal vectors. Beware that the computations can now be very lengthy when "x" has many
minimal vectors.
The library syntax is perf"(x)".
qfsign"(x)"
signature of the quadratic form represented by the symmetric matrix "x". The result is a
two-component vector.
The library syntax is signat"(x)".
setintersect"(x,y)"
intersection of the two sets "x" and "y".
The library syntax is setintersect"(x,y)".
setisset"(x)"
returns true (1) if "x" is a set, false (0) if not. In PARI, a set is simply a row vector
whose entries are strictly increasing. To convert any vector (and other objects) into a
set, use the function "Set".
The library syntax is setisset"(x)", and this returns a "long".
setminus"(x,y)"
difference of the two sets "x" and "y", i.e. set of elements of "x" which do not belong to
"y".
The library syntax is setminus"(x,y)".
setsearch"(x,y,{flag = 0})"
searches if "y" belongs to the set "x". If it does and "flag" is zero or omitted, returns
the index "j" such that "x[j] = y", otherwise returns 0. If "flag" is non-zero returns the
index "j" where "y" should be inserted, and 0 if it already belongs to "x" (this is meant
to be used in conjunction with "listinsert").
This function works also if "x" is a sorted list (see "listsort").
The library syntax is setsearch"(x,y,flag)" which returns a "long" integer.
setunion"(x,y)"
union of the two sets "x" and "y".
The library syntax is setunion"(x,y)".
trace"(x)"
this applies to quite general "x". If "x" is not a matrix, it is equal to the sum of "x"
and its conjugate, except for polmods where it is the trace as an algebraic number.
For "x" a square matrix, it is the ordinary trace. If "x" is a non-square matrix (but not
a vector), an error occurs.
The library syntax is gtrace"(x)".
vecextract"(x,y,{z})"
extraction of components of the vector or matrix "x" according to "y". In case "x" is a
matrix, its components are as usual the columns of "x". The parameter "y" is a component
specifier, which is either an integer, a string describing a range, or a vector.
If "y" is an integer, it is considered as a mask: the binary bits of "y" are read from
right to left, but correspond to taking the components from left to right. For example, if
"y = 13 = (1101)_2" then the components 1,3 and 4 are extracted.
If "y" is a vector, which must have integer entries, these entries correspond to the
component numbers to be extracted, in the order specified.
If "y" is a string, it can be
"*" a single (non-zero) index giving a component number (a negative index means we start
counting from the end).
"*" a range of the form "a..b", where "a" and "b" are indexes as above. Any of "a" and "b"
can be omitted; in this case, we take as default values "a = 1" and "b = -1", i.e. the
first and last components respectively. We then extract all components in the interval
"[a,b]", in reverse order if "b < a".
In addition, if the first character in the string is "^", the complement of the given set
of indices is taken.
If "z" is not omitted, "x" must be a matrix. "y" is then the line specifier, and "z" the
column specifier, where the component specifier is as explained above.
? v = [a, b, c, d, e];
? vecextract(v, 5) \\ mask
%1 = [a, c]
? vecextract(v, [4, 2, 1]) \\ component list
%2 = [d, b, a]
? vecextract(v, "2..4") \\ interval
%3 = [b, c, d]
? vecextract(v, "-1..-3") \\ interval + reverse order
%4 = [e, d, c]
? vecextract([1,2,3], "^2") \\ complement
%5 = [1, 3]
? vecextract(matid(3), "2..", "..")
%6 =
[0 1 0]
[0 0 1]
The library syntax is extract"(x,y)" or "matextract(x,y,z)".
vecsort"(x,{k},{flag = 0})"
sorts the vector "x" in ascending order, using the heapsort method. "x" must be a vector,
and its components integers, reals, or fractions.
If "k" is present and is an integer, sorts according to the value of the "k"-th
subcomponents of the components of "x". "k" can also be a vector, in which case the
sorting is done lexicographically according to the components listed in the vector "k".
For example, if "k = [2,1,3]", sorting will be done with respect to the second component,
and when these are equal, with respect to the first, and when these are equal, with
respect to the third.
The binary digits of flag mean:
"*" 1: indirect sorting of the vector "x", i.e. if "x" is an "n"-component vector, returns
a permutation of "[1,2,...,n]" which applied to the components of "x" sorts "x" in
increasing order. For example, "vecextract(x, vecsort(x,,1))" is equivalent to
vecsort(x).
"*" 2: sorts "x" by ascending lexicographic order (as per the "lex" comparison function).
"*" 4: use decreasing instead of ascending order.
The library syntax is vecsort0"(x,k,flag)". To omit "k", use "NULL" instead. You can also
use the simpler functions
"sort(x)" ( = "vecsort0(x,NULL,0)").
"indexsort(x)" ( = "vecsort0(x,NULL,1)").
"lexsort(x)" ( = "vecsort0(x,NULL,2)").
Also available are sindexsort and sindexlexsort which return a vector of C-long integers
(private type "t_VECSMALL") "v", where "v[1]...v[n]" contain the indices. Note that the
resulting "v" is not a generic PARI object, but is in general easier to use in C programs!
vector"(n,{X},{expr = 0})"
creates a row vector (type "t_VEC") with "n" components whose components are the
expression expr evaluated at the integer points between 1 and "n". If one of the last two
arguments is omitted, fill the vector with zeroes.
The library syntax is vecteur"(GEN nmax, entree *ep, char *expr)".
vectorv"(n,X,expr)"
as vector, but returns a column vector (type "t_COL").
The library syntax is vvecteur"(GEN nmax, entree *ep, char *expr)".
Sums, products, integrals and similar functions
Although the GP calculator is programmable, it is useful to have preprogrammed a number of
loops, including sums, products, and a certain number of recursions. Also, a number of
functions from numerical analysis like numerical integration and summation of series will
be described here.
One of the parameters in these loops must be the control variable, hence a simple variable
name. The last parameter can be any legal PARI expression, including of course expressions
using loops. Since it is much easier to program directly the loops in library mode, these
functions are mainly useful for GP programming. The use of these functions in library mode
is a little tricky and its explanation will be mostly omitted, although the reader can try
and figure it out by himself by checking the example given for the "sum" function. In this
section we only give the library syntax, with no semantic explanation.
The letter "X" will always denote any simple variable name, and represents the formal
parameter used in the function.
(numerical) integration: A number of Romberg-like integration methods are implemented (see
"intnum" as opposed to "intformal" which we already described). The user should not
require too much accuracy: 18 or 28 decimal digits is OK, but not much more. In addition,
analytical cleanup of the integral must have been done: there must be no singularities in
the interval or at the boundaries. In practice this can be accomplished with a simple
change of variable. Furthermore, for improper integrals, where one or both of the limits
of integration are plus or minus infinity, the function must decrease sufficiently rapidly
at infinity. This can often be accomplished through integration by parts. Finally, the
function to be integrated should not be very small (compared to the current precision) on
the entire interval. This can of course be accomplished by just multiplying by an
appropriate constant.
Note that infinity can be represented with essentially no loss of accuracy by 1e4000.
However beware of real underflow when dealing with rapidly decreasing functions. For
example, if one wants to compute the "int_0^ oo e^{-x^2}dx" to 28 decimal digits, then one
should set infinity equal to 10 for example, and certainly not to 1e4000.
The integrand may have values belonging to a vector space over the real numbers; in
particular, it can be complex-valued or vector-valued.
See also the discrete summation methods below (sharing the prefix "sum").
intnum"(X = a,b,expr,{flag = 0})"
numerical integration of expr (smooth in "]a,b["), with respect to "X".
Set "flag = 0" (or omit it altogether) when "a" and "b" are not too large, the function is
smooth, and can be evaluated exactly everywhere on the interval "[a,b]".
If "flag = 1", uses a general driver routine for doing numerical integration, making no
particular assumption (slow).
"flag = 2" is tailored for being used when "a" or "b" are infinite. One must have "ab >
0", and in fact if for example "b = + oo ", then it is preferable to have "a" as large as
possible, at least "a >= 1".
If "flag = 3", the function is allowed to be undefined (but continuous) at "a" or "b", for
example the function " sin (x)/x" at "x = 0".
The library syntax is intnum0"(entree*e,GEN a,GEN b,char*expr,long flag,long prec)".
prod"(X = a,b,expr,{x = 1})"
product of expression expr, initialized at "x", the formal parameter "X" going from "a" to
"b". As for "sum", the main purpose of the initialization parameter "x" is to force the
type of the operations being performed. For example if it is set equal to the integer 1,
operations will start being done exactly. If it is set equal to the real 1., they will be
done using real numbers having the default precision. If it is set equal to the power
series "1+O(X^k)" for a certain "k", they will be done using power series of precision at
most "k". These are the three most common initializations.
As an extreme example, compare
? prod(i=1, 100, 1 - X^i); \\ this has degree 5050 !!
time = 3,335 ms.
? prod(i=1, 100, 1 - X^i, 1 + O(X^101))
time = 43 ms.
%2 = 1 - X - X^2 + X^5 + X^7 - X^12 - X^15 + X^22 + X^26 - X^35 - X^40 + \
X^51 + X^57 - X^70 - X^77 + X^92 + X^100 + O(X^101)
The library syntax is produit"(entree *ep, GEN a, GEN b, char *expr, GEN x)".
prodeuler"(X = a,b,expr)"
product of expression expr, initialized at 1. (i.e. to a real number equal to 1 to the
current "realprecision"), the formal parameter "X" ranging over the prime numbers between
"a" and "b".
The library syntax is prodeuler"(entree *ep, GEN a, GEN b, char *expr, long prec)".
prodinf"(X = a,expr,{flag = 0})"
infinite product of expression expr, the formal parameter "X" starting at "a". The
evaluation stops when the relative error of the expression minus 1 is less than the
default precision. The expressions must always evaluate to an element of C.
If "flag = 1", do the product of the ("1+expr") instead.
The library syntax is prodinf"(entree *ep, GEN a, char *expr, long prec)" ("flag = 0"), or
prodinf1 with the same arguments ("flag = 1").
solve"(X = a,b,expr)"
find a real root of expression expr between "a" and "b", under the condition "expr(X = a)
* expr(X = b) <= 0". This routine uses Brent's method and can fail miserably if expr is
not defined in the whole of "[a,b]" (try "solve(x = 1, 2, tan(x)").
The library syntax is zbrent"(entree *ep, GEN a, GEN b, char *expr, long prec)".
sum"(X = a,b,expr,{x = 0})"
sum of expression expr, initialized at "x", the formal parameter going from "a" to "b". As
for "prod", the initialization parameter "x" may be given to force the type of the
operations being performed.
As an extreme example, compare
? sum(i=1, 5000, 1/i); \\ rational number: denominator has 2166 digits.
time = 1,241 ms.
? sum(i=1, 5000, 1/i, 0.)
time = 158 ms.
%2 = 9.094508852984436967261245533
The library syntax is somme"(entree *ep, GEN a, GEN b, char *expr, GEN x)". This is to be
used as follows: "ep" represents the dummy variable used in the expression "expr"
/* compute a^2 + ... + b^2 */
{
/* define the dummy variable "i" */
entree *ep = is_entry("i");
/* sum for a <= i <= b */
return somme(ep, a, b, "i^2", gzero);
}
sumalt"(X = a,expr,{flag = 0})"
numerical summation of the series expr, which should be an alternating series, the formal
variable "X" starting at "a".
If "flag = 0", use an algorithm of F. Villegas as modified by D. Zagier. This is much
better than Euler-Van Wijngaarden's method which was used formerly. Beware that the
stopping criterion is that the term gets small enough, hence terms which are equal to 0
will create problems and should be removed.
If "flag = 1", use a variant with slightly different polynomials. Sometimes faster.
Divergent alternating series can sometimes be summed by this method, as well as series
which are not exactly alternating (see for example "Label se:user_defined").
Important hint: a significant speed gain can be obtained by writing the "(-1)^X" which may
occur in the expression as "(1. - X%2*2)".
The library syntax is sumalt"(entree *ep, GEN a, char *expr, long flag, long prec)".
sumdiv"(n,X,expr)"
sum of expression expr over the positive divisors of "n".
Arithmetic functions like sigma use the multiplicativity of the underlying expression to
speed up the computation. In the present version 2.2.0, there is no way to indicate that
expr is multiplicative in "n", hence specialized functions should be prefered whenever
possible.
The library syntax is divsum"(entree *ep, GEN num, char *expr)".
suminf"(X = a,expr)"
infinite sum of expression expr, the formal parameter "X" starting at "a". The evaluation
stops when the relative error of the expression is less than the default precision. The
expressions must always evaluate to a complex number.
The library syntax is suminf"(entree *ep, GEN a, char *expr, long prec)".
sumpos"(X = a,expr,{flag = 0})"
numerical summation of the series expr, which must be a series of terms having the same
sign, the formal variable "X" starting at "a". The algorithm used is Van Wijngaarden's
trick for converting such a series into an alternating one, and is quite slow. Beware
that the stopping criterion is that the term gets small enough, hence terms which are
equal to 0 will create problems and should be removed.
If "flag = 1", use slightly different polynomials. Sometimes faster.
The library syntax is sumpos"(entree *ep, GEN a, char *expr, long flag, long prec)".
Plotting functions
Although plotting is not even a side purpose of PARI, a number of plotting functions are
provided. Moreover, a lot of people felt like suggesting ideas or submitting huge patches
for this section of the code. Among these, special thanks go to Klaus-Peter Nischke who
suggested the recursive plotting and the forking/resizing stuff under X11, and Ilya
Zakharevich who undertook a complete rewrite of the graphic code, so that most of it is
now platform-independent and should be relatively easy to port or expand.
These graphic functions are either
"*" high-level plotting functions (all the functions starting with "ploth") in which the
user has little to do but explain what type of plot he wants, and whose syntax is similar
to the one used in the preceding section (with somewhat more complicated flags).
"*" low-level plotting functions, where every drawing primitive (point, line, box, etc.)
must be specified by the user. These low-level functions (called rectplot functions,
sharing the prefix "plot") work as follows. You have at your disposal 16 virtual windows
which are filled independently, and can then be physically ORed on a single window at
user-defined positions. These windows are numbered from 0 to 15, and must be initialized
before being used by the function "plotinit", which specifies the height and width of the
virtual window (called a rectwindow in the sequel). At all times, a virtual cursor
(initialized at "[0,0]") is associated to the window, and its current value can be
obtained using the function "plotcursor".
A number of primitive graphic objects (called rect objects) can then be drawn in these
windows, using a default color associated to that window (which can be changed under X11,
using the "plotcolor" function, black otherwise) and only the part of the object which is
inside the window will be drawn, with the exception of polygons and strings which are
drawn entirely (but the virtual cursor can move outside of the window). The ones sharing
the prefix "plotr" draw relatively to the current position of the virtual cursor, the
others use absolute coordinates. Those having the prefix "plotrecth" put in the rectwindow
a large batch of rect objects corresponding to the output of the related "ploth" function.
Finally, the actual physical drawing is done using the function "plotdraw". Note that the
windows are preserved so that further drawings using the same windows at different
positions or different windows can be done without extra work. If you want to erase a
window (and free the corresponding memory), use the function "plotkill". It is not
possible to partially erase a window. Erase it completely, initialize it again and then
fill it with the graphic objects that you want to keep.
In addition to initializing the window, you may want to have a scaled window to avoid
unnecessary conversions. For this, use the function "plotscale" below. As long as this
function is not called, the scaling is simply the number of pixels, the origin being at
the upper left and the "y"-coordinates going downwards.
Note that in the present version 2.2.0 all these plotting functions (both low and high
level) have been written for the X11-window system (hence also for GUI's based on X11 such
as Openwindows and Motif) only, though very little code remains which is actually
platform-dependent. A Suntools/Sunview, Macintosh, and an Atari/Gem port were provided for
previous versions. These may be adapted in future releases.
Under X11/Suntools, the physical window (opened by "plotdraw" or any of the "ploth*"
functions) is completely separated from GP (technically, a "fork" is done, and the non-
graphical memory is immediately freed in the child process), which means you can go on
working in the current GP session, without having to kill the window first. Under X11,
this window can be closed, enlarged or reduced using the standard window manager
functions. No zooming procedure is implemented though (yet).
"*" Finally, note that in the same way that "printtex" allows you to have a TeX output
corresponding to printed results, the functions starting with "ps" allow you to have
"PostScript" output of the plots. This will not be absolutely identical with the screen
output, but will be sufficiently close. Note that you can use PostScript output even if
you do not have the plotting routines enabled. The PostScript output is written in a file
whose name is derived from the "psfile" default ("./pari.ps" if you did not tamper with
it). Each time a new PostScript output is asked for, the PostScript output is appended to
that file. Hence the user must remove this file, or change the value of "psfile", first if
he does not want unnecessary drawings from preceding sessions to appear. On the other
hand, in this manner as many plots as desired can be kept in a single file.
None of the graphic functions are available within the PARI library, you must be under GP
to use them. The reason for that is that you really should not use PARI for heavy-duty
graphical work, there are much better specialized alternatives around. This whole set of
routines was only meant as a convenient, but simple-minded, visual aid. If you really
insist on using these in your program (we warned you), the source ("plot*.c") should be
readable enough for you to achieve something.
plot"(X = a,b,expr,{Ymin},{Ymax})"
crude (ASCII) plot of the function represented by expression expr from "a" to "b", with Y
ranging from Ymin to Ymax. If Ymin (resp. Ymax) is not given, the minima (resp. the
maxima) of the computed values of the expression is used instead.
plotbox"(w,x2,y2)"
let "(x1,y1)" be the current position of the virtual cursor. Draw in the rectwindow "w"
the outline of the rectangle which is such that the points "(x1,y1)" and "(x2,y2)" are
opposite corners. Only the part of the rectangle which is in "w" is drawn. The virtual
cursor does not move.
plotclip"(w)"
`clips' the content of rectwindow "w", i.e remove all parts of the drawing that would not
be visible on the screen. Together with "plotcopy" this function enables you to draw on a
scratchpad before commiting the part you're interested in to the final picture.
plotcolor"(w,c)"
set default color to "c" in rectwindow "w". In present version 2.2.0, this is only
implemented for X11 window system, and you only have the following palette to choose from:
1 = black, 2 = blue, 3 = sienna, 4 = red, 5 = cornsilk, 6 = grey, 7 = gainsborough.
Note that it should be fairly easy for you to hardwire some more colors by tweaking the
files "rect.h" and "plotX.c". User-defined colormaps would be nice, and may be available
in future versions.
plotcopy"(w1,w2,dx,dy)"
copy the contents of rectwindow "w1" to rectwindow "w2", with offset "(dx,dy)".
plotcursor"(w)"
give as a 2-component vector the current (scaled) position of the virtual cursor
corresponding to the rectwindow "w".
plotdraw"(list)"
physically draw the rectwindows given in "list" which must be a vector whose number of
components is divisible by 3. If "list = [w1,x1,y1,w2,x2,y2,...]", the windows "w1", "w2",
etc. are physically placed with their upper left corner at physical position "(x1,y1)",
"(x2,y2)",...respectively, and are then drawn together. Overlapping regions will thus be
drawn twice, and the windows are considered transparent. Then display the whole drawing in
a special window on your screen.
plotfile"(s)"
set the output file for plotting output. Special filename "-" redirects to the same place
as PARI output.
ploth"(X = a,b,expr,{flag = 0},{n = 0})"
high precision plot of the function "y = f(x)" represented by the expression expr, "x"
going from "a" to "b". This opens a specific window (which is killed whenever you click on
it), and returns a four-component vector giving the coordinates of the bounding box in the
form "[xmin,xmax,ymin,ymax]".
Important note: Since this may involve a lot of function calls, it is advised to keep the
current precision to a minimum (e.g. 9) before calling this function.
"n" specifies the number of reference point on the graph (0 means use the hardwired
default values, that is: 1000 for general plot, 1500 for parametric plot, and 15 for
recursive plot).
If no "flag" is given, expr is either a scalar expression f(X), in which case the plane
curve "y = f(X)" will be drawn, or a vector "[f_1(X),...,f_k(X)]", and then all the curves
"y = f_i(X)" will be drawn in the same window.
The binary digits of "flag" mean:
"*" 1: parametric plot. Here expr must be a vector with an even number of components.
Successive pairs are then understood as the parametric coordinates of a plane curve. Each
of these are then drawn.
For instance:
"ploth(X = 0,2*Pi,[sin(X),cos(X)],1)" will draw a circle.
"ploth(X = 0,2*Pi,[sin(X),cos(X)])" will draw two entwined sinusoidal curves.
"ploth(X = 0,2*Pi,[X,X,sin(X),cos(X)],1)" will draw a circle and the line "y = x".
"*" 2: recursive plot. If this flag is set, only one curve can be drawn at time, i.e. expr
must be either a two-component vector (for a single parametric curve, and the parametric
flag has to be set), or a scalar function. The idea is to choose pairs of successive
reference points, and if their middle point is not too far away from the segment joining
them, draw this as a local approximation to the curve. Otherwise, add the middle point to
the reference points. This is very fast, and usually more precise than usual plot. Compare
the results of
"ploth(X = -1,1,sin(1/X),2) and ploth(X = -1,1,sin(1/X))"
for instance. But beware that if you are extremely unlucky, or choose too few reference
points, you may draw some nice polygon bearing little resemblance to the original curve.
For instance you should never plot recursively an odd function in a symmetric interval
around 0. Try
ploth(x = -20, 20, sin(x), 2)
to see why. Hence, it's usually a good idea to try and plot the same curve with slightly
different parameters.
The other values toggle various display options:
"*" 4: do not rescale plot according to the computed extrema. This is meant to be used
when graphing multiple functions on a rectwindow (as a "plotrecth" call), in conjuction
with "plotscale".
"*" 8: do not print the "x"-axis.
"*" 16: do not print the "y"-axis.
"*" 32: do not print frame.
"*" 64: only plot reference points, do not join them.
"*" 256: use splines to interpolate the points.
"*" 512: plot no "x"-ticks.
"*" 1024: plot no "y"-ticks.
"*" 2048: plot all ticks with the same length.
plothraw"(listx,listy,{flag = 0})"
given listx and listy two vectors of equal length, plots (in high precision) the points
whose "(x,y)"-coordinates are given in listx and listy. Automatic positioning and scaling
is done, but with the same scaling factor on "x" and "y". If "flag" is 1, join points,
other non-0 flags toggle display options and should be combinations of bits "2^k", "k
>= 3" as in "ploth".
plothsizes"()"
return data corresponding to the output window in the form of a 6-component vector: window
width and height, sizes for ticks in horizontal and vertical directions (this is intended
for the "gnuplot" interface and is currently not significant), width and height of
characters.
plotinit"(w,x,y)"
initialize the rectwindow "w" to width "x" and height "y", and position the virtual cursor
at "(0,0)". This destroys any rect objects you may have already drawn in "w".
The plotting device imposes an upper bound for "x" and "y", for instance the number of
pixels for screen output. These bounds are available through the "plothsizes" function.
The following sequence initializes in a portable way (i.e independant of the output
device) a window of maximal size, accessed through coordinates in the "[0,1000] x
[0,1000]" range :
s = plothsizes();
plotinit(0, s[1]-1, s[2]-1);
plotscale(0, 0,1000, 0,1000);
plotkill"(w)"
erase rectwindow "w" and free the corresponding memory. Note that if you want to use the
rectwindow "w" again, you have to use "initrect" first to specify the new size. So it's
better in this case to use "initrect" directly as this throws away any previous work in
the given rectwindow.
plotlines"(w,X,Y,{flag = 0})"
draw on the rectwindow "w" the polygon such that the (x,y)-coordinates of the vertices are
in the vectors of equal length "X" and "Y". For simplicity, the whole polygon is drawn,
not only the part of the polygon which is inside the rectwindow. If "flag" is non-zero,
close the polygon. In any case, the virtual cursor does not move.
"X" and "Y" are allowed to be scalars (in this case, both have to). There, a single
segment will be drawn, between the virtual cursor current position and the point "(X,Y)".
And only the part thereof which actually lies within the boundary of "w". Then move the
virtual cursor to "(X,Y)", even if it is outside the window. If you want to draw a line
from "(x1,y1)" to "(x2,y2)" where "(x1,y1)" is not necessarily the position of the virtual
cursor, use "plotmove(w,x1,y1)" before using this function.
plotlinetype"(w,type)"
change the type of lines subsequently plotted in rectwindow "w". type "-2" corresponds to
frames, "-1" to axes, larger values may correspond to something else. "w = -1" changes
highlevel plotting. This is only taken into account by the "gnuplot" interface.
plotmove"(w,x,y)"
move the virtual cursor of the rectwindow "w" to position "(x,y)".
plotpoints"(w,X,Y)"
draw on the rectwindow "w" the points whose "(x,y)"-coordinates are in the vectors of
equal length "X" and "Y" and which are inside "w". The virtual cursor does not move. This
is basically the same function as "plothraw", but either with no scaling factor or with a
scale chosen using the function "plotscale".
As was the case with the "plotlines" function, "X" and "Y" are allowed to be
(simultaneously) scalar. In this case, draw the single point "(X,Y)" on the rectwindow "w"
(if it is actually inside "w"), and in any case move the virtual cursor to position
"(x,y)".
plotpointsize"(w,size)"
changes the ``size'' of following points in rectwindow "w". If "w = -1", change it in all
rectwindows. This only works in the "gnuplot" interface.
plotpointtype"(w,type)"
change the type of points subsequently plotted in rectwindow "w". "type = -1" corresponds
to a dot, larger values may correspond to something else. "w = -1" changes highlevel
plotting. This is only taken into account by the "gnuplot" interface.
plotrbox"(w,dx,dy)"
draw in the rectwindow "w" the outline of the rectangle which is such that the points
"(x1,y1)" and "(x1+dx,y1+dy)" are opposite corners, where "(x1,y1)" is the current
position of the cursor. Only the part of the rectangle which is in "w" is drawn. The
virtual cursor does not move.
plotrecth"(w,X = a,b,expr,{flag = 0},{n = 0})"
writes to rectwindow "w" the curve output of "ploth""(w,X = a,b,expr,flag,n)".
plotrecthraw"(w,data,{flag = 0})"
plot graph(s) for data in rectwindow "w". "flag" has the same significance here as in
"ploth", though recursive plot is no more significant.
data is a vector of vectors, each corresponding to a list a coordinates. If parametric
plot is set, there must be an even number of vectors, each successive pair corresponding
to a curve. Otherwise, the first one containe the "x" coordinates, and the other ones
contain the "y"-coordinates of curves to plot.
plotrline"(w,dx,dy)"
draw in the rectwindow "w" the part of the segment "(x1,y1)-(x1+dx,y1+dy)" which is inside
"w", where "(x1,y1)" is the current position of the virtual cursor, and move the virtual
cursor to "(x1+dx,y1+dy)" (even if it is outside the window).
plotrmove"(w,dx,dy)"
move the virtual cursor of the rectwindow "w" to position "(x1+dx,y1+dy)", where "(x1,y1)"
is the initial position of the cursor (i.e. to position "(dx,dy)" relative to the initial
cursor).
plotrpoint"(w,dx,dy)"
draw the point "(x1+dx,y1+dy)" on the rectwindow "w" (if it is inside "w"), where
"(x1,y1)" is the current position of the cursor, and in any case move the virtual cursor
to position "(x1+dx,y1+dy)".
plotscale"(w,x1,x2,y1,y2)"
scale the local coordinates of the rectwindow "w" so that "x" goes from "x1" to "x2" and
"y" goes from "y1" to "y2" ("x2 < x1" and "y2 < y1" being allowed). Initially, after the
initialization of the rectwindow "w" using the function "plotinit", the default scaling is
the graphic pixel count, and in particular the "y" axis is oriented downwards since the
origin is at the upper left. The function "plotscale" allows to change all these defaults
and should be used whenever functions are graphed.
plotstring"(w,x,{flag = 0})"
draw on the rectwindow "w" the String "x" (see "Label se:strings"), at the current
position of the cursor.
flag is used for justification: bits 1 and 2 regulate horizontal alignment: left if 0,
right if 2, center if 1. Bits 4 and 8 regulate vertical alignment: bottom if 0, top if 8,
v-center if 4. Can insert additional small gap between point and string: horizontal if bit
16 is set, vertical if bit 32 is set (see the tutorial for an example).
plotterm"(term)"
sets terminal where high resolution plots go (this is currently only taken into account by
the "gnuplot" graphical driver). Using the "gnuplot" driver, possible terminals are the
same as in gnuplot. If term is "?", lists possible values.
Terminal options can be appended to the terminal name and space; terminal size can be put
immediately after the name, as in "gif = 300,200". Positive return value means success.
psdraw"(list)"
same as "plotdraw", except that the output is a PostScript program appended to the
"psfile".
psploth"(X = a,b,expr)"
same as "ploth", except that the output is a PostScript program appended to the "psfile".
psplothraw"(listx,listy)"
same as "plothraw", except that the output is a PostScript program appended to the
"psfile".
Programming under GP
=head2 Control statements.
A number of control statements are available under GP. They are simpler and have a syntax
slightly different from their C counterparts, but are quite powerful enough to write any
kind of program. Some of them are specific to GP, since they are made for number
theorists. As usual, "X" will denote any simple variable name, and seq will always denote
a sequence of expressions, including the empty sequence.
break"({n = 1})"
interrupts execution of current seq, and immediately exits from the "n" innermost
enclosing loops, within the current function call (or the top level loop). "n" must be
bigger than 1. If "n" is greater than the number of enclosing loops, all enclosing
loops are exited.
for"(X = a,b,seq)"
the formal variable "X" going from "a" to "b", the seq is evaluated. Nothing is done
if "a > b". "a" and "b" must be in R.
fordiv"(n,X,seq)"
the formal variable "X" ranging through the positive divisors of "n", the sequence seq
is evaluated. "n" must be of type integer.
forprime"(X = a,b,seq)"
the formal variable "X" ranging over the prime numbers between "a" to "b" (including
"a" and "b" if they are prime), the seq is evaluated. More precisely, the value of "X"
is incremented to the smallest prime strictly larger than "X" at the end of each
iteration. Nothing is done if "a > b". Note that "a" and "b" must be in R.
? { forprime(p = 2, 12,
print(p);
if (p == 3, p = 6);
)
}
2
3
7
11
forstep"(X = a,b,s,seq)"
the formal variable "X" going from "a" to "b", in increments of "s", the seq is
evaluated. Nothing is done if "s > 0" and "a > b" or if "s < 0" and "a < b". "s" must
be in "R^*" or a vector of steps "[s_1,...,s_n]". In the latter case, the successive
steps are used in the order they appear in "s".
? forstep(x=5, 20, [2,4], print(x))
5
7
11
13
17
19
forsubgroup"(H = G,{B},seq)"
executes seq for each subgroup "H" of the abelian group "G" (given in SNF form or as a
vector of elementary divisors), whose index is bounded by bound. The subgroups are not
ordered in any obvious way, unless "G" is a "p"-group in which case Birkhoff's
algorithm produces them by decreasing index. A subgroup is given as a matrix whose
columns give its generators on the implicit generators of "G". For example, the
following prints all subgroups of index less than 2 in "G = Z/2Z g_1 x Z/2Z g_2" :
? G = [2,2]; forsubgroup(H=G, 2, print(H))
[1; 1]
[1; 2]
[2; 1]
[1, 0; 1, 1]
The last one, for instance is generated by "(g_1, g_1 + g_2)". This routine is
intended to treat huge groups, when subgrouplist is not an option due to the sheer
size of the output.
For maximal speed the subgroups have been left as produced by the algorithm. To print
them in canonical form (as left divisors of "G" in HNF form), one can for instance use
? G = matdiagonal([2,2]); forsubgroup(H=G, 2, print(mathnf(concat(G,H))))
[2, 1; 0, 1]
[1, 0; 0, 2]
[2, 0; 0, 1]
[1, 0; 0, 1]
Note that in this last representation, the index "[G:H]" is given by the determinant.
forvec"(X = v,seq,{flag = 0})"
"v" being an "n"-component vector (where "n" is arbitrary) of two-component vectors
"[a_i,b_i]" for "1 <= i <= n", the seq is evaluated with the formal variable "X[1]"
going from "a_1" to "b_1",...,"X[n]" going from "a_n" to "b_n". The formal variable
with the highest index moves the fastest. If "flag = 1", generate only nondecreasing
vectors "X", and if "flag = 2", generate only strictly increasing vectors "X".
if"(a,{seq1},{seq2})"
if "a" is non-zero, the expression sequence seq1 is evaluated, otherwise the
expression seq2 is evaluated. Of course, seq1 or seq2 may be empty, so "if (a,seq)"
evaluates seq if "a" is not equal to zero (you don't have to write the second comma),
and does nothing otherwise, whereas "if (a,,seq)" evaluates seq if "a" is equal to
zero, and does nothing otherwise. You could get the same result using the "!" ("not")
operator: "if (!a,seq)".
Note that the boolean operators "&&" and "||" are evaluated according to operator
precedence as explained in "Label se:operators", but that, contrary to other
operators, the evaluation of the arguments is stopped as soon as the final truth value
has been determined. For instance
if (reallydoit && longcomplicatedfunction(), ...)%
is a perfectly safe statement.
Recall that functions such as "break" and "next" operate on loops (such as "forxxx",
"while", "until"). The "if" statement is not a loop (obviously!).
next"({n = 1})"
interrupts execution of current "seq", resume the next iteration of the innermost
enclosing loop, within the current fonction call (or top level loop). If "n" is
specified, resume at the "n"-th enclosing loop. If "n" is bigger than the number of
enclosing loops, all enclosing loops are exited.
return"({x = 0})"
returns from current subroutine, with result "x".
until"(a,seq)"
evaluates expression sequence seq until "a" is not equal to 0 (i.e. until "a" is
true). If "a" is initially not equal to 0, seq is evaluated once (more generally, the
condition on "a" is tested after execution of the seq, not before as in "while").
while"(a,seq)"
while "a" is non-zero evaluate the expression sequence seq. The test is made before
evaluating the "seq", hence in particular if "a" is initially equal to zero the seq
will not be evaluated at all.
Specific functions used in GP programming
In addition to the general PARI functions, it is necessary to have some functions which
will be of use specifically for GP, though a few of these can be accessed under library
mode. Before we start describing these, we recall the difference between strings and
keywords (see "Label se:strings"): the latter don't get expanded at all, and you can type
them without any enclosing quotes. The former are dynamic objects, where everything
outside quotes gets immediately expanded.
We need an additional notation for this chapter. An argument between braces, followed by a
star, like "{str}*", means that any number of such arguments (possibly none) can be given.
addhelp"(S,str)"
changes the help message for the symbol "S". The string str is expanded on the spot
and stored as the online help for "S". If "S" is a function you have defined, its
definition will still be printed before the message str. It is recommended that you
document global variables and user functions in this way. Of course GP won't protest
if you don't do it.
There's nothing to prevent you from modifying the help of built-in PARI functions (but
if you do, we'd like to hear why you needed to do it!).
alias"(newkey,key)"
defines the keyword newkey as an alias for keyword key. key must correspond to an
existing function name. This is different from the general user macros in that alias
expansion takes place immediately upon execution, without having to look up any
function code, and is thus much faster. A sample alias file "misc/gpalias" is provided
with the standard distribution. Alias commands are meant to be read upon startup from
the ".gprc" file, to cope with function names you are dissatisfied with, and should be
useless in interactive usage.
allocatemem"({x = 0})"
this is a very special operation which allows the user to change the stack size after
initialization. "x" must be a non-negative integer. If "x! = 0", a new stack of size
"16*\lceil x/16\rceil" bytes will be allocated, all the PARI data on the old stack
will be moved to the new one, and the old stack will be discarded. If "x = 0", the
size of the new stack will be twice the size of the old one.
Although it is a function, this must be the last instruction in any GP sequence. The
technical reason is that this routine usually moves the stack, so objects from the
current sequence might not be correct anymore. Hence, to prevent such problems, this
routine terminates by a "longjmp" (just as an error would) and not by a return.
The library syntax is allocatemoremem"(x)", where "x" is an unsigned long, and the
return type is void. GP uses a variant which ends by a "longjmp".
default"({key},{val},{flag})"
sets the default corresponding to keyword key to value val. val is a string (which of
course accepts numeric arguments without adverse effects, due to the expansion
mechanism). See "Label se:defaults" for a list of available defaults, and "Label
se:meta" for some shortcut alternatives. Typing "default()" (or "\d") yields the
complete default list as well as their current values.
If val is omitted, prints the current value of default key. If "flag" is set, returns
the result instead of printing it.
error"({str}*)"
outputs its argument list (each of them interpreted as a string), then interrupts the
running GP program, returning to the input prompt.
Example: "error("n = ", n, " is not squarefree !")".
Note that, due to the automatic concatenation of strings, you could in fact use only
one argument, just by suppressing the commas.
extern"(str)"
the string str is the name of an external command (i.e. one you would type from your
UNIX shell prompt). This command is immediately run and its input fed into GP, just
as if read from a file.
getheap"()"
returns a two-component row vector giving the number of objects on the heap and the
amount of memory they occupy in long words. Useful mainly for debugging purposes.
The library syntax is getheap"()".
getrand"()"
returns the current value of the random number seed. Useful mainly for debugging
purposes.
The library syntax is getrand"()", returns a C long.
getstack"()"
returns the current value of "top-avma", i.e. the number of bytes used up to now on
the stack. Should be equal to 0 in between commands. Useful mainly for debugging
purposes.
The library syntax is getstack"()", returns a C long.
gettime"()"
returns the time (in milliseconds) elapsed since either the last call to "gettime", or
to the beginning of the containing GP instruction (if inside GP), whichever came last.
The library syntax is gettime"()", returns a C long.
global"({list of variables})"
declares the corresponding variables to be global. From now on, you will be forbidden
to use them as formal parameters for function definitions or as loop indexes. This is
especially useful when patching together various scripts, possibly written with
different naming conventions. For instance the following situation is dangerous:
p = 3 \\ fix characteristic
...
forprime(p = 2, N, ...)
f(p) = ...
since within the loop or within the function's body (even worse: in the subroutines
called in that scope), the true global value of "p" will be hidden. If the statement
"global(p = 3)" appears at the beginning of the script, then both expressions will
trigger syntax errors.
Calling "global" without arguments prints the list of global variables in use. In
particular, "eval(global)" will output the values of all local variables.
input"()"
reads a string, interpreted as a GP expression, from the input file, usually standard
input (i.e. the keyboard). If a sequence of expressions is given, the result is the
result of the last expression of the sequence. When using this instruction, it is
useful to prompt for the string by using the "print1" function. Note that in the
present version 2.19 of "pari.el", when using GP under GNU Emacs (see "Label
se:emacs") one must prompt for the string, with a string which ends with the same
prompt as any of the previous ones (a "? " will do for instance).
install"(name,code,{gpname},{lib})"
loads from dynamic library lib the function name. Assigns to it the name gpname in
this GP session, with argument code code (see "Label se:gp.interface" for an
explanation of those). If lib is omitted, uses "libpari.so". If gpname is omitted,
uses name.
This function is useful for adding custom functions to the GP interpreter, or picking
useful functions from unrelated libraries. For instance, it makes the function
"system" obsolete:
? install(system, vs, sys, "libc.so")
? sys("ls gp*")
gp.c gp.h gp_rl.c
But it also gives you access to all (non static) functions defined in the PARI
library. For instance, the function "GEN addii(GEN x, GEN y)" adds two PARI integers,
and is not directly accessible under GP (it's eventually called by the "+" operator of
course):
? install("addii", "GG")
? addii(1, 2)
%1 = 3
Caution: This function may not work on all systems, especially when GP has been
compiled statically. In that case, the first use of an installed function will provoke
a Segmentation Fault, i.e. a major internal blunder (this should never happen with a
dynamically linked executable). Hence, if you intend to use this function, please
check first on some harmless example such as the ones above that it works properly on
your machine.
kill"(s)"
kills the present value of the variable, alias or user-defined function "s". The
corresponding identifier can now be used to name any GP object (variable or function).
This is the only way to replace a variable by a function having the same name (or the
other way round), as in the following example:
? f = 1
%1 = 1
? f(x) = 0
*** unused characters: f(x)=0
^----
? kill(f)
? f(x) = 0
? f()
%2 = 0
When you kill a variable, all objects that used it become invalid. You can still
display them, even though the killed variable will be printed in a funny way
(following the same convention as used by the library function "fetch_var", see "Label
se:vars"). For example:
? a^2 + 1
%1 = a^2 + 1
? kill(a)
? %1
%2 = #<1>^2 + 1
If you simply want to restore a variable to its ``undefined'' value (monomial of
degree one), use the quote operator: "a = 'a". Predefined symbols ("x" and GP
function names) cannot be killed.
print"({str}*)"
outputs its (string) arguments in raw format, ending with a newline.
print1"({str}*)"
outputs its (string) arguments in raw format, without ending with a newline (note that
you can still embed newlines within your strings, using the "\n" notation !).
printp"({str}*)"
outputs its (string) arguments in prettyprint (beautified) format, ending with a
newline.
printp1"({str}*)"
outputs its (string) arguments in prettyprint (beautified) format, without ending with
a newline.
printtex"({str}*)"
outputs its (string) arguments in TeX format. This output can then be used in a TeX
manuscript. The printing is done on the standard output. If you want to print it to a
file you should use "writetex" (see there).
Another possibility is to enable the "log" default (see "Label se:defaults"). You
could for instance do:
default(logfile, "new.tex");
default(log, 1);
printtex(result);
(You can use the automatic string expansion/concatenation process to have dynamic file
names if you wish).
quit"()"
exits GP.
read"({str})"
reads in the file whose name results from the expansion of the string str. If str is
omitted, re-reads the last file that was fed into GP. The return value is the result
of the last expression evaluated.
reorder"({x = []})"
"x" must be a vector. If "x" is the empty vector, this gives the vector whose
components are the existing variables in increasing order (i.e. in decreasing
importance). Killed variables (see "kill") will be shown as 0. If "x" is non-empty, it
must be a permutation of variable names, and this permutation gives a new order of
importance of the variables, for output only. For example, if the existing order is
"[x,y,z]", then after "reorder([z,x])" the order of importance of the variables, with
respect to output, will be "[z,y,x]". The internal representation is unaffected.
setrand"(n)"
reseeds the random number generator to the value "n". The initial seed is "n = 1".
The library syntax is setrand"(n)", where "n" is a "long". Returns "n".
system"(str)"
str is a string representing a system command. This command is executed, its output
written to the standard output (this won't get into your logfile), and control returns
to the PARI system. This simply calls the C "system" command.
trap"({e}, {rec}, {seq})"
tries to execute seq, trapping error "e", that is effectively preventing it from
aborting computations in the usual way; the recovery sequence rec is executed if the
error occurs and the evaluation of rec becomes the result of the command. If "e" is
omitted, all exceptions are trapped. Note in particular that hitting "^C" (Control-C)
raises an exception.
? \\ trap division by 0
? inv(x) = trap (gdiver2, INFINITY, 1/x)
? inv(2)
%1 = 1/2
? inv(0)
%2 = INFINITY
If seq is omitted, defines rec as a default action when encountering exception "e".
The error message is printed, as well as the result of the evaluation of rec, and the
control is given back to the GP prompt. In particular, current computation is then
lost.
The following error handler prints the list of all user variables, then stores in a
file their name and their values:
? { trap( ,
print(reorder);
write("crash", reorder);
write("crash", eval(reorder))) }
If no recovery code is given (rec is omitted) a so-called break loop will be started.
During a break loop, all commands are read and evaluated as during the main GP loop
(except that no history of results is kept).
To get out of the break loop, you can use "next", "break" or "return"; reading in a
file by "\r" will also terminate the loop once the file has been read ("read" will
remain in the break loop). If the error is not fatal ("^C" is the only non-fatal
error), "next" will continue the computation as if nothing had happened (except of
course, you may have changed GP state during the break loop); otherwise control will
come back to the GP prompt. After a user interrupt ("^C"), entering an empty input
line (i.e hitting the return key) has the same effect as "next".
Break loops are useful as a debugging tool to inspect the values of GP variables to
understand why a problem occurred, or to change GP behaviour (increase debugging
level, start storing results in a logfile, modify parameters...) in the middle of a
long computation (hit "^C", type in your modifications, then type "next").
If rec is the empty string "" the last default handler is popped out, and replaced by
the previous one for that error.
Note: The interface is currently not adequate for trapping individual exceptions. In
the current version 2.2.0, the following keywords are recognized, but the name list
will be expanded and changed in the future (all library mode errors can be trapped:
it's a matter of defining the keywords to GP, and there are currently far too many
useless ones):
"accurer": accuracy problem
"gdiver2": division by 0
"archer": not available on this architecture or operating system
"typeer": wrong type
"errpile": the PARI stack overflows
type"(x,{t})"
this is useful only under GP. If "t" is not present, returns the internal type number
of the PARI object "x". Otherwise, makes a copy of "x" and sets its type equal to
type "t", which can be either a number or, preferably since internal codes may
eventually change, a symbolic name such as "t_FRACN" (you can skip the "t_" part here,
so that "FRACN" by itself would also be all right). Check out existing type names with
the metacommand "\t".
GP won't let you create meaningless objects in this way where the internal structure
doesn't match the type. This function can be useful to create reducible rationals
(type "t_FRACN") or rational functions (type "t_RFRACN"). In fact it's the only way to
do so in GP. In this case, the created object, as well as the objects created from it,
will not be reduced automatically, making some operations a bit faster.
There is no equivalent library syntax, since the internal functions "typ" and "settyp"
are available. Note that "settyp" does not create a copy of "x", contrary to most PARI
functions. It also doesn't check for consistency. "settyp" just changes the type in
place and returns nothing. "typ" returns a C long integer. Note also the different
spellings of the internal functions ("set")"typ" and of the GP function "type", which
is due to the fact that "type" is a reserved identifier for some C compilers.
whatnow"(key)"
if keyword key is the name of a function that was present in GP version 1.39.15 or
lower, outputs the new function name and syntax, if it changed at all (387 out of 560
did).
write"(filename,{str*})"
writes (appends) to filename the remaining arguments, and appends a newline (same
output as "print").
write1"(filename,{str*})"
writes (appends) to filename the remaining arguments without a trailing newline (same
output as "print1").
writetex"(filename,{str*})"
as "write", in TeX format.
POD ERRORS
Hey! The above document had some coding errors, which are explained below:
Around line 7476:
'=item' outside of any '=over'
Around line 7628:
You forgot a '=back' before '=head2'
Around line 7643:
'=item' outside of any '=over'
perl v5.10.0 2011-06-20 libPARI(3pm)
Generated by $Id: phpMan.php,v 4.49 2006/02/26 13:18:18 chedong Exp $ Author: Che Dong
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