pari-doc 2.9.4-1 / usr / share / pari / doc / refcard.tex

% Copyright (c) 2007-2016 Karim Belabas.
% Permission is granted to copy, distribute and/or modify this document
% under the terms of the GNU General Public License

% Reference Card for PARI-GP.
% Author:
%  Karim Belabas
%  Universite de Bordeaux, 351 avenue de la Liberation, F-33405 Talence
%  email: Karim.Belabas@math.u-bordeaux.fr
%
% Based on an earlier version by Joseph H. Silverman who kindly let me
% use his original file.
% Thanks to Bill Allombert, Henri Cohen, Gerhard Niklasch, and Joe Silverman
% for comments and corrections.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% The original copyright notice read:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% Copyright (c) 1993,1994 Joseph H. Silverman. May be freely distributed.
%% Created Tuesday, July 27, 1993
%% Thanks to Stephen Gildea for the multicolumn macro package
%% which I modified from his GNU emacs reference card
%%
%% Original Thanks:
%%  I would like to thank Jim Delaney, Kevin Buzzard, Dan Lieman,
%%  and Jaap Top for sending me corrections.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% This file is intended to be processed by plain TeX (TeX82).
\def\TITLE{Pari-GP reference card}
\input refmacro.tex

Note: optional arguments are surrounded by braces $\{\}$.\hfill\break
To start the calculator, type its name in the terminal: \kbd{gp}\hfill\break
To exit \kbd{gp}, type \kbd{quit}, \kbd{\\q}, or \kbd{<C-D>} at prompt.\hfill
\section{Help}
\li{describe function}{?{\it function}}
\li{extended description}{??{\it keyword}}
\li{list of relevant help topics}{???{\it pattern}}
\li{name of GP-1.39 function $f$ in GP-2.*}{whatnow$(f)$}

\section{Input/Output}
\li{previous result, the result before}
  {\%{\rm, }\%`{\rm, }\%``{\rm, etc.}}
\li{$n$-th result since startup}{\%$n$}
\li{separate multiple statements on line}{;}
\li{extend statement on additional lines}{\\}
\li{extend statements on several lines}{\{$\seq_1$; $\seq_2$;\}}
\li{comment}{/* $\dots$ */}
\li{one-line comment, rest of line ignored}{\\\\ \dots}

\section{Metacommands \& Defaults}
\li{set default $d$ to \var{val}} {default$(\{d\},\{\var{val}\})$}
\li{toggle timer on/off}{\#}
\li{print time for last result}{\#\#}
\li{print defaults}{\\d}
\li{set debug level to $n$}{\\g $n$}
\li{set memory debug level to $n$}{\\gm $n$}
\li{set $n$ significant digits / bits}{\\p $n$, \\pb $n$}
\li{set $n$ terms in series}{\\ps $n$}
\li{quit GP}{\\q}
\li{print the list of PARI types}{\\t}
\li{print the list of user-defined functions}{\\u}
\li{read file into GP}{\\r {\it filename}}

\section{Debugger / break loop}
\li{get out of break loop}{break {\rm or} <C-D>}
\li{go up/down $n$ frames}{dbg\_up$(\{n\})${\rm, }dbg\_down}
\li{set break point}{breakpoint$()$}
\li{examine object $o$}{dbg\_x$(o)$}
\li{current error data}{dbg\_err$()$}
\li{number of objects on heap and their size}{getheap$()$}
\li{total size of objects on PARI stack}{getstack$()$}

\section{PARI Types \& Input Formats}
\li{\typ{INT}. Integers; hex, binary}{$\pm 31$; $\pm $0x1F, $\pm$0b101}
\li{\typ{REAL}. Reals}{$\pm 3.14$, $6.022$ E$23$}
\li{\typ{INTMOD}. Integers modulo $m$}{Mod$(n,m)$}
\li{\typ{FRAC}. Rational Numbers}{$n/m$}
\li{\typ{FFELT}. Elt in finite field $\F_q$}{ffgen(q)}
\li{\typ{COMPLEX}. Complex Numbers}{$x +y\,*\;$I}
\li{\typ{PADIC}. $p$-adic Numbers}{$x\;+\;$O$(p$\pow$k)$}
\li{\typ{QUAD}. Quadratic Numbers}{$x + y\,*\;$quadgen$(D)$}
\li{\typ{POLMOD}. Polynomials modulo $g$}{Mod$(f,g)$}
\li{\typ{POL}. Polynomials}{$a*x$\pow$n+\cdots+b$}
\li{\typ{SER}. Power Series}{$f\;+\;$O$(x$\pow$k)$}
\li{\typ{RFRAC}. Rational Functions}{$f/g$}
\li{\typ{QFI}/\typ{QFR}. Imag/Real binary quad.\ form}{Qfb$(a,b,c,\{d\})$}
\li{\typ{VEC}/\typ{COL}. Row/Column Vectors}
  {[$x,y,z$]{\rm, }[$x,y,z$]\til}
\li{\typ{VEC} integer range}{[1..10]}

% This goes at the bottom of page 1
\shortcopyrightnotice
\newcolumn

\li{\typ{VECSMALL}. Vector of small ints}{Vecsmall([$x,y,z$])}
\li{\typ{MAT}. Matrices}{[$a,b$;$c,d$]}
\li{\typ{LIST}. Lists}{List$($[$x,y,z$]$)$}
\li{\typ{STR}. Strings}{"abc"}
\li{\typ{INFINITY}. $\pm\infty$}{+oo, -oo}

\section{Reserved Variable Names}
\li{$\pi=3.14\dots$, $\gamma=0.57\dots$, $C=0.91\dots$}{Pi{\rm, }Euler{\rm, }Catalan}
\li{square root of $-1$}{I}
\li{Landau's big-oh notation}{O}


\section{Information about an Object}
\li{PARI type of object $x$}{type$(x)$}
\li{length of $x$ / size of $x$ in memory}{\#$x${\rm, }sizebyte$(x)$}
\li{real precision / bit precision of $x$}{precision$(x)${\rm, }bitprecision}
\li{$p$-adic, series prec. of $x$}{padicprec$(x)${\rm, }serprec}

\section{Operators}
\li{basic operations}{+{\rm,} - {\rm,} *{\rm,} /{\rm,} \pow{\rm,} sqr}
\li{\kbd{i=i+1}, \kbd{i=i-1}, \kbd{i=i*j}, \dots}
  {i++{\rm,} i--{\rm,} i*=j{\rm,}\dots}
\li{euclidean quotient, remainder}{$x$\bs/$y${\rm,} $x$\bs$y${\rm,}
$x$\%$y${\rm,} divrem$(x,y)$}
\li{shift $x$ left or right $n$ bits}{ $x$<<$n$, $x$>>$n$
  {\rm or} shift$(x,\pm n)$}
\li{multiply by $2^n$}{shiftmul$(x,n)$}
\li{comparison operators}
   {<={\rm, }<{\rm, }>={\rm, }>{\rm, }=={\rm, }!={\rm, }==={\rm, }lex{\rm, }cmp}
\li{boolean operators (or, and, not)}{||{\rm, } \&\&{\rm ,} !}
\li{bit operations}
   {bitand{\rm, }bitneg{\rm, }bitor{\rm, }bitxor{\rm, }bitnegimply}
\li{sign of $x=-1,0,1$}{sign$(x)$}
\li{maximum/minimum of $x$ and $y$}{max{\rm,} min$(x,y)$}
\li{derivative of $f$}{$f$'}
\li{differential operator}{diffop$(f,v,d,\{n=1\})$}
\li{quote operator (formal variable)}{'x}
\li{assignment}{x = \var{value}}
\li{simultaneous assignment $x\leftarrow v_1$, $y\leftarrow v_2$}{[x,y] = v}
\section{Select Components}
\li{$n$-th component of $x$}{component$(x,n)$}
\li{$n$-th component of vector/list $x$}{$x$[$n$]}
\li{components $a,a+1,\dots,b$ of vector $x$}{$x$[$a$..$b$]}
\li{$(m,n)$-th component of matrix $x$}{$x$[$m,n$]}
\li{row $m$ or column $n$ of matrix $x$}{$x$[$m,$]{\rm,} $x$[$,n$]}
\li{numerator/denominator of $x$}{numerator$(x)${\rm, }denominator}

\section{Random Numbers}
\li{random integer/prime in $[0,N[$}{random$(N)${\rm, }randomprime}
\li{get/set random seed}{getrand{\rm, }setrand$(s)$}

\section{Conversions}
\li{to vector, matrix, vec. of small ints}{Col{\rm/}Vec{\rm,}Mat{\rm,}Vecsmall}
\li{to list, set, map, string}{List{\rm,} Set{\rm,} Map{\rm,} Str}
\li{create PARI object $(x\mod y)$}{Mod$(x,y)$}
\li{make $x$ a polynomial of $v$}{Pol$(x,\{v\})$}
\li{as \kbd{Pol}, etc., starting with constant term}
   {Polrev{\rm, }Vecrev{\rm, }Colrev}
\li{make $x$ a power series of $v$}{Ser$(x,\{v\})$}
\li{string from bytes / from format+args}{Strchr{\rm, }Strprintf}
\li{TeX string}{Strtex$(x)$}
\li{convert $x$ to simplest possible type}{simplify$(x)$}
\li{object $x$ with real precision $n$}{precision$(x,n)$}
\li{object $x$ with bit precision $n$}{bitprecision$(x,n)$}
\li{set precision to $p$ digits in dynamic scope}{localprec$(p)$}
\li{set precision to $p$ bits in dynamic scope}{localbitprec$(p)$}

\subsec{Conjugates and Lifts}
\li{conjugate of a number $x$}{conj$(x)$}
\li{norm of $x$, product with conjugate}{norm$(x)$}
\li{$L^p$ norm of $x$ ($L^\infty$ if no $p$)}{normlp$(x,\{p\})$}
\li{square of $L^2$ norm of $x$}{norml2$(x)$}
\li{lift of $x$ from Mods and $p$-adics}{lift{\rm,} centerlift$(x)$}
\li{recursive lift}{liftall}
\li{lift all \typ{INT} and \typ{PADIC} ($\to$\typ{INT})}{liftint}
\li{lift all \typ{POLMOD} ($\to$\typ{POL})}{liftpol}

\section{Lists, Sets \& Maps}
  {\bf Sets} (= row vector with strictly increasing entries w.r.t. \kbd{cmp})\hfill\break
%
\li{intersection of sets $x$ and $y$}{setintersect$(x,y)$}
\li{set of elements in $x$ not belonging to $y$}{setminus$(x,y)$}
\li{union of sets $x$ and $y$}{setunion$(x,y)$}
\li{does $y$ belong to the set $x$}{setsearch$(x,y,\{\fl\})$}
\li{set of all $f(x,y)$, $x\in X$, $y\in Y$}{setbinop$(f,X,Y)$}
\li{is $x$ a set ?}{setisset$(x)$}

\subsec{Lists. {\rm create empty list: $L$ = \kbd{List}$()$}}
\li{append $x$ to list $L$}{listput$(L,x,\{i\})$}
\li{remove $i$-th component from list $L$}{listpop$(L,\{i\})$}
\li{insert $x$ in list $L$ at position $i$}{listinsert$(L,x,i)$}
\li{sort the list $L$ in place}{listsort$(L,\{\fl\})$}

\subsec{Maps. {\rm create empty dictionnary: $M$ = \kbd{Map}$()$}}
\li{attach value $v$ to key $k$}{mapput$(M,k,v)$}
\li{recover value attach to key $k$ or error}{mapget$(M,k)$}
\li{is key $k$ in the dict ? (set $v$ to $M(k)$)}
   {mapisdefined$(M,k,\{\&v\})$}
\li{remove $k$ from map domain}{mapdelete$(M,k)$}

\section{GP Programming}
\subsec{User functions and closures}
$x,y$ are formal parameters; $y$ defaults to \kbd{Pi} if parameter opitted;
$z,t$ are local variables (lexical scope), $z$ initialized to 1.
\hfil\break
 {\tt fun(x, y=Pi) = my(z=1, t); \var{seq}\hfil\break}
 {\tt fun = (x, y=Pi) -> my(z=1, t); \var{seq}\hfill\break}
\li{attach a help message to $f$}{addhelp$(f)$}
\li{undefine symbol $s$ (also kills help)}{kill$(s)$}
\subsec{Control Statements {\rm ($X$: formal parameter in expression \seq)}}
\li{if $a\ne0$, evaluate $\seq_1$, else $\seq_2$}{if$(a,\{\seq_1\},\{\seq_2\})$}
\smallskip

\li{eval.\ \seq\ for $a\le X\le b$}{for$(X=a,b,\seq)$}
\li{\dots for primes $a\le X\le b$}{forprime$(X=a,b,\seq)$}
\li{\dots for composites $a\le X\le b$}{forcomposite$(X=a,b,\seq)$}
\li{\dots for $a\le X\le b$ stepping $s$}{forstep$(X=a,b,s,\seq)$}
\li{\dots for $X$ dividing $n$}{fordiv$(n,X,\seq)$}
\li{multivariable {\tt for}, lex ordering}{forvec$(X=v,\seq)$}
\li{loop over partitions of $n$}{forpart$(p=n,\seq)$}
\li{loop over vectors $v$, $q(v)\leq B$; $q > 0$}{forqfvec$(v, q, b, \seq)$}
\li{loop over $H < G$ finite abelian group}{forsubgroup$(H=G)$}
\smallskip

\li{evaluate \seq\ until $a\ne0$}{until$(a,\seq)$}
\li{while $a\ne0$, evaluate \seq}{while$(a,\seq)$}
\li{exit $n$ innermost enclosing loops}{break$(\{n\})$}
\li{start new iteration of $n$-th enclosing loop}{next$(\{n\})$}
\li{return $x$ from current subroutine}{return$(\{x\})$}

\subsec{Exceptions, warnings}
\li{raise an exception / warn}{error$()$, warning$()$}
\li{type of error message $E$}{errname$(E)$}
\li{try $\seq_1$, evaluate $\seq_2$ on error}{iferr$(\seq_1, E, \seq_2)$}

\subsec{Functions with closure arguments / results}
\li{select from $v$ according to $f$}{select$(f, v)$}
\li{apply $f$ to all entries in $v$}{apply$(f, v)$}
\li{evaluate $f(a_1,\dots,a_n)$}{call$(f,a)$}
\li{evaluate $f(\dots f(f(a_1,a_2),a_3)\dots,a_n)$}{fold$(f,a)$}
\li{calling function as closure}{self$()$}

\subsec{Sums \& Products}
\li{sum $X=a$ to $X=b$, initialized at $x$}{sum$(X=a,b,\expr,\{x\})$}
\li{sum entries of vector $v$}{vecsum$(v)$}
\li{sum \expr\ over divisors of $n$}{sumdiv$(n,X,\expr)$}
\li{\dots assuming \expr\ multiplicative}{sumdivmult$(n,X,\expr)$}
\li{product $a\le X\le b$, initialized at $x$}{prod$(X=a,b,\expr,\{x\})$}
\li{product over primes $a\le X\le b$}{prodeuler$(X=a,b,\expr)$}

\subsec{Sorting}
\li{sort $x$ by $k$-th component}{vecsort$(x,\{k\},\{fl=0\})$}
\li{min.~$m$ of $x$ ($m=x[i]$), max.}{vecmin$(x,\{\&i\})${\rm, }vecmax}
\li{does $y$ belong to $x$, sorted wrt. $f$}{vecsearch$(x,y,\{f\})$}

\subsec{Input/Output}
\li{print with/without \kbd{\bs n}, \TeX\ format}
   {print{\rm, }print1{\rm, }printtex}
\li{print fields with separator}{printsep$(\var{sep},\dots)$,{\rm, }printsep1}
\li{formatted printing}{printf$()$}
\li{write \args\ to file}{write{\rm,} write1{\rm,} writetex$(\file,\args)$}
\li{write $x$ in binary format}{writebin$(\file,x)$}
\li{read file into GP}{read($\{\file\}$)}
\li{\dots return as vector of lines}{readvec($\{\file\}$)}
\li{\dots return as vector of strings}{readstr($\{\file\}$)}
\li{read a string from keyboard}{input$()$}

\subsec{Timers}
\li{CPU time in \var{ms} and reset timer}{gettime$()$}
\li{CPU time in \var{ms} since gp startup}{getabstime$()$}
\li{time in \var{ms} since UNIX Epoch}{getwalltime$()$}
\li{timeout command after $s$ seconds}{alarm$(s, \expr)$}

\subsec{Interface with system}
\li{allocates a new stack of $s$ bytes}{allocatemem$(\{s\})$}
\li{alias \var{old}\ to \var{new}}{alias$(\var{new},\var{old})$}
\li{install function from library}{install$(f,code,\{\var{gpf\/}\},\{\var{lib}\})$}
\li{execute system command $a$}{system$(a)$}
\li{as above, feed result to GP}{extern$(a)$}
\li{as above, return GP string}{externstr$(a)$}
\li{get \kbd{\$VAR} from environment}{getenv$($\kbd{"VAR"}$)$}
\li{expand env. variable in string}{Strexpand$(x)$}

%
\section{Parallel evaluation}
These functions evaluate their arguments in parallel (pthreads or MPI);
args.~must not access global variables and must be free of side
effects. Enabled if threading engine is not \emph{single} in gp
header.\hfil\break
\li{evaluate $f$ on $x[1],\dots,x[n]$}{parapply$(f,x)$}
\li{evaluate closures $f[1],\dots,f[n]$}{pareval$(f)$}
\li{as \kbd{select}}{parselect$(f,A,\{\fl\})$}
\li{as \kbd{sum}}{parsum$(i = a,b,\var{expr},\{x\})$}
\li{as \kbd{vector}}{parvector$(n,i,\{\var{expr}\})$}
\li{eval $f$ for $i=a,\dots,b$}
   {parfor$(i = a, \{b\}, f,\{r\}, \{f_2\})$}
\li{\dots for $p$ prime in $[a,b]$}
   {parforprime$(p = a, \{b\}, f,\{r\}, \{f_2\})$}
\li{\dots multivariate}
   {parforvec$(X = v, f,\{r\}, \{f_2\}, \{\fl\})$}
\li{declare $x$ as inline (allows to use as global)}{inline$(x)$}
\li{stop inlining}{uninline$()$}

\newcolumn
\title{\TITLE}

\centerline{(PARI-GP version \PARIversion)}

\section{Linear Algebra}
%
\li{dimensions of matrix $x$}{matsize$(x)$}
\li{concatenation of $x$ and $y$}{concat$(x,\{y\})$}
\li{extract components of $x$}{vecextract$(x,y,\{z\})$}
\li{transpose of vector or matrix $x$}{mattranspose$(x)$ {\rm or} $x$\til}
\li{adjoint of the matrix $x$}{matadjoint$(x)$}
\li{eigenvectors/values of matrix $x$}{mateigen$(x)$}
\li{characteristic/minimal polynomial of $x$}{charpoly$(x)${\rm, }minpoly}
\li{trace/determinant of matrix $x$}{trace$(x)${\rm, }matdet}
\li{Frobenius form of $x$}{matfrobenius$(x)$}
\li{QR decomposition}{matqr$(x)$}
\li{apply \kbd{matqr}'s transform to $v$}{mathouseholder$(Q,v)$}

\subsec{Constructors \& Special Matrices}
\li{$\{g(x)\colon x \in v~{\rm s.t.}~f(x)\}$}{[g(x) | x <- v, f(x)]}
\li{$\{x\colon x \in v~{\rm s.t.}~f(x)\}$}{[x | x <- v, f(x)]}
\li{$\{g(x)\colon x \in v\}$}{[g(x) | x <- v]}
\li{row vec.\ of \expr\ eval'ed at $1\le i\le n$}{vector$(n,\{i\},\{\expr\})$}
\li{col.\ vec.\ of \expr\ eval'ed at $1\le i\le n$}{vectorv$(n,\{i\},\{\expr\})$}
\li{vector of small ints}{vectorsmall$(n,\{i\},\{\expr\})$}
\li{$[c, c\cdot x, \dots, c\cdot x^n]$}{powers$(x,n,\{c = 1\})$}
\li{matrix $1\le i\le m$, $1\le j\le n$}{matrix$(m,n,\{i\},\{j\},\{\expr\})$}
\li{define matrix by blocks}{matconcat$(B)$}
\li{diagonal matrix with diagonal $x$}{matdiagonal$(x)$}
\li{is $x$ diagonal?}{matisdiagonal$(x)$}
\li{$x\,\cdot\, $\kbd{matdiagonal}$(d)$}{matmuldiagonal$(x,d)$}
\li{$n\times n$ identity matrix}{matid$(n)$}
\li{Hessenberg form of square matrix $x$}{mathess$(x)$}
\li{$n\times n$ Hilbert matrix $H_{ij}=(i+j-1)^{-1}$}{mathilbert$(n)$}
\li{$n\times n$ Pascal triangle}{matpascal$(n-1)$}
\li{companion matrix to polynomial $x$}{matcompanion$(x)$}
\li{Sylvester matrix of $x$}{polsylvestermatrix$(x)$}

\subsec{Gaussian elimination}
\li{kernel of matrix $x$}{matker$(x,\{\fl\})$}
\li{intersection of column spaces of $x$ and $y$}{matintersect$(x,y)$}
\li{solve $M*X = B$ ($M$ invertible)}{matsolve$(M,B)$}
\li{as solve, modulo $D$ (col. vector)}{matsolvemod$(M,D,B)$}
\li{one sol of $M*X = B$}{matinverseimage$(M,B)$}
\li{basis for image of matrix $x$}{matimage$(x)$}
\li{columns of $x$ \emph{not} in \kbd{matimage}}{matimagecompl$(x)$}
\li{supplement columns of $x$ to get basis}{matsupplement$(x)$}
\li{rows, cols to extract invertible matrix}{matindexrank$(x)$}
\li{rank of the matrix $x$}{matrank$(x)$}

\section{Lattices \& Quadratic Forms}
\subsec{Quadratic forms}
\li{evaluate $^tx Q y$}{qfeval$(\{Q=\var{id}\},x,y)$}
\li{evaluate $^tx Q x$}{qfeval$(\{Q=\var{id}\},x)$}
\li{signature of quad form $^ty*x*y$}{qfsign$(x)$}
\li{decomp into squares of $^ty*x*y$}{qfgaussred$(x)$}
\li{eigenvalues/vectors for real symmetric $x$}{qfjacobi$(x)$}

\subsec{HNF and SNF}
\li{upper triangular Hermite Normal Form}{mathnf$(x)$}
\li{HNF of $x$ where $d$ is a multiple of det$(x)$}{mathnfmod$(x,d)$}
\li{multiple of det$(x)$}{matdetint$(x)$}
\li{HNF of $(x \mid\,$\kbd{diagonal}$(D))$}{mathnfmodid$(x,D)$}
\li{elementary divisors of $x$}{matsnf$(x)$}
\li{elementary divisors of $\ZZ[a]/(f'(a))$}{poldiscreduced$(f)$}
\li{integer kernel of $x$}{matkerint$(x)$}
\li{$\ZZ$-module $\leftrightarrow$ $\QQ$-vector space}{matrixqz$(x,p)$}

\subsec{Lattices}
\li{LLL-algorithm applied to columns of $x$}{qflll$(x,\{\fl\})$}
\li{\dots for Gram matrix of lattice}{qflllgram$(x,\{\fl\})$}
\li{find up to $m$ sols of \kbd{qfnorm}$(x,y)\le b$}{qfminim$(x,b,m)$}
\li{$v$, $v[i]:=$number of $y$ s.t. \kbd{qfnorm}$(x,y)= i$}
    {qfrep$(x,B,\{\fl\})$}
\li{perfection rank of $x$}{qfperfection$(x)$}
\li{find isomorphism between $q$ and $Q$}{qfisom$(q,Q)$}
\li{precompute for isomorphism test with $q$}{qfisominit$(q)$}
\li{automorphism group of $q$}{qfauto$(q)$}
\li{convert \kbd{qfauto} for GAP/Magma}{qfautoexport$(G,\{\fl\})$}
\li{orbits of $V$ under $G\subset \text{GL}(V)$}
   {qforbits$(G,V)$}

\section{Polynomials \& Rational Functions}
\li{all defined polynomial variables}{variables$()$}
\li{get var.~of highest priority (higher than $v$)}
   {varhigher$(\var{name},\{v\})$}
\li{\dots of lowest priority (lower than $v$)}
   {varlower$(\var{name},\{v\})$}

\subsec{Coefficients, variables and basic operators}
\li{degree of $f$}{poldegree$(f)$}
\li{coeff. of degree $n$ of $f$, leading coeff.}{polcoeff$(f,n)${\rm, }pollead}
\li{main variable / all variables in $f$ }{variable$(f)${\rm, }variables$(f)$}
\li{replace $x$ by $y$ in $f$}{subst$(f,x,y)$}
\li{evaluate $f$ replacing vars by their value}{eval$(f)$}
\li{replace polynomial expr.~$T(x)$ by $y$ in $f$}{substpol$(f,T,y)$}
\li{replace $x_1,\dots,x_n$ by $y_1,\dots,y_n$ in $f$}{substvec$(f,x,y)$}
\li{reciprocal polynomial $x^{\deg f}f(1/x)$}{polrecip$(f)$}
\smallskip

\li{gcd of coefficients of $f$}{content$(f)$}
\li{derivative of $f$ w.r.t. $x$}{deriv$(f,\{x\})$}
\li{formal integral of $f$ w.r.t. $x$}{intformal$(f,\{x\})$}
\li{formal sum of $f$ w.r.t. $x$}{sumformal$(f,\{x\})$}

\subsec{Constructors \& Special Polynomials}
\li{interpolating pol.~eval.~at $a$}{polinterpolate$(X,\{Y\},\{a\})$}
\li{$P_n$, $T_n/U_n$, $H_n$}{pollegendre{\rm, }polchebyshev{\rm, }polhermite}
\li{$n$-th cyclotomic polynomial $\Phi_n$}{polcyclo$(n,\{v\})$}
\li{return $n$ if $f=\Phi_n$, else $0$}{poliscyclo$(f)$}
\li{is $f$ a product of cyclotomic polynomials?}{poliscycloprod$(f)$}
\li{Zagier's polynomial of index $(n,m)$}{polzagier$(n,m)$}

\subsec{Resultant, elimination}
\li{discriminant of polynomial $f$}{poldisc$(f)$}
\li{resultant $R = \text{Res}_v(f,g)$}{polresultant$(f,g,\{v\})$}
\li{$[u,v,R]$, $xu + yv = \text{Res}_v(f,g)$}{polresultantext$(x,y,\{v\})$}
\li{solve Thue equation $f(x,y)=a$}{thue$(t,a,\{sol\})$}
\li{initialize $t$ for Thue equation solver}{thueinit$(f)$}
\copyrightnotice

\newcolumn
\title{\TITLE}

\centerline{(PARI-GP version \PARIversion)}
\bigskip

\subsec{Roots and Factorization}
\li{complex roots of $f$}{polroots$(f)$}
\li{number of real roots of $f$ (in $[a,b]$)}{polsturm$(f,\{[a,b]\})$}
\li{real roots of $f$ (in $[a,b]$)}{polrootsreal$(f,\{[a,b]\})$}
\li{symmetric powers of roots of $f$ up to $n$}{polsym$(f,n)$}
\li{Graeffe transform of $f$, $g(x^2)=f(x)f(-x)$}{polgraeffe$(f)$}
\li{factor $f$}{factor$(f)$}
\li{factor $f\mod p$ / roots}{factormod$(f,p)${\rm, }polrootsmod}
\li{\dots using Cantor-Zassenhaus}{factorcantor$(f,p)$}
\li{factor $f$ over $\FF_{p^a}$ / roots}{factorff$(f,p,a)${\rm, }polrootsff}
\li{factor $f$ over $\QQ_p$ / roots}{factorpadic$(f,p,r)${\rm, }polrootspadic}
\li{cyclotomic factors of $f\in\QQ[X]$}{polcyclofactors$(f)$}


\li{find irreducible $T\in \FF_p[x]$, $\deg T = n$}{ffinit$(p,n,\{x\})$}
\li{$\#\{{\rm monic\ irred.}\ T\in \FF_q[x], \deg T = n\}$}{ffnbirred$(q,n)$}
\li{$p$-adic root of $f$ congruent to $a\mod p$}{padicappr$(f,a)$}
\li{Newton polygon of $f$ for prime $p$}{newtonpoly$(f,p)$}
\li{Hensel lift $A/\text{lc}(A) = \prod_i B[i]$ mod $p^e$}
   {polhensellift$(A,B,p,e)$}
\li{extensions of $\QQ_p$ of degree $N$}{padicfields$(p,N)$}

\section{Formal \& p-adic Series}
\li{truncate power series or $p$-adic number}{truncate$(x)$}
\li{valuation of $x$ at $p$}{valuation$(x,p)$}
\subsec{Dirichlet and Power Series}
\li{Taylor expansion around $0$ of $f$ w.r.t. $x$}{taylor$(f,x)$}
\li{$\sum a_kb_k t^k$ from $\sum a_kt^k$ and $\sum b_kt^k$}{serconvol$(a,b)$}
\li{$f=\sum a_k t^k$ from $\sum (a_k/k!)t^k$}{serlaplace$(f)$}
\li{reverse power series $F$ so $F(f(x))=x$}{serreverse$(f)$}
\li{Dirichlet series multiplication / division}{dirmul{\rm,} dirdiv$(x,y)$}
\li{Dirichlet Euler product ($b$ terms)}{direuler$(p=a,b,\expr)$}


\section{Transcendental and $p$-adic Functions}
\li{real, imaginary part of $x$}{real$(x)$, imag$(x)$}
\li{absolute value, argument of $x$}{abs$(x)$, arg$(x)$}
\li{square/nth root of $x$}{sqrt$(x)$, sqrtn$(x,n,\{$\&$z\})$}
\li{trig functions}{sin, cos, tan, cotan, sinc}
\li{inverse trig functions}{asin, acos, atan}
\li{hyperbolic functions}{sinh, cosh, tanh, cotanh}
\li{inverse hyperbolic functions}{asinh, acosh, atanh}
\li{log$(x)$, $e^x$, $e^x-1$ }{log{\rm, }exp{\rm, }expm1}
\li{Euler $\Gamma$ function, $\log \Gamma$, $\Gamma'/\Gamma$}
   {gamma{\rm, }lngamma{\rm, }psi}
\li{half-integer gamma function $\Gamma(n+1/2)$}{gammah$(n)$}
\li{Riemann's zeta $\zeta(s)=\sum n^{-s}$}{zeta$(s)$}
\li{multiple zeta value (MZV), $\zeta(s_1,\dots,s_k)$}{zetamult$(s)$}
\li{incomplete $\Gamma$ function ($y=\Gamma(s)$)}{incgam$(s,x,\{y\})$}
\li{complementary incomplete $\Gamma$}{incgamc$(s,x)$}
\li{exponential integral $\int_x^\infty e^{-t}/t\,dt$}{eint1$(x)$}
\li{error function $2/\sqrt\pi\int_x^\infty e^{-t^2}dt$}{erfc$(x)$}
\li{dilogarithm of $x$}{dilog$(x)$}
\li{$m$-th polylogarithm of $x$}{polylog$(m,x,\{\fl\})$}
\li{$U$-confluent hypergeometric function}{hyperu$(a,b,u)$}
\li{Bessel $J_n(x)$, $J_{n+1/2}(x)$}{besselj$(n,x)$, besseljh$(n,x)$}
\li{Bessel $I_\nu$, $K_\nu$, $H^1_\nu$, $H^2_\nu$, $N_\nu$}
{(bessel)i{\rm, }k{\rm, }h1{\rm, }h2{\rm, }n}
\li{Lambert $W$: $x$ s.t. $xe^x =y$}{lambertw$(y)$}
\li{Teichmuller character of $p$-adic $x$}{teichmuller$(x)$}

\newcolumn

\section{Iterations, Sums \& Products}

\subsec{Numerical integration for meromorphic functions}
Behaviour at endpoint for Double Exponential methods: either a scalar
($a\in\CC$, regular) or $\pm$\kbd{oo} (decreasing at least as $x^{-2}$) or
\beginindentedkeys
  \li{$(x-a)^{-\alpha}$ singularity}{$[a,\alpha]$}
  \li{exponential decrease $e^{-\alpha|x|}$}{$[\pm\infty,\alpha]$, $\alpha > 0$}
  \li{slow decrease $|x|^{\alpha}$}{$\dots \alpha < -1$}
  \li{oscillating as $\cos(kx))$}{$\alpha = k$\kbd{I}, $k > 0$}
  \li{oscillating as $\sin(kx))$}{$\alpha = -k$\kbd{I}, $k > 0$}
\endindentedkeys
\li{numerical integration}{intnum$(x=a,b,f,\{T\})$}
\li{weights $T$ for \kbd{intnum}}{intnuminit$(a,b,\{m\})$}
\li{weights $T$ incl.~kernel $K$}{intfuncinit$(a,b,K,\{m\})$}
\li{integrate $(2i\pi)^{-1}f$ on circle $|z - a| = R$}
   {intcirc$(x=a,R,f,\{T\})$}

\subsec{Other integration methods}
\li{$n$-point Gauss-Legendre}{intnumgauss$(x=a,b,f,\{n\})$}
\li{weights for $n$-point Gauss-Legendre}{intnumgaussinit$(\{n\})$}
\li{Romberg integration (low accuracy)}{intnumromb$(x=a,b,f,\{\fl\})$}

\subsec{Numerical summation}
\li{sum of series $f(n)$, $n\geq a$ (low accuracy)}{suminf$(n=a,\expr)$}
\li{sum of alternating/positive series}{sumalt{\rm,} sumpos}
\li{sum of series using Euler-Maclaurin}{sumnum$(n=a,f,\{T\})$}
\li{weights for \kbd{sumnum}, $a$ as in DE}
   {sumnuminit$(\{\infty, a\})$}
\li{sum of series by Monien summation}{sumnummonien$(n=a,f,\{T\})$}
\li{weights for \kbd{sumnummonien}}
   {sumnummonieninit$(\{\infty, a\})$}

\subsec{Products}
\li{product $a\le X\le b$, initialized at $x$}{prod$(X=a,b,\expr,\{x\})$}
\li{product over primes $a\le X\le b$}{prodeuler$(X=a,b,\expr)$}
\li{infinite product $a\le X\le\infty$}{prodinf$(X=a,\expr)$}

\subsec{Other numerical methods}
\li{real root of $f$ in $[a,b]$; bracketed root}{solve$(X=a,b,f)$}
\li{\dots by interval splitting}{solvestep$(X=a,b,f,\{\fl=0\})$}
\li{limit of $f(t)$, $t\to\infty$}{limitnum(f, \{k\}, \{alpha\})}
\li{asymptotic expansion of $f$ at $\infty$}{asympnum(f, \{k\}, \{alpha\})}
\li{numerical derivation w.r.t $x$: $f'(a)$}{derivnum$(x=a, f)$}
\li{evaluate continued fraction $F$ at $t$}
   {contfraceval$(F,t,\{L\})$}
\li{power series to cont.~fraction ($L$ terms)}
   {contfracinit$(S,\{L\})$}
\li{Pad\'{e} approximant (deg.~denom.~$\leq B$)}{bestapprPade$(S,\{B\})$}

\section{Elementary Arithmetic Functions}
\li{vector of binary digits of $|x|$}{binary$(x)$}
\li{bit number $n$ of integer $x$}{bittest$(x,n)$}
\li{Hamming weight of integer $x$}{hammingweight$(x)$}
\li{digits of integer $x$ in base $B$}{digits$(x,\{B=10\})$}
\li{sum of digits of integer $x$ in base $B$}{sumdigits$(x,\{B=10\})$}
\li{integer from digits}{fromdigits$(v,\{B=10\})$}
\li{ceiling/floor/fractional part}{ceil{\rm, }floor{\rm, }frac}
\li{round $x$ to nearest integer}{round$(x,\{$\&$e\})$}
\li{truncate $x$}{truncate$(x,\{$\&$e\})$}
\li{gcd/LCM of $x$ and $y$}{gcd$(x,y)$, lcm$(x,y)$}
\li{gcd of entries of a vector/matrix}{content$(x)$}

\subsec{Primes and Factorization}
\li{extra prime table}{addprimes$()$}
\li{add primes in $v$ to prime table}{addprimes$(v)$}
\li{remove primes from prime table}{removeprimes$(v)$}
\li{Chebyshev $\pi(x)$, $n$-th prime $p_n$}{primepi$(x)$, prime$(n)$}
\li{vector of first $n$ primes}{primes$(n)$}
\li{smallest prime $\ge x$}{nextprime$(x)$}
\li{largest prime $\le x$}{precprime$(x)$}
\li{factorization of $x$}{factor$(x,\{lim\})$}
\li{\dots selecting specific algorithms}{factorint$(x,\{\fl=0\})$}
\li{$n=df^2$, $d$ squarefree/fundamental}{core$(n,\{fl\})${\rm, }coredisc}
\li{recover $x$ from its factorization}{factorback$(f,\{e\})$}
\li{$x\in\ZZ$, $|x|\leq X$, $\gcd(N,P(x)) \geq N$}
   {zncoppersmith$(P,N,X,\{B\})$}

\subsec{Divisors and multiplicative functions}
\li{number of prime divisors $\omega(n)$ / $\Omega(n)$}
   {omega$(n)${\rm, }bigomega}
\li{divisors of $n$ / number of divisors $\tau(n)$}{divisors$(n)${\rm, }numdiv}
\li{sum of ($k$-th powers of) divisors of $n$}{sigma$(n,\{k\})$}
\li{M\"obius $\mu$-function}{moebius$(x)$}
\li{Ramanujan's $\tau$-function}{ramanujantau$(x)$}

\subsec{Combinatorics}
\li{factorial of $x$}{$x$!~{\rm or} factorial$(x)$}
\li{binomial coefficient $x\choose y$}{binomial$(x,y)$}
\li{Bernoulli number $B_n$ as real/rational}{bernreal$(n)${\rm, }bernfrac}
\li{Bernoulli polynomial $B_n(x)$}{bernpol$(n,\{x\})$}
\li{$n$-th Fibonacci number}{fibonacci$(n)$}
\li{Stirling numbers $s(n,k)$ and $S(n,k)$}{stirling$(n,k,\{\fl\})$}
\li{number of partitions of $n$}{numbpart$(n)$}
\li{$k$-th permutation on $n$ letters}{numtoperm$(n,k)$}
\li{convert permutation to $(n,k)$ form}{permtonum$(v)$}

\subsec{Multiplicative groups $(\ZZ/N\ZZ)^*$, $\FF_q^*$}
\li{Euler $\phi$-function}{eulerphi$(x)$}
\li{multiplicative order of $x$ (divides $o$)}{znorder$(x,\{o\})${\rm, }fforder}
\li{primitive root mod $q$ / $x$\kbd{.mod}}{znprimroot$(q)${\rm, }ffprimroot$(x)$}
\li{structure of $(\ZZ/n\ZZ)^*$}{znstar$(n)$}
\li{discrete logarithm of $x$ in base $g$}{znlog$(x,g,\{o\})${\rm, }fflog}
\li{Kronecker-Legendre symbol $({x\over y})$}{kronecker$(x,y)$}
\li{quadratic Hilbert symbol (at $p$)}{hilbert$(x,y,\{p\})$}

\subsec{Miscellaneous}
\li{integer square / $n$-th root of $x$}{sqrtint$(x)$, sqrtnint$(x,n)$}
\li{largest integer $e$ s.t. $b^e \leq b$, $e = \lfloor \log_b(x)\rfloor$}
   {logint$(x,b,\{\&z\})$}
\li{CRT: solve $z\equiv x$ and $z\equiv y$}{chinese$(x,y)$}
\li{minimal $u,v$ so $xu+yv=\gcd(x,y)$}{gcdext$(x,y)$}
\li{continued fraction of $x$}{contfrac$(x,\{b\},\{lmax\})$}
\li{last convergent of continued fraction $x$}{contfracpnqn$(x)$}
\li{rational approximation to $x$ (den. $\leq B$)}{bestappr$(x,\{B\}k)$}

\section{Characters}
Let \var{cyc} $=[d_1,\dots,d_k]$ represent an abelian group $G = \oplus
(\ZZ/d_j\ZZ)\cdot g_j$ or any
structure $G$ affording a \kbd{.cyc} method; e.g. \kbd{idealstar}$(,q)$
for Dirichlet characters. A character $\chi$ is coded by
$[c_1,\dots,c_k]$ such that $\chi(g_j) = e(n_j/d_j)$.
\hfil\break
\li{$\chi\cdot \psi$; $\chi^{-1}$; $\chi\cdot \psi^{-1}$}
   {charmul{\rm, }charconj{\rm, }chardiv}
\li{order of $\chi$}{charorder$(\var{cyc}, \chi)$}
\li{kernel of $\chi$}{charker$(\var{cyc},\chi)$}
\li{$\chi(x)$, $G$ a GP group structure}{chareval$(G, \chi, x, \{z\})$}

\subsec{Dirichlet Characters}
\li{initialize $G = (\ZZ/q\ZZ)^*$}{G = idealstar$(,q)$}
\li{is $\chi$ odd?}{zncharisodd$(G,\chi)$}
\li{real $\chi \to $ Kronecker symbol $(D/.)$}{znchartokronecker$(G,\chi)$}
\li{induce $\chi\in \hat{G}$ to $\ZZ/N\ZZ$}{zncharinduce$(G,\var{chi},N)$}

\subsec{Conrey labelling}
\li{Conrey label $m\in (\ZZ/q\ZZ)^* \to\;$ character}{znconreychar$(G,m)$}
\li{character $\to$ Conrey label}{znconreyexp$(G,\chi)$}
\li{log on Conrey generators}{znconreylog$(G, m)$}
\li{conductor of $\chi$ ($\chi_0$ primitive)}
   {znconreyconductor$(G,\chi, \{\chi_0\})$}

\section{True-False Tests}
\li{is $x$ the disc. of a quadratic field?}{isfundamental$(x)$}
\li{is $x$ a prime?}{isprime$(x)$}
\li{is $x$ a strong pseudo-prime?}{ispseudoprime$(x)$}
\li{is $x$ square-free?}{issquarefree$(x)$}
\li{is $x$ a square?}{issquare$(x,\{\&n\})$}
\li{is $x$ a perfect power?}{ispower$(x,\{k\},\{\&n\})$}
\li{is $x$ a perfect power of a prime? ($x=p^n$)}{isprimepower$(x,\&n\})$}
\li{\dots of a pseudoprime?}{ispseudoprimepower$(x,\&n\})$}
\li{is $x$ powerful?}{ispowerful$(x)$}
\li{is $x$ a totient? ($x=\varphi(n)$)}{istotient$(x,\{\&n\})$}
\li{is $x$ a polygonal number? ($x = P(s,n)$)}{ispolygonal$(x, s, \{\&n\})$}

\li{is \var{pol}\ irreducible?}{polisirreducible$(\var{pol})$}

\section{Graphic Functions}
\li{crude graph of \expr\ between $a$ and $b$}{plot$(X=a,b,expr)$}
\subsec{High-resolution plot {\rm (immediate plot)}}
\li{plot \expr\ between $a$ and $b$}{ploth$(X=a,b,expr,\{\fl\},\{n\})$}
\li{plot points given by lists $lx$, $ly$}{plothraw$(lx,ly,\{\fl\})$}
\li{terminal dimensions}{plothsizes$()$}
%
\subsec{Rectwindow functions}
\li{init window $w$, with size $x$,$y$}{plotinit$(w,x,y)$}
\li{erase window $w$}{plotkill$(w)$}
\li{copy $w$ to $w_2$ with offset $(dx,dy)$}{plotcopy$(w,w_2,dx,dy)$}
\li{clips contents of $w$}{plotclip$(w)$}
\li{scale coordinates in $w$}{plotscale$(w,x_1,x_2,y_1,y_2)$}
\li{\kbd{ploth} in $w$}{plotrecth$(w,X=a,b,expr,\{\fl\},\{n\})$}
\li{\kbd{plothraw} in $w$}{plotrecthraw$(w,data,\{\fl\})$}
\li{draw window $w_1$ at $(x_1,y_1)$, \dots} {plotdraw$($[[$w_1,x_1,y_1$]$,\dots$]$)$}
%
\subsec{Low-level Rectwindow Functions}
%\li{}{plotlinetype$(w,)$}
%\li{}{plotpointtype$(w,)$}
%\li{}{plotterm$(w,)$}
\li{set current drawing color in $w$ to $c$}{plotcolor$(w,c)$}
\li{current position of cursor in $w$}{plotcursor$(w)$}
%
\li{write $s$ at cursor's position}{plotstring$(w,s)$}
\li{move cursor to $(x,y)$}{plotmove$(w,x,y)$}
\li{move cursor to $(x+dx,y+dy)$}{plotrmove$(w,dx,dy)$}
\li{draw a box to $(x_2,y_2)$}{plotbox$(w,x_2,y_2)$}
\li{draw a box to $(x+dx,y+dy)$}{plotrbox$(w,dx,dy)$}
\li{draw polygon}{plotlines$(w,lx,ly,\{\fl\})$}
\li{draw points}{plotpoints$(w,lx,ly)$}
\li{draw line to $(x+dx,y+dy)$}{plotrline$(w,dx,dy)$}
\li{draw point $(x+dx,y+dy)$}{plotrpoint$(w,dx,dy)$}
\li{draw point $(x+dx,y+dy)$}{plotrpoint$(w,dx,dy)$}
%
\subsec{Postscript Functions}
\li{as {\tt ploth}}{psploth$(X=a,b,expr,\{\fl\},\{n\})$}
\li{as {\tt plothraw}}{psplothraw$(lx,ly,\{\fl\})$}
\li{as {\tt plotdraw}}{psdraw$($[[$w_1,x_1,y_1$]$,\dots$]$)$}

% This goes at the bottom of the second page (column 6)
\copyrightnotice
\bye