Annotation of OpenXM_contrib/pari-2.2/doc/refcard.tex, Revision 1.1
1.1 ! noro 1: % $Id: refcard.tex,v 1.8 2001/08/28 17:00:00 karim Exp $
! 2: % This file is intended to be processed by plain TeX (TeX82).
! 3: % Reference Card for PARI-GP version 2.1
! 4:
! 5: % Copyright (c) 1997-2000 Karim Belabas.
! 6: % Permission is granted to copy, distribute and/or modify this document
! 7: % under the terms of the GNU Free Documentation License
! 8:
! 9: % Based on an earlier version by Joseph H. Silverman who kindly let me
! 10: % use his original file.
! 11: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
! 12: % The original copyright notice read:
! 13: %
! 14: %% Copyright (c) 1993,1994 Joseph H. Silverman. May be freely distributed.
! 15: %% Created Tuesday, July 27, 1993
! 16: %% Thanks to Stephen Gildea for the multicolumn macro package
! 17: %% which I modified from his GNU emacs reference card
! 18: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
! 19: \def\TITLE{Pari-GP reference card}
! 20: % ignore parimacro.tex's \magnification setting
! 21: \let\oldmagnification\magnification
! 22: \catcode`@=11
! 23: \def\magnification{\count@}%
! 24: \catcode`@=12
! 25: \input parimacro.tex
! 26: \let\magnification\oldmagnification
! 27: \ifPDF
! 28: \input pdfmacs.tex
! 29: \pdfpagewidth=11.69in
! 30: \pdfpageheight=8.26in
! 31: \fi
! 32:
! 33: %**start of header
! 34: \newcount\columnsperpage
! 35: % The final reference card has six columns, three on each side.
! 36: % This file can be used to produce it in any of three ways:
! 37: % 1 column per page
! 38: % produces six separate pages, each of which needs to be reduced to 80%.
! 39: % This gives the best resolution.
! 40: % 2 columns per page
! 41: % produces three already-reduced pages.
! 42: % You will still need to cut and paste.
! 43: % 3 columns per page
! 44: % produces two pages which must be printed sideways to make a
! 45: % ready-to-use 8.5 x 11 inch reference card.
! 46: % For this you need a dvi device driver that can print sideways.
! 47: % [For 2 or 3 columns, you'll need 6 and 8 point fonts.]
! 48: % Which mode to use is controlled by setting \columnsperpage above.
! 49: %
! 50: % Specify how many columns per page you want here:
! 51: \columnsperpage=3
! 52:
! 53: % You shouldn't need to modify anything below this line.
! 54: %
! 55: % Author:
! 56: % Karim Belabas
! 57: % Universite Paris Sud
! 58: % Departement de Mathematiques (bat. 425)
! 59: % F-91405 Orsay
! 60: % Internet: Karim.Belabas@math.u-psud.fr
! 61: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
! 62: % (original reference card by Joseph H. Silverman)
! 63: % (original reference card macros due to Stephen Gildea)
! 64: % Original Thanks and History:
! 65: %
! 66: %% Thanks:
! 67: %% I would like to thank Jim Delaney, Kevin Buzzard, Dan Lieman,
! 68: %% and Jaap Top for sending me corrections.
! 69: %%
! 70: %% History:
! 71: %% Version 1.0 - July 1993, first general distribution
! 72: %% Version 1.1 - April 1994, corrected six typos
! 73: %% Version 1.2 - January 1995, minor corrections and additions
! 74: %% Version 1.3 - January 1996, minor corrections and additions
! 75: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
! 76: % Version 2.0 - November 1997, general distribution for GP 2.0
! 77: % Version 2.1 - January 1998, set nf,bnf,etc in a sensible font, updated default
! 78: % Version 2.2 - March 1998, some new functions (modpr, bnrstark), updated
! 79: % concat, removed spurious tabs.
! 80: % Version 2.3 - May 1998, added write1, corrected my email address.
! 81: % Version 2.4 - July 1998, removed vecindexsort, added ellrootno, updated
! 82: % elllseries
! 83: % Version 2.5 - October 1998, updated elliptic functions. Added quadray and
! 84: % user member functions
! 85: % Version 2.6 - December 1998, added local() keyword
! 86: % Version 2.7 - February 1999, added some pointer '&' arguments. Removed
! 87: % rounderror
! 88: % Version 2.8 - April 1999, removed \k, added \l filename
! 89: % Version 2.9 - April 2000, added \o¸ updated control statements
! 90: % Version 2.10 - June 2000, updated polinterpolate
! 91: % Version 2.11 - November 2000, changed Copyright
! 92: %% Thanks to Bill Allombert, Henri Cohen, Gerhard Niklasch, and Joe
! 93: %% Silverman for many comments and corrections.
! 94:
! 95: \def\versionnumber{2.11}% Version of this reference card
! 96: \def\PARIversion{2.1.0}% Version of PARI described on this reference card
! 97: \def\year{2000}
! 98: \def\month{November}
! 99: \def\version{\month\ \year\ v\versionnumber}
! 100:
! 101: \def\shortcopyrightnotice{\vskip .5ex plus 2 fill
! 102: \centerline{\small \copyright\ \year\ Karim Belabas.
! 103: Permissions on back. v\versionnumber}}
! 104:
! 105: \def\<#1>{$\langle${#1}$\rangle$}
! 106: \def\copyrightnotice{\vskip 1ex plus 2 fill
! 107: \begingroup\small
! 108: \centerline{Based on an earlier version by Joseph H. Silverman}
! 109: \centerline{\version. Copyright \copyright\ \year\ K. Belabas}
! 110: \centerline{GP copyright by The PARI Group}
! 111:
! 112: Permission is granted to make and distribute copies of this card provided the
! 113: copyright and this permission notice are preserved on all copies.
! 114:
! 115: Send comments and corrections to \<Karim.BELABAS@math.u-psud.fr>
! 116: \endgroup}
! 117:
! 118: % make \bye not \outer so that the \def\bye in the \else clause below
! 119: % can be scanned without complaint.
! 120: \def\bye{\par\vfill\supereject\end}
! 121:
! 122: \newdimen\intercolumnskip
! 123: \newbox\columna
! 124: \newbox\columnb
! 125:
! 126: \def\ncolumns{\the\columnsperpage}
! 127:
! 128: \message{[\ncolumns\space
! 129: column\if 1\ncolumns\else s\fi\space per page]}
! 130:
! 131: \def\scaledmag#1{ scaled \magstep #1}
! 132:
! 133: % This multi-way format was designed by Stephen Gildea
! 134: % October 1986.
! 135: \if 1\ncolumns
! 136: \hsize 4in
! 137: \vsize 10in
! 138: \voffset -.7in
! 139: \font\titlefont=\fontname\tenbf \scaledmag3
! 140: \font\headingfont=\fontname\tenbf \scaledmag2
! 141: \font\smallfont=\fontname\sevenrm
! 142: \font\smallsy=\fontname\sevensy
! 143:
! 144: \footline{\hss\folio}
! 145: \def\makefootline{\baselineskip10pt\hsize6.5in\line{\the\footline}}
! 146: \else
! 147: \hsize 3.2in
! 148: % \vsize 7.95in
! 149: \vsize 7.90in
! 150: \hoffset -.75in
! 151: % \voffset -.745in
! 152: \voffset -.815in
! 153: \font\titlefont=cmbx10 \scaledmag2
! 154: \font\headingfont=cmbx10 \scaledmag1
! 155: \font\smallfont=cmr6
! 156: \font\smallsy=cmsy6
! 157: \font\eightrm=cmr8
! 158: \font\eightbf=cmbx8
! 159: \font\eightit=cmti8
! 160: \font\eighttt=cmtt8
! 161: \font\eightsy=cmsy8
! 162: \font\eightsl=cmsl8
! 163: \font\eighti=cmmi8
! 164: \font\eightex=cmex10 at 8pt
! 165: \textfont0=\eightrm
! 166: \textfont1=\eighti
! 167: \textfont2=\eightsy
! 168: \textfont3=\eightex
! 169: \def\rm{\fam0 \eightrm}
! 170: \def\bf{\eightbf}
! 171: \def\it{\eightit}
! 172: \def\tt{\eighttt}
! 173: \normalbaselineskip=.8\normalbaselineskip
! 174: \normallineskip=.8\normallineskip
! 175: \normallineskiplimit=.8\normallineskiplimit
! 176: \normalbaselines\rm %make definitions take effect
! 177:
! 178: \if 2\ncolumns
! 179: \let\maxcolumn=b
! 180: \footline{\hss\rm\folio\hss}
! 181: \def\makefootline{\vskip 2in \hsize=6.86in\line{\the\footline}}
! 182: \else \if 3\ncolumns
! 183: \let\maxcolumn=c
! 184: \nopagenumbers
! 185: \else
! 186: \errhelp{You must set \columnsperpage equal to 1, 2, or 3.}
! 187: \errmessage{Illegal number of columns per page}
! 188: \fi\fi
! 189:
! 190: \intercolumnskip=.46in
! 191: \def\abc{a}
! 192: \output={%
! 193: % This next line is useful when designing the layout.
! 194: %\immediate\write16{Column \folio\abc\space starts with \firstmark}
! 195: \if \maxcolumn\abc \multicolumnformat \global\def\abc{a}
! 196: \else\if a\abc
! 197: \global\setbox\columna\columnbox \global\def\abc{b}
! 198: %% in case we never use \columnb (two-column mode)
! 199: \global\setbox\columnb\hbox to -\intercolumnskip{}
! 200: \else
! 201: \global\setbox\columnb\columnbox \global\def\abc{c}\fi\fi}
! 202: \def\multicolumnformat{\shipout\vbox{\makeheadline
! 203: \hbox{\box\columna\hskip\intercolumnskip
! 204: \box\columnb\hskip\intercolumnskip\columnbox}
! 205: \makefootline}\advancepageno}
! 206: \def\columnbox{\leftline{\pagebody}}
! 207:
! 208: \def\bye{\par\vfill\supereject
! 209: \if a\abc \else\null\vfill\eject\fi
! 210: \if a\abc \else\null\vfill\eject\fi
! 211: \end}
! 212: \fi
! 213:
! 214: % we won't be using math mode much, so redefine some of the characters
! 215: % we might want to talk about
! 216: %\catcode`\^=12
! 217: %\catcode`\_=12
! 218: %\catcode`\~=12
! 219:
! 220: \chardef\\=`\\
! 221: \chardef\{=`\{
! 222: \chardef\}=`\}
! 223:
! 224: \hyphenation{}
! 225:
! 226: \parindent 0pt
! 227: \parskip 0pt
! 228:
! 229: \def\small{\smallfont\textfont2=\smallsy\baselineskip=.8\baselineskip}
! 230:
! 231: \outer\def\newcolumn{\vfill\eject}
! 232:
! 233: \outer\def\title#1{{\titlefont\centerline{#1}}}
! 234:
! 235: \outer\def\section#1{\par\filbreak
! 236: \vskip 1.4ex plus .4ex minus .5ex
! 237: {\headingfont #1}\mark{#1}%
! 238: \vskip .7ex plus .3ex minus .5ex
! 239: }
! 240:
! 241: \outer\def\subsec#1{\filbreak
! 242: \vskip 0.1ex plus 0.05ex
! 243: {\bf #1}
! 244: \vskip 0.04ex plus 0.05ex
! 245: }
! 246:
! 247: \newdimen\keyindent
! 248: \def\beginindentedkeys{\keyindent=1em}
! 249: \def\endindentedkeys{\keyindent=0em}
! 250: \def\begindoubleindentedkeys{\keyindent=2em}
! 251: \def\enddoubleindentedkeys{\keyindent=1em}
! 252: \endindentedkeys
! 253:
! 254: \def\kbd#1{{\tt#1}\null} %\null so not an abbrev even if period follows
! 255: \def\var#1{\hbox{\it #1}}
! 256: \def\fl{\{\var{f{}l\/}\}}
! 257:
! 258: \def\key#1#2{\leavevmode\hbox to \hsize{\vtop
! 259: {\hsize=.75\hsize\rightskip=1em
! 260: \hskip\keyindent\relax#1}\kbd{#2}\hfil}}
! 261:
! 262: \newbox\libox
! 263: \setbox\libox\hbox{\kbd{M-x }}
! 264: \newdimen\liwidth
! 265: \liwidth=\wd\libox
! 266:
! 267: \def\li#1#2{\leavevmode\hbox to \hsize{\hbox to .75\hsize
! 268: {\hskip\keyindent\relax#1\hfil}%
! 269: \hskip -\liwidth minus 1fil
! 270: \kbd{#2}\hfil}}
! 271:
! 272: \def\threecol#1#2#3{\hskip\keyindent\relax#1\hfil&\kbd{#2}\quad
! 273: &\kbd{#3}\quad\cr}
! 274:
! 275: \def\mod{\;\hbox{\rm mod}\;}
! 276: \def\expr{\hbox{\it expr}}
! 277: \def\seq{\hbox{\it seq}}
! 278: \def\args{\hbox{\it args}}
! 279: \def\file{\hbox{\it file}}
! 280: \def\QQ{\hbox{\bf Q}}
! 281: \def\ZZ{\hbox{\bf Z}}
! 282: \def\RR{\hbox{\bf R}}
! 283: \def\FF{\hbox{\bf F}}
! 284: \def\CC{\hbox{\bf C}}
! 285: \def\deg{\mathop{\rm deg}}
! 286: \def\bs{\char'134}
! 287: \def\pow{\^{}\hskip0pt}
! 288: \def\til{\raise-0.3em\hbox{\~{}}}
! 289: \def\typ#1{\kbd{t\_#1}}
! 290: %**end of header
! 291:
! 292: \title{PARI-GP Reference Card}
! 293: \centerline{(PARI-GP version \PARIversion)}
! 294: Note: optional arguments are surrounded by braces {\tt \{\}}.
! 295:
! 296: \section{Starting \& Stopping GP}
! 297: \key{to enter GP, just type its name:}{gp}
! 298: \key{to exit GP, type}{\\q {\rm or }quit}
! 299:
! 300: \section{Help}
! 301: \li{describe function}{?{\rm function}}
! 302: \li{extended description}{??{\rm keyword}}
! 303: \li{list of relevant help topics}{???{\rm pattern}}
! 304:
! 305: \section{Input/Output \& Defaults}
! 306: \li{output previous line, the lines before}
! 307: {\%{\rm, }\%`{\rm, }\%``{\rm, etc.}}
! 308: \key{output from line $n$}{\%$n$}
! 309: \key{separate multiple statements on line}{;}
! 310: \key{extend statement on additional lines}{\\}
! 311: \key{extend statements on several lines}{\{\seq1; \seq2;\}}
! 312: \key{comment}{/* \dots */}
! 313: \key{one-line comment, rest of line ignored}{\\\\ \dots}
! 314: \li{set default $d$ to \var{val}} {default$(\{d\},\{\var{val}\},\fl)$}
! 315: \li{mimic behaviour of GP 1.39} {default(compatible,3)}
! 316:
! 317: \section{Metacommands}
! 318: \key{toggle timer on/off}{\#}
! 319: \key{print time for last result}{\#\#}
! 320: \key{print \%$n$ in raw format}{\\a $n$}
! 321: \key{print \%$n$ in pretty format}{\\b $n$}
! 322: \key{print defaults}{\\d}
! 323: \key{set debug level to $n$}{\\g $n$}
! 324: \key{set memory debug level to $n$}{\\gm $n$}
! 325: \key{enable/disable logfile}{\\l \{filename\}}
! 326: \key{print \%$n$ in pretty matrix format}{\\m}
! 327: \key{set output mode (raw, default, prettyprint)}{\\o $n$}
! 328: \key{set $n$ significant digits}{\\p $n$}
! 329: \key{set $n$ terms in series}{\\ps $n$}
! 330: \key{quit GP}{\\q}
! 331: \key{print the list of PARI types}{\\t}
! 332: \key{print the list of user-defined functions}{\\u}
! 333: \li{read file into GP}{\\r {\rm filename}}
! 334: \li{write \%$n$ to file}{\\w $n$ {\rm filename}}
! 335:
! 336: \section{GP Within Emacs}
! 337: \li{to enter GP from within Emacs:}{M-x gp{\rm,} C-u M-x gp}
! 338: \li{word completion}{<TAB>}
! 339: \li{help menu window}{M-\\c}
! 340: \li{describe function}{M-?}
! 341: \li{display \TeX'd PARI manual}{M-x gpman}
! 342: \li{set prompt string}{M-\\p}
! 343: \li{break line at column 100, insert \kbd{\\}}{M-\\\\}
! 344: \li{PARI metacommand \kbd{\\}{\it letter}}{M-\\\hbox{\it letter}}
! 345:
! 346: \section{Reserved Variable Names}
! 347: \li{$\pi=3.14159\cdots$}{Pi}
! 348: \li{Euler's constant ${}=.57721\cdots$}{Euler}
! 349: \li{square root of $-1$}{I}
! 350: \li{big-oh notation}{O}
! 351:
! 352: % ****************************************
! 353: % This goes at the bottom of page 1
! 354: \shortcopyrightnotice
! 355: \newcolumn
! 356:
! 357: \section{PARI Types \& Input Formats}
! 358: \li{\typ{INT}. Integers}{$\pm n$}
! 359: \li{\typ{REAL}. Real Numbers}{$\pm n.ddd$}
! 360: \li{\typ{INTMOD}. Integers modulo $m$}{Mod$(n,m)$}
! 361: \li{\typ{FRAC}. Rational Numbers}{$n/m$}
! 362: \li{\typ{COMPLEX}. Complex Numbers}{$x+\kbd{I}*y$}
! 363: \li{\typ{PADIC}. $p$-adic Numbers}{$x+O(p$\pow$k)$}
! 364: \li{\typ{QUAD}. Quadratic Numbers}{$x + y\,*\;$quadgen$(D)$}
! 365: \li{\typ{POLMOD}. Polynomials modulo $g$}{Mod$(f,g)$}
! 366: \li{\typ{POL}. Polynomials}{$a*x$\pow$n+\cdots+b$}
! 367: \li{\typ{SER}. Power Series}{$f+O(x$\pow$k)$}
! 368: \li{\typ{QFI}/\typ{QFR}. Imag/Real bin.\ quad.\ forms}
! 369: {Qfb$(a,b,c,\{d\})$}
! 370: \li{\typ{RFRAC}. Rational Functions}{$f/g$}
! 371: \li{\typ{VEC}/\typ{COL}. Row/Column Vectors}
! 372: {$[x,y,z]${\rm,} $[x,y,z]$\til}
! 373: %\li{\typ{COL}. Column Vectors}{$[x,y,z]$\til}
! 374: \li{\typ{MAT}. Matrices}{$[x,y;z,t;u,v]$}
! 375: \li{\typ{LIST}. Lists}{List$([x,y,z])$}
! 376: \li{\typ{STR}. Strings}{"aaa"}
! 377:
! 378: \section{Standard Operators}
! 379: \li{basic operations}{+{\rm,} - {\rm,} *{\rm,} /{\rm,} \pow}
! 380: \li{\kbd{i=i+1}, \kbd{i=i-1}, \kbd{i=i*j}, \dots}
! 381: {i++{\rm,} i--{\rm,} i*=j{\rm,}\dots}
! 382: \li{euclidean quotient, remainder}{$x$\bs/$y${\rm,} $x$\bs$y${\rm,}
! 383: $x$\%$y${\rm,} divrem$(x,y)$}
! 384: \li{shift $x$ left or right $n$ bits}{ $x$<<$n$, $x$>>$n$
! 385: {\rm or} shift$(x,n)$}
! 386: \li{comparison operators}{<={\rm, }<{\rm, }>={\rm, }>{\rm, }=={\rm, }!=}
! 387: \li{boolean operators (or, and, not)}{||{\rm, } \&\&{\rm ,} !}
! 388: \li{sign of $x=-1,0,1$}{sign$(x)$}
! 389: \li{maximum/minimum of $x$ and $y$}{max{\rm,} min$(x,y)$}
! 390: \li{integer or real factorial of $x$}{$x$!~{\rm or} fact$(x)$}
! 391:
! 392: \section{Conversions}
! 393: %
! 394: \subsec{Change Objects}
! 395: \li{make $x$ a vector, matrix, set, list, string}
! 396: {Vec{\rm,}Mat{\rm,}Set{\rm,}List{\rm,}Str}
! 397: \li{create PARI object $(x\mod y)$}{Mod$(x,y)$}
! 398: \li{make $x$ a polynomial of $v$}{Pol$(x,\{v\})$}
! 399: \li{as above, starting with constant term}{Polrev$(x,\{v\})$}
! 400: \li{make $x$ a power series of $v$}{Ser$(x,\{v\})$}
! 401: \li{PARI type of object $x$}{type$(x, \{t\})$}
! 402: \li{object $x$ with precision $n$}{prec$(x,\{n\})$}
! 403: \li{evaluate $f$ replacing vars by their value}{eval$(f)$}
! 404: %
! 405: \subsec{Select Pieces of an Object}
! 406: \li{length of $x$}{length$(x)$}
! 407: \li{$n$-th component of $x$}{component$(x,n)$}
! 408: \li{$n$-th component of vector/list $x$}{$x$[n]}
! 409: \li{$(m,n)$-th component of matrix $x$}{$x$[m,n]}
! 410: \li{row $m$ or column $n$ of matrix $x$}{$x$[m,]{\rm,} $x$[,n]}
! 411: \li{numerator of $x$}{numerator$(x)$}
! 412: \li{lowest denominator of $x$}{denominator$(x)$}
! 413: %
! 414: \subsec{Conjugates and Lifts}
! 415: \li{conjugate of a number $x$}{conj$(x)$}
! 416: \li{conjugate vector of algebraic number $x$}{conjvec$(x)$}
! 417: \li{norm of $x$, product with conjugate}{norm$(x)$}
! 418: \li{square of $L^2$ norm of vector $x$}{norml2$(x)$}
! 419: \li{lift of $x$ from Mods}{lift{\rm,} centerlift$(x)$}
! 420:
! 421: \section{Random Numbers}
! 422: \li{random integer between $0$ and $N-1$}{random$(\{N\})$}
! 423: \li{get random seed}{getrand$()$}
! 424: \li{set random seed to $s$}{setrand$(s)$}
! 425:
! 426: \begingroup
! 427: \outer\def\subsec#1{\filbreak
! 428: \vskip 0.05ex plus 0.05ex
! 429: {\bf #1}
! 430: \vskip 0.05ex plus 0.05ex
! 431: }
! 432:
! 433: \section{Lists, Sets \& Sorting}
! 434: \li{sort $x$ by $k$th component}{vecsort$(x,\{k\},\{\fl=0\})$}
! 435: {\bf Sets} (= row vector of strings with strictly increasing entries)\hfill\break
! 436: %
! 437: \li{intersection of sets $x$ and $y$}{setintersect$(x,y)$}
! 438: \li{set of elements in $x$ not belonging to $y$}{setminus$(x,y)$}
! 439: \li{union of sets $x$ and $y$}{setunion$(x,y)$}
! 440: \li{look if $y$ belongs to the set $x$}{setsearch$(x,y,\fl)$}
! 441: %
! 442: \subsec{Lists}
! 443: \li{create empty list of maximal length $n$}{listcreate$(n)$}
! 444: \li{delete all components of list $l$}{listkill$(l)$}
! 445: \li{append $x$ to list $l$}{listput$(l,x,\{i\})$}
! 446: \li{insert $x$ in list $l$ at position $i$}{listinsert$(l,x,i)$}
! 447: \li{sort the list $l$}{listsort$(l,\fl)$}
! 448:
! 449: \section{Programming \& User Functions}
! 450: \subsec{Control Statements {\rm ($X$: formal parameter in expression \seq)}}
! 451: \li{eval.\ \seq\ for $a\le X\le b$}{for$(X=a,b,\seq)$}
! 452: \li{eval.\ \seq\ for $X$ dividing $n$}{fordiv$(n,X,\seq)$}
! 453: \li{eval.\ \seq\ for primes $a\le X\le b$}{forprime$(X=a,b,\seq)$}
! 454: \li{eval.\ \seq\ for $a\le X\le b$ stepping $s$}{forstep$(X=a,b,s,\seq)$}
! 455: \li{multivariable {\tt for}}{forvec$(X=v,\seq)$}
! 456: \li{if $a\ne0$, evaluate \seq1, else \seq2}{if$(a,\{\seq1\},\{\seq2\})$}
! 457: \li{evaluate \seq\ until $a\ne0$}{until$(a,\seq)$}
! 458: \li{while $a\ne0$, evaluate \seq}{while$(a,\seq)$}
! 459: \li{exit $n$ innermost enclosing loops}{break$(\{n\})$}
! 460: \li{start new iteration of $n$th enclosing loop}{next$(\{n\})$}
! 461: \li{return $x$ from current subroutine}{return$(x)$}
! 462: \li{error recovery (try \seq1)}{trap$(\{err\},\{\seq2\},\{\seq1\})$}
! 463: %
! 464: \subsec{Input/Output}
! 465: \li{prettyprint args with/without newline}{printp(){\rm,} printp1()}
! 466: \li{print args with/without newline}{print(){\rm,} print1()}
! 467: \li{read a string from keyboard}{input$()$}
! 468: \li{reorder priority of variables $[x,y,z]$}{reorder$(\{[x,y,z]\})$}
! 469: \li{output \args\ in \TeX\ format}{printtex$(\args)$}
! 470: \li{write \args\ to file}{write{\rm,} write1{\rm,} writetex$(\file,\args)$}
! 471: \li{read file into GP}{read(\{\file\})}
! 472: %
! 473: \subsec{Interface with User and System}
! 474: \li{allocates a new stack of $s$ bytes}{allocatemem$(\{s\})$}
! 475: \li{execute system command $a$}{system$(a)$}
! 476: \li{as above, feed result to GP}{extern$(a)$}
! 477: \li{install function from library}{install$(f,code,\{\var{gpf\/}\},\{\var{lib}\})$}
! 478: \li{alias \var{old}\ to \var{new}}{alias$(\var{new},\var{old})$}
! 479: \li{new name of function $f$ in GP 2.0}{whatnow$(f)$}
! 480: %
! 481: \subsec{User Defined Functions}
! 482: \leavevmode
! 483: {\tt name(formal vars) = local(local vars); \var{seq}}\hfill\break
! 484: {\tt struct.member = \var{seq}}\hfill\break
! 485: \li{kill value of variable or function $x$}{kill$(x)$}
! 486: \li{declare global variables}{global$(x,...)$}
! 487:
! 488: \section{Iterations, Sums \& Products}
! 489: \li{numerical integration}{intnum$(X=a,b,\expr,\fl)$}
! 490: \li{sum \expr\ over divisors of $n$}{sumdiv$(n,X,\expr)$}
! 491: \li{sum $X=a$ to $X=b$, initialized at $x$}{sum$(X=a,b,\expr,\{x\})$}
! 492: \li{sum of series \expr}{suminf$(X=a,\expr)$}
! 493: \li{sum of alternating/positive series}{sumalt{\rm,} sumpos}
! 494: \li{product $a\le X\le b$, initialized at $x$}{prod$(X=a,b,\expr,\{x\})$}
! 495: \li{product over primes $a\le X\le b$}{prodeuler$(X=a,b,\expr)$}
! 496: \li{infinite product $a\le X\le\infty$}{prodinf$(X=a,\expr)$}
! 497: \li{real root of \expr\ between $a$ and $b$}{solve$(X=a,b,\expr)$}
! 498: \endgroup
! 499:
! 500: % This goes at the top of page 4 (=1st column on back of reference card)
! 501:
! 502: \section{Vectors \& Matrices}
! 503: %
! 504: \li{dimensions of matrix $x$}{matsize$(x)$}
! 505: \li{concatenation of $x$ and $y$}{concat$(x,\{y\})$}
! 506: \li{extract components of $x$}{vecextract$(x,y,\{z\})$}
! 507: \li{transpose of vector or matrix $x$}{mattranspose$(x)$ {\rm or} $x$\til}
! 508: \li{adjoint of the matrix $x$}{matadj$(x)$}
! 509: \li{eigenvectors of matrix $x$}{mateigen$(x)$}
! 510: \li{characteristic polynomial of $x$}{charpoly$(x,\{v\},\fl)$}
! 511: \li{trace of matrix $x$}{trace$(x)$}
! 512: %
! 513: \subsec{Constructors \& Special Matrices}
! 514: \li{row vec.\ of \expr\ eval'ed at $1\le X\le n$}{vector$(n,\{X\},\{\expr\})$}
! 515: \li{col.\ vec.\ of \expr\ eval'ed at $1\le X\le n$}{vectorv$(n,\{X\},\{\expr\})$}
! 516: \li{matrix $1\le X\le m$, $1\le Y\le n$}{matrix$(m,n,\{X\},\{Y\},\{\expr\})$}
! 517: \li{diagonal matrix whose diag. is $x$}{matdiagonal$(x)$}
! 518: \li{$n\times n$ identity matrix}{matid$(n)$}
! 519: \li{Hessenberg form of square matrix $x$}{mathess$(x)$}
! 520: \li{$n\times n$ Hilbert matrix $H_{ij}=(i+j-1)^{-1}$}{mathilbert$(n)$}
! 521: \li{$n\times n$ Pascal triangle $P_{ij}={i\choose j}$}{matpascal$(n-1)$}
! 522: \li{companion matrix to polynomial $x$}{matcompanion$(x)$}
! 523: %
! 524: \subsec{Gaussian elimination}
! 525: \li{determinant of matrix $x$}{matdet$(x,\fl)$}
! 526: \li{kernel of matrix $x$}{matker$(x,\fl)$}
! 527: \li{intersection of column spaces of $x$ and $y$}{matintersect$(x,y)$}
! 528: \li{solve $M*X = B$ ($M$ invertible)}{matsolve$(M,B)$}
! 529: \li{as solve, modulo $D$ (col. vector)}{matsolvemod$(M,D,B)$}
! 530: \li{one sol of $M*X = B$}{matinverseimage$(M,B)$}
! 531: \li{basis for image of matrix $x$}{matimage$(x)$}
! 532: \li{supplement columns of $x$ to get basis}{matsupplement$(x)$}
! 533: \li{rows, cols to extract invertible matrix}{matindexrank$(x)$}
! 534: \li{rank of the matrix $x$}{matrank$(x)$}
! 535:
! 536: \section{Lattices \& Quadratic Forms}
! 537: \li{upper triangular Hermite Normal Form}{mathnf$(x)$}
! 538: \li{HNF of $x$ where $d$ is a multiple of det$(x)$}{mathnfmod$(x,d)$}
! 539: \li{vector of elementary divisors of $x$}{matsnf$(x)$}
! 540: \li{LLL-algorithm applied to columns of $x$}{qflll$(x,\fl)$}
! 541: \li{like \kbd{qflll}, $x$ is Gram matrix of lattice}
! 542: {qflllgram$(x,\fl)$}
! 543: \li{LLL-reduced basis for kernel of $x$}{matkerint$(x)$}
! 544: \li{$\ZZ$-lattice $\longleftrightarrow$ $\QQ$-vector space}{matrixqz$(x,p)$}
! 545: %
! 546: \subsec{Quadratic Forms}
! 547: \li{signature of quad form $^ty*x*y$}{qfsign$(x)$}
! 548: \li{decomp into squares of $^ty*x*y$}{qfgaussred$(x)$}
! 549: \li{find up to $m$ sols of $^ty*x*y\le b$}{qfminim$(x,b,m)$}
! 550: %\li{perfection rank of $x$}{qfperfection$(x)$}
! 551: \li{eigenvals/eigenvecs for real symmetric $x$}{qfjacobi$(x)$}
! 552:
! 553: \section{Formal \& p-adic Series}
! 554: \li{truncate power series or $p$-adic number}{truncate$(x)$}
! 555: \li{valuation of $x$ at $p$}{valuation$(x,p)$}
! 556: \subsec{Dirichlet and Power Series}
! 557: \li{Taylor expansion around $0$ of $f$ w.r.t. $x$}{taylor$(f,x)$}
! 558: \li{$\sum a_kb_kt^k$ from $\sum a_kt^k$ and $\sum b_kt^k$}{serconvol$(x,y)$}
! 559: \li{$f=\sum a_k*t^k$ from $\sum (a_k/k!)*t^k$}{serlaplace$(f)$}
! 560: \li{reverse power series $F$ so $F(f(x))=x$}{serreverse$(f)$}
! 561: \li{Dirichlet series multiplication / division}{dirmul{\rm,} dirdiv$(x,y)$}
! 562: \li{Dirichlet Euler product ($b$ terms)}{direuler$(p=a,b,\expr)$}
! 563: \subsec{$p$-adic Functions}
! 564: \li{square of $x$, good for $2$-adics}{sqr$(x)$}
! 565: \li{Teichmuller character of $x$}{teichmuller$(x)$}
! 566: \li{Newton polygon of $f$ for prime $p$}{newtonpoly$(f,p)$}
! 567:
! 568: \newcolumn
! 569: \title{PARI-GP Reference Card}
! 570: \centerline{(PARI-GP version \PARIversion)}
! 571:
! 572: \section{Polynomials \& Rational Functions}
! 573: %
! 574: \li{degree of $f$}{poldegree$(f)$}
! 575: \li{coefficient of degree $n$ of $f$}{polcoeff$(f,n)$}
! 576: \li{round coeffs of $f$ to nearest integer}{round$(f,\{\&e\})$}
! 577: \li{gcd of coefficients of $f$}{content$(f)$}
! 578: \li{replace $x$ by $y$ in $f$}{subst$(f,x,y)$}
! 579: \li{discriminant of polynomial $f$}{poldisc$(f)$}
! 580: %\li{elementary divisors of Z[a]/f'(a)Z[a]}{poldiscreduced$(f)$}
! 581: \li{resultant of $f$ and $g$}{polresultant$(f,g,\fl)$}
! 582: \li{as above, give $[u,v,d]$, $xu + yv = d$}{bezoutres$(x,y)$}
! 583: \li{derivative of $f$ w.r.t. $x$}{deriv$(f,x)$}
! 584: \li{formal integral of $f$ w.r.t. $x$}{intformal$(f,x)$}
! 585: \li{reciprocal poly $x^{\deg f}f(1/x)$}{polrecip$(f)$}
! 586: \li{interpolating poly evaluated at $a$}{polinterpolate$(X,\{Y\},\{a\},\{\&e\})$}
! 587: \li{initialize $t$ for Thue equation solver}{thueinit(f)}
! 588: \li{solve Thue equation $f(x,y)=a$}{thue$(t,a,\{sol\})$}
! 589: %
! 590: \subsec{Roots and Factorization}
! 591: \li{number of real roots of $f$, $a < x\le b$}{polsturm$(f,\{a\},\{b\})$}
! 592: \li{complex roots of $f$}{polroots$(f)$}
! 593: \li{symmetric powers of roots of $f$ up to $n$}{polsym$(f,n)$}
! 594: \li{roots of $f \mod p$}{polrootsmod$(f,p,\fl)$}
! 595: \li{factor $f$}{factor$(f,\{lim\})$}
! 596: \li{factorization of $f\mod p$}{factormod$(f,p,\fl)$}
! 597: \li{factorization of $f$ over $\FF_{p^a}$}{factorff$(f,p,a)$}
! 598: \li{$p$-adic fact. of $f$ to prec. $r$}{factorpadic$(f,p,r,\fl)$}
! 599: \li{$p$-adic roots of $f$ to prec. $r$}{polrootspadic$(f,p,r)$}
! 600: \li{$p$-adic root of $f$ cong. to $a\mod p$}{padicappr$(f,a)$}
! 601: \li{Newton polygon of $f$ for prime $p$}{newtonpoly$(f,p)$}
! 602: %
! 603: \subsec{Special Polynomials}
! 604: \li{$n$th cyclotomic polynomial in var. $v$}{polcyclo$(n,\{v\})$}
! 605: \li{$d$-th degree subfield of $\QQ(\zeta_n)$} {polsubcyclo$(n,d,\{v\})$}
! 606: \li{$n$-th Legendre polynomial}{pollegendre$(n)$}
! 607: \li{$n$-th Tchebicheff polynomial}{poltchebi$(n)$}
! 608: \li{Zagier's polynomial of index $n$,$m$}{polzagier$(n,m)$}
! 609:
! 610: \section{Transcendental Functions}
! 611: \li{real, imaginary part of $x$}{real$(x)$, imag$(x)$}
! 612: \li{absolute value, argument of $x$}{abs$(x)$, arg$(x)$}
! 613: \li{square/nth root of $x$}{sqrt$(x)$, sqrtn$(x,n,\&z)$}
! 614: \li{trig functions}{sin, cos, tan, cotan}
! 615: \li{inverse trig functions}{asin, acos, atan}
! 616: \li{hyperbolic functions}{sinh, cosh, tanh}
! 617: \li{inverse hyperbolic functions}{asinh, acosh, atanh}
! 618: \li{exponential of $x$}{exp$(x)$}
! 619: \li{natural log of $x$}{ln$(x)$ {\rm or} log$(x)$}
! 620: %
! 621: \li{gamma function $\Gamma(x)=\int_0^\infty e^{-t}t^{x-1}dt$}{gamma$(x)$}
! 622: %\li{half-integer gamma function $\Gamma(n+1/2)$}{gammah$(n)$}
! 623: \li{logarithm of gamma function}{lngamma$(x)$}
! 624: \li{$\psi(x)=\Gamma'(x)/\Gamma(x)$}{psi$(x)$}
! 625: \li{incomplete gamma function ($y=\Gamma(s)$)}{incgam$(s,x,\{y\})$}
! 626: \li{exponential integral $\int_x^\infty e^{-t}/t\,dt$}{eint1$(x)$}
! 627: \li{error function $2/\sqrt\pi\int_x^\infty e^{-t^2}dt$}{erfc$(x)$}
! 628: \li{dilogarithm of $x$}{dilog$(x)$}
! 629: \li{$m$th polylogarithm of $x$}{polylog$(m,x,\fl)$}
! 630: \li{$U$-confluent hypergeometric function}{hyperu$(a,b,u)$}
! 631: \li{$J$-Bessel function $J_{n+1/2}(x)$}{besseljh$(n,x)$}
! 632: \li{$K$-Bessel function of index \var{nu}}{besselk$(\var{nu},x)$}
! 633:
! 634: \section{Elementary Arithmetic Functions}
! 635: \li{vector of binary digits of $|x|$}{binary$(x)$}
! 636: \li{give bit number $n$ of integer $x$}{bittest$(x,n)$}
! 637: \li{ceiling of $x$}{ceil$(x)$}
! 638: \li{floor of $x$}{floor$(x)$}
! 639: \li{fractional part of $x$}{frac$(x)$}
! 640: \li{round $x$ to nearest integer}{round$(x,\{\&e\})$}
! 641: \li{truncate $x$}{truncate$(x,\{\&e\})$}
! 642: \li{gcd of $x$ and $y$}{gcd$(x,y)$}
! 643: \li{LCM of $x$ and $y$}{lcm$(x,y)$}
! 644: \li{gcd of entries of a vector/matrix}{content$(x)$}
! 645: \par
! 646: \subsec{Primes and Factorization}
! 647: \li{add primes in $v$ to the prime table}{addprimes$(v)$}
! 648: \li{the $n$th prime}{prime$(n)$}
! 649: \li{vector of first $n$ primes}{primes$(n)$}
! 650: \li{smallest prime $\ge x$}{nextprime$(x)$}
! 651: \li{largest prime $\le x$}{precprime$(x)$}
! 652: \li{factorization of $x$}{factor$(x,\{lim\})$}
! 653: \li{reconstruct $x$ from its factorization}{factorback$(fa,\{nf\})$}
! 654: \par
! 655: \subsec{Divisors}
! 656: \li{number of distinct prime divisors}{omega$(x)$}
! 657: \li{number of prime divisors with mult}{bigomega$(x)$}
! 658: \li{number of divisors of $x$}{numdiv$(x)$}
! 659: \li{row vector of divisors of $x$}{divisors$(x)$}
! 660: \li{sum of ($k$-th powers of) divisors of $x$}{sigma$(x,\{k\})$}
! 661: \par
! 662: \subsec{Special Functions and Numbers}
! 663: \li{binomial coefficient $x\choose y$}{binomial$(x,y)$}
! 664: \li{Bernoulli number $B_n$ as real}{bernreal$(n)$}
! 665: \li{Bernoulli vector $B_0,B_2,\ldots,B_{2n}$}{bernvec$(n)$}
! 666: \li{$n$th Fibonacci number}{fibonacci$(n)$}
! 667: \li{Euler $\phi$-function}{eulerphi$(x)$}
! 668: \li{M\"obius $\mu$-function}{moebius$(x)$}
! 669: \li{Hilbert symbol of $x$ and $y$ (at $p$)}{hilbert$(x,y,\{p\})$}
! 670: \li{Kronecker-Legendre symbol $({x\over y})$}{kronecker$(x,y)$}
! 671: \par
! 672: \subsec{Miscellaneous}
! 673: \li{integer or real factorial of $x$}{$x!$ {\rm or} fact$(x)$}
! 674: \li{integer square root of $x$}{sqrtint$(x)$}
! 675: \li{solve $z\equiv x$ and $z\equiv y$}{chinese$(x,y)$}
! 676: \li{minimal $u,v$ so $xu+yv=\gcd(x,y)$}{bezout$(x,y)$}
! 677: \li{multiplicative order of $x$ (intmod)}{znorder$(x)$}
! 678: \li{primitive root mod prime power $q$}{znprimroot$(q)$}
! 679: \li{structure of $(\ZZ/n\ZZ)^*$}{znstar$(n)$}
! 680: \li{continued fraction of $x$}{contfrac$(x,\{b\},\{lmax\})$}
! 681: \li{last convergent of continued fraction $x$}{contfracpnqn$(x)$}
! 682: \li{best rational approximation to $x$}{bestappr$(x,k)$}
! 683:
! 684: \section{True-False Tests}
! 685: \li{is $x$ the disc. of a quadratic field?}{isfundamental$(x)$}
! 686: \li{is $x$ a prime?}{isprime$(x)$}
! 687: \li{is $x$ a strong pseudo-prime?}{ispseudoprime$(x)$}
! 688: \li{is $x$ square-free?}{issquarefree$(x)$}
! 689: \li{is $x$ a square?}{issquare$(x,\{\&n\})$}
! 690: \li{is \var{pol}\ irreducible?}{polisirreducible$(\var{pol})$}
! 691:
! 692: % This goes at the bottom of the second page (column 6)
! 693: \copyrightnotice
! 694: %
! 695:
! 696: %%%%%%%%%%% Extra Material (part II)
! 697: %
! 698: \newcolumn
! 699: \title{PARI-GP Reference Card (2)}
! 700: \centerline{(PARI-GP version \PARIversion)}
! 701:
! 702: \section{Elliptic Curves}
! 703: %
! 704: Elliptic curve initially given by $5$-tuple $E=$\kbd{[a1,a2,a3,a4,a6]}.
! 705: Points are \kbd{[x,y]}, the origin is \kbd{[0]}.
! 706: \hfill\break
! 707: \li{Initialize elliptic struct. $\var{ell}$, i.e create}{ellinit$(E,\fl)$}
! 708: \leavevmode\strut\hskip1em
! 709: $a_1,a_2,a_3,a_4,a_6,b_2,b_4,b_6,b_8,c_4,c_6,disc,j$. This data can be
! 710: recovered by typing \kbd{\var{ell}.a1},$\dots$,\kbd{\var{ell}.j}.
! 711: If $\var{fl}$ omitted, also
! 712: \hfill\break
! 713: \beginindentedkeys
! 714: \li{$E$ defined over $\RR$}{}
! 715: \begindoubleindentedkeys
! 716: \key{$x$-coords. of points of order $2$}{\var{ell}.roots}
! 717: \key{real and complex periods}{\var{ell}.omega}
! 718: \key{associated quasi-periods}{\var{ell}.eta}
! 719: \key{volume of complex lattice}{\var{ell}.area}
! 720: \enddoubleindentedkeys
! 721: \li{$E$ defined over $\QQ_p$, $|j|_p>1$}{}
! 722: \begindoubleindentedkeys
! 723: \key{$x$-coord. of unit $2$ torsion point}{\var{ell}.roots}
! 724: \key{Tate's $[u^2, u, q]$}{\var{ell}.tate}
! 725: \key{Mestre's $w$}{\var{ell}.w}
! 726: \endindentedkeys
! 727: \li{change curve $E$ using $v=[u,r,s,t]$}{ellchangecurve$(ell,v)$}
! 728: \li{change point $z$ using $v=[u,r,s,t]$}{ellchangepoint$(z,v)$}
! 729: \li{cond, min mod, Tamgawa nmbr $[N,v,c]$}{ellglobalred$(ell)$}
! 730: \li{Kodaira type of $p$ fiber of $E$}{elllocalred$(ell,p)$}
! 731: \li{add points $z1+z2$}{elladd$(ell,z1,z2)$}
! 732: \li{subtract points $z1-z2$}{ellsub$(ell,z1,z2)$}
! 733: \li{compute $n\cdot z$}{ellpow$(ell,z,n)$}
! 734: \li{check if $z$ is on $E$}{ellisoncurve$(ell,z)$}
! 735: \li{order of torsion point $z$}{ellorder$(ell,z)$}
! 736: \li{torsion subgroup with generators}{elltors$(ell)$}
! 737: \li{$y$-coordinates of point(s) for $x$}{ellordinate$(ell,x)$}
! 738: \li{canonical bilinear form taken at $z1$, $z2$}{ellbil$(ell,z1,z2)$}
! 739: \li{canonical height of $z$}{ellheight$(ell,z,\fl)$}
! 740: \li{height regulator matrix for pts in $x$}{ellheightmatrix$(ell,x)$}
! 741: \li{$p$th coeff $a_p$ of $L$-function, $p$ prime}{ellap$(ell,p)$}
! 742: \li{$k$th coeff $a_k$ of $L$-function}{ellak$(ell,k)$}
! 743: \li{vector of first $n$ $a_k$'s in $L$-function}{ellan$(ell,n)$}
! 744: \li{$L(E,s)$, set $A\approx1$}{elllseries$(ell,s,\{A\})$}
! 745: \li{root number for $L(E,.)$ at $p$}{ellrootno$(ell,\{p\})$}
! 746: \li{modular parametrization of $E$}{elltaniyama$(ell)$}
! 747: \li{point $[\wp(z),\wp'(z)]$ corresp. to $z$}{ellztopoint$(ell,z)$}
! 748: \li{complex $z$ such that $p=[\wp(z),\wp'(z)]$}{ellpointtoz$(ell,p)$}
! 749:
! 750: \section{Elliptic \& Modular Functions}
! 751: %
! 752: \li{arithmetic-geometric mean}{agm$(x,y)$}
! 753: \li{elliptic $j$-function $1/q+744+\cdots$}{ellj$(x)$}
! 754: \li{Weierstrass $\sigma$ function}{ellsigma$(ell,z,\fl)$}
! 755: \li{Weierstrass $\wp$ function}{ellwp$(ell,\{z\},\fl)$}
! 756: \li{Weierstrass $\zeta$ function}{ellzeta$(ell,z)$}
! 757: \li{modified Dedekind $\eta$ func. $\prod(1-q^n)$}{eta$(x,\fl)$}
! 758: \li{Jacobi sine theta function}{theta$(q,z)$}
! 759: \li{k-th derivative at z=0 of \kbd{theta}$(q,z)$}{thetanullk$(q,k)$}
! 760: \li{Weber's $f$ functions}{weber$(x,\fl)$}
! 761: \li{Riemann's zeta $\zeta(s)=\sum n^{-s}$}{zeta$(s)$}
! 762: %
! 763: \shortcopyrightnotice
! 764: \newcolumn
! 765:
! 766: \section{Graphic Functions}
! 767: \li{crude graph of \expr\ between $a$ and $b$}{plot$(X=a,b,expr)$}
! 768: \subsec{High-resolution plot {\rm (immediate plot)}}
! 769: \li{plot \expr\ between $a$ and $b$}{ploth$(X=a,b,expr,\fl,\{n\})$}
! 770: \li{plot points given by lists $lx$, $ly$}{plothraw$(lx,ly,\fl)$}
! 771: \li{terminal dimensions}{plothsizes$()$}
! 772: %
! 773: \subsec{Rectwindow functions}
! 774: \li{init window $w$, with size $x$,$y$}{plotinit$(w,x,y)$}
! 775: \li{erase window $w$}{plotkill$(w)$}
! 776: \li{copy $w$ to $w2$ with offset $(dx,dy)$}{plotcopy$(w,w2,dx,dy)$}
! 777: \li{scale coordinates in $w$}{plotscale$(w,x_1,x_2,y_1,y_2)$}
! 778: \li{\kbd{ploth} in $w$}{plotrecth$(w,X=a,b,expr,\fl,\{n\})$}
! 779: \li{\kbd{plothraw} in $w$}{plotrecthraw$(w,data,\fl)$}
! 780: \li{draw window $w_1$ at $(x_1,y_1)$, \dots} {plotdraw$([[w_1,x_1,y_1],\dots])$}
! 781: %
! 782: \subsec{Low-level Rectwindow Functions}
! 783: %\li{}{plotlinetype$(w,)$}
! 784: %\li{}{plotpointtype$(w,)$}
! 785: %\li{}{plotterm$(w,)$}
! 786: \li{set current drawing color in $w$ to $c$}{plotcolor$(w,c)$}
! 787: \li{current position of cursor in $w$}{plotcursor$(w)$}
! 788: %
! 789: \li{write $s$ at cursor's position}{plotstring$(w,s)$}
! 790: \li{move cursor to $(x,y)$}{plotmove$(w,x,y)$}
! 791: \li{move cursor to $(x+dx,y+dy)$}{plotrmove$(w,dx,dy)$}
! 792: \li{draw a box to $(x_2,y_2)$}{plotbox$(w,x_2,y_2)$}
! 793: \li{draw a box to $(x+dx,y+dy)$}{plotrbox$(w,dx,dy)$}
! 794: \li{draw polygon}{plotlines$(w,lx,ly,\fl)$}
! 795: \li{draw points}{plotpoints$(w,lx,ly)$}
! 796: \li{draw line to $(x+dx,y+dy)$}{plotrline$(w,dx,dy)$}
! 797: \li{draw point $(x+dx,y+dy)$}{plotrpoint$(w,dx,dy)$}
! 798: %
! 799: \subsec{Postscript Functions}
! 800: \li{as {\tt ploth}}{psploth$(X=a,b,expr,\fl,\{n\})$}
! 801: \li{as {\tt plothraw}}{psplothraw$(lx,ly,\fl)$}
! 802: \li{as {\tt plotdraw}}{psdraw$([[w_1,x_1,y_1],\dots])$}
! 803: \newcolumn
! 804:
! 805: \section{Binary Quadratic Forms}
! 806: %
! 807: \li{create $ax^2+bxy+cy^2$ (distance $d$) }{Qfb$(a,b,c,\{d\})$}
! 808: \li{reduce $x$ ($s =\sqrt{D}$, $l=\lfloor s\rfloor$)}
! 809: {qfbred$(x,\fl,\{D\},\{l\},\{s\})$}
! 810: \li{composition of forms}{$x*y$ {\rm or }qfbnucomp$(x,y,l)$}
! 811: \li{$n$-th power of form}{$x$\pow$n$ {\rm or }qfbnupow$(x,n)$}
! 812: \li{composition without reduction}{qfbcompraw$(x,y)$}
! 813: \li{$n$-th power without reduction}{qfbpowraw$(x,n)$}
! 814: \li{prime form of disc. $x$ above prime $p$}{qfbprimeform$(x,p)$}
! 815: \li{class number of disc. $x$}{qfbclassno$(x)$}
! 816: \li{Hurwitz class number of disc. $x$}{qfbhclassno$(x)$}
! 817:
! 818: \section{Quadratic Fields}
! 819: %
! 820: \li{quadratic number $\omega=\sqrt x$ or $(1+\sqrt x)/2$}{quadgen$(x)$}
! 821: \li{minimal polynomial of $\omega$}{quadpoly$(x)$}
! 822: \li{discriminant of $\QQ(\sqrt{D})$}{quaddisc$(x)$}
! 823: \li{regulator of real quadratic field}{quadregulator$(x)$}
! 824: \li{fundamental unit in real $\QQ(x)$}{quadunit$(x)$}
! 825: \li{class group of $\QQ(\sqrt{D})$}{quadclassunit$(D,\fl,\{t\})$}
! 826: \li{Hilbert class field of $\QQ(\sqrt{D})$}{quadhilbert$(D,\fl)$}
! 827: \li{ray class field modulo $f$ of $\QQ(\sqrt{D})$}{quadray$(D,f,\fl)$}
! 828:
! 829: \section{General Number Fields: Initializations}
! 830: A number field $K$ is given by a monic irreducible $f\in\ZZ[X]$.\hfill\break
! 831: \li{init number field structure \var{nf}}{nfinit$(f,\fl)$}
! 832: \subsec{nf members:}
! 833: \beginindentedkeys
! 834: \key{polynomial defining \var{nf}, $f(\theta)=0$}{\var{nf}.pol}
! 835: \key{number of [real,complex] places}{\var{nf}.sign}
! 836: \key{discriminant of \var{nf}}{\var{nf}.disc}
! 837: \key{$T_2$ matrix}{\var{nf}.t2}
! 838: \key{vector of roots of $f$}{\var{nf}.roots}
! 839: \key{integral basis of $\ZZ_K$ as powers of $\theta$}{\var{nf}.zk}
! 840: \key{different}{\var{nf}.diff}
! 841: \key{codifferent}{\var{nf}.codiff}
! 842: \endindentedkeys
! 843: \li{recompute \var{nf}\ using current precision}{nfnewprec$(nf)$}
! 844: \li{init relative \var{rnf}\ given by $g=0$ over $K$}{rnfinit$(\var{nf},g)$}
! 845: %
! 846: \li{init big number field structure \var{bnf}}{bnfinit$(f,\fl)$}
! 847: \subsec{bnf members: {\rm same as \var{nf}, plus}}
! 848: \beginindentedkeys
! 849: \key{underlying \var{nf}}{\var{bnf}.nf}
! 850: \key{classgroup}{\var{bnf}.clgp}
! 851: \key{regulator}{\var{bnf}.reg}
! 852: \key{fundamental units}{\var{bnf}.fu}
! 853: \key{torsion units}{\var{bnf}.tu}
! 854: \key{$[tu,fu]$, $[fu,tu]$}{\var{bnf}.tufu{\rm/}futu}
! 855: \endindentedkeys
! 856: \li{compute a \var{bnf}\ from small \var{bnf}}{bnfmake$(\var{sbnf})$}
! 857: %
! 858: \li{add $S$-class group and units, yield \var{bnfs}}{bnfsunit$(\var{nf},S)$}
! 859: \li{init class field structure \var{bnr}}{bnrinit$(\var{bnf},m,\fl)$}
! 860: %
! 861: \subsec{bnr members: {\rm same as \var{bnf}, plus}}
! 862: \beginindentedkeys
! 863: \key{underlying \var{bnf}}{\var{bnr}.bnf}
! 864: \key{structure of $(\ZZ_K/m)^*$}{\var{bnr}.zkst}
! 865: \endindentedkeys
! 866:
! 867: \section{Simple Arithmetic Invariants (nf)}
! 868: Elements are rational numbers, polynomials, polmods, or column vectors (on
! 869: integral basis \kbd{\var{nf}.zk}).\hfill\break
! 870: \li{integral basis of field def. by $f=0$}{nfbasis$(f)$}
! 871: \li{field discriminant of field $f=0$}{nfdisc$(f)$}
! 872: \li{reverse polmod $a=A(X)\mod T(X)$}{modreverse$(a)$}
! 873: \li{Galois group of field $f=0$, $\deg f\le 11$}{polgalois$(f)$}
! 874: \li{smallest poly defining $f=0$}{polredabs$(f,\fl)$}
! 875: \li{small polys defining subfields of $f=0$}{polred$(f,\fl,\{p\})$}
! 876: \li{small polys defining suborders of $f=0$}{polredord$(f)$}
! 877: \li{poly of degree $\le k$ with root $x\in\CC$}{algdep$(x,k)$}
! 878: \li{small linear rel.\ on coords of vector $x$}{lindep$(x)$}
! 879: \li{are fields $f=0$ and $g=0$ isomorphic?}{nfisisom$(f,g)$}
! 880: \li{is field $f=0$ a subfield of $g=0$?}{nfisincl$(f,g)$}
! 881: \li{compositum of $f=0$, $g=0$}{polcompositum$(f,g,\fl)$}
! 882: %
! 883: \li{basic element operations (prefix \kbd{nfelt}):}{}
! 884: \leavevmode\strut\hskip1em
! 885: $($\kbd{nfelt}$)$\kbd{mul}, \kbd{pow}, \kbd{div}, \kbd{diveuc},
! 886: \kbd{mod}, \kbd{divrem}, \kbd{val}
! 887: \hfill\break
! 888: %
! 889: \li{express $x$ on integer basis}{nfalgtobasis$(\var{nf},x)$}
! 890: \li{express element\ $x$ as a polmod}{nfbasistoalg$(\var{nf},x)$}
! 891: \li{quadratic Hilbert symbol (at $p$)}{nfhilbert$(\var{nf},a,b,\{p\})$}
! 892: \li{roots of $g$ belonging to {\tt nf}}{nfroots$(\var{nf},g)$}
! 893: \li{factor $g$ in {\tt nf}}{nffactor$(\var{nf},g)$}
! 894: \li{factor $g$ mod prime $pr$ in {\tt nf}}{nffactormod$(\var{nf},g,pr)$}
! 895: \li{number of roots of $1$ in {\tt nf}}{nfrootsof1$(nf)$}
! 896: \li{conjugates of a root $\theta$ of {\tt nf}}{nfgaloisconj$(\var{nf},\fl)$}
! 897: \li{apply Galois automorphism $s$ to $x$}{nfgaloisapply$(\var{nf},s,x)$}
! 898: \li{subfields (of degree $d$) of {\tt nf}}{nfsubfields$(\var{nf},\{d\})$}
! 899: %
! 900: \subsec{Dedekind Zeta Function $\zeta_K$}
! 901: \li{$\zeta_K$ as Dirichlet series, $N(I)<b$}{dirzetak$(\var{nf},b)$}
! 902: \li{init \var{nfz}\ for field $f=0$}{zetakinit$(f)$}
! 903: \li{compute $\zeta_K(s)$}{zetak$(\var{nfz},s,\fl)$}
! 904: \li{Artin root number of $K$}{bnrrootnumber$(\var{bnr},\var{chi},\fl)$}
! 905:
! 906: \section{Class Groups \& Units (\var{bnf}, bnr)}
! 907: \leavevmode
! 908: $a1,\{a2\},\{a3\}$ usually $bnr,subgp$ or $\var{bnf},module,\{subgp\}$
! 909: \hfill\break
! 910: %
! 911: \li{remove GRH assumption from \var{bnf}}{bnfcertify$(\var{bnf})$}
! 912: \li{expo.~of ideal $x$ on class gp}{bnfisprincipal$(\var{bnf},x,\fl)$}
! 913: \li{expo.~of ideal $x$ on ray class gp}{bnrisprincipal$(\var{bnr},x,\fl)$}
! 914: \li{expo.~of $x$ on fund.~units}{bnfisunit$(\var{bnf},x)$}
! 915: \li{as above for $S$-units}{bnfissunit$(\var{bnfs},x)$}
! 916: \li{fundamental units of \var{bnf}}{bnfunit$(\var{bnf})$}
! 917: \li{signs of real embeddings of \kbd{\var{bnf}.fu}}{bnfsignunit$(\var{bnf})$}
! 918: %
! 919: \subsec{Class Field Theory}
! 920: \li{ray class group structure for mod.~$m$}{bnrclass$(\var{bnf},m,\fl)$}
! 921: \li{ray class number for mod.~$m$}{bnrclassno$(\var{bnf},m)$}
! 922: \li{discriminant of class field ext}{bnrdisc$(a1,\{a2\},\{a3\})$}
! 923: \li{ray class numbers, $l$ list of mods}{bnrclassnolist$(\var{bnf},l)$}
! 924: \li{discriminants of class fields}{bnrdisclist$(\var{bnf},l,\{arch\},\fl)$}
! 925: \li{decode output from \kbd{bnrdisclist}}{bnfdecodemodule$(\var{nf},fa)$}
! 926: \li{is modulus the conductor?}{bnrisconductor$(a1,\{a2\},\{a3\})$}
! 927: \li{conductor of character $chi$}{bnrconductorofchar$(\var{bnr},chi)$}
! 928: \li{conductor of extension}{bnrconductor$(a1,\{a2\},\{a3\},\fl)$}
! 929: \li{conductor of extension def.\ by $g$}{rnfconductor$(\var{bnf},g)$}
! 930: \li{Artin group of ext.\ def'd by $g$}{rnfnormgroup$(\var{bnr},g)$}
! 931: \li{subgroups of {\tt bnr}, index $<=b$}{subgrouplist$(\var{bnr},b,\fl)$}
! 932: \li{rel.\ eq.\ for class field def'd by $sub$}{rnfkummer$(\var{bnr},sub,\{d\})$}
! 933: \li{same, using Stark units (real field)}{bnrstark$(\var{bnr},sub,\fl)$}
! 934:
! 935: \newcolumn
! 936: \title{PARI-GP Reference Card (2)}
! 937: \centerline{(PARI-GP version \PARIversion)}
! 938:
! 939: \section{Ideals}
! 940: Ideals are elements, primes, or matrix of generators in HNF.\hfill\break
! 941: \li{is $id$ an ideal in {\tt nf}?}{nfisideal$(\var{nf},id)$}
! 942: \li{is $x$ principal in {\tt bnf}?}{bnfisprincipal$(\var{bnf},x)$}
! 943: \li{principal ideal generated by $x$}{idealprincipal$(\var{nf},x)$}
! 944: \li{principal idele generated by $x$}{ideleprincipal$(\var{nf},x)$}
! 945: \li{give $[a,b]$, s.t.~ $a\ZZ_K+b\ZZ_K = x$}{idealtwoelt$(\var{nf},x,\{a\})$}
! 946: \li{put ideal $a$ ($a\ZZ_K+b\ZZ_K$) in HNF form}{idealhnf$(\var{nf},a,\{b\})$}
! 947: \li{norm of ideal $x$}{idealnorm$(\var{nf},x)$}
! 948: \li{minimum of ideal $x$ (direction $v$)}{idealmin$(\var{nf},x,v)$}
! 949: \li{LLL-reduce the ideal $x$ (direction $v$)}{idealred$(\var{nf},x,\{v\})$}
! 950: %
! 951: \subsec{Ideal Operations}
! 952: \li{add ideals $x$ and $y$}{idealadd$(\var{nf},x,y)$}
! 953: \li{multiply ideals $x$ and $y$}{idealmul$(\var{nf},x,y,\fl)$}
! 954: \li{intersection of ideals $x$ and $y$}{idealintersect$(\var{nf},x,y,\fl)$}
! 955: \li{$n$-th power of ideal $x$}{idealpow$(\var{nf},x,n,\fl)$}
! 956: \li{inverse of ideal $x$}{idealinv$(\var{nf},x)$}
! 957: \li{divide ideal $x$ by $y$}{idealdiv$(\var{nf},x,y,\fl)$}
! 958: \li{Find $[a,b]\in x\times y$, $a+b=1$}{idealaddtoone$(\var{nf},x,\{y\})$}
! 959: %
! 960: \subsec{Primes and Multiplicative Structure}
! 961: \li{factor ideal $x$ in {\tt nf}}{idealfactor$(\var{nf},x)$}
! 962: \li{recover $x$ from its factorization in {\tt nf}}{factorback$(x,nf)$}
! 963: \li{decomposition of prime $p$ in {\tt nf}}{idealprimedec$(\var{nf},p)$}
! 964: \li{valuation of $x$ at prime ideal $pr$}{idealval$(\var{nf},x,pr)$}
! 965: \li{weak approximation theorem in {\tt nf}}{idealchinese$(\var{nf},x,y)$}
! 966: \li{give $bid=$structure of $(\ZZ_K/id)^*$}{idealstar$(\var{nf},id,\fl)$}
! 967: \li{discrete log of $x$ in $(\ZZ_K/bid)^*$}{ideallog$(\var{nf},x,bid)$}
! 968: \li{\kbd{idealstar} of all ideals of norm $\le b$}{ideallist$(\var{nf},b,\fl)$}
! 969: \li{add archimedean places}{ideallistarch$(\var{nf},b,\{ar\},\fl)$}
! 970: \li{init \kbd{prmod} structure}{nfmodprinit$(\var{nf},pr)$}
! 971: \li{kernel of matrix $M$ in $(\ZZ_K/pr)^*$}{nfkermodpr$(\var{nf},M,prmod)$}
! 972: \li{solve $M x = B$ in $(\ZZ_K/pr)^*$}{nfsolvemodpr$(\var{nf},M,B,prmod)$}
! 973:
! 974: \section{Relative Number Fields (rnf)}
! 975: Extension $L/K$ is defined by $g\in K[x]$. We have $order\subset L$.
! 976: \hfill\break
! 977: %
! 978: \li{absolute equation of $L$}{rnfequation$(\var{nf},g,\fl)$}
! 979: %
! 980: \subsec{Lifts and Push-downs}
! 981: \li{absolute $\rightarrow$ relative repres.\ for $x$}
! 982: {rnfeltabstorel$(\var{rnf},x)$}
! 983: \li{relative $\rightarrow$ absolute repres.\ for $x$}
! 984: {rnfeltreltoabs$(\var{rnf},x)$}
! 985: \li{lift $x$ to the relative field}{rnfeltup$(\var{rnf},x)$}
! 986: \li{push $x$ down to the base field}{rnfeltdown$(\var{rnf},x)$}
! 987: \leavevmode idem for $x$ ideal:
! 988: \kbd{$($rnfideal$)$reltoabs}, \kbd{abstorel}, \kbd{up}, \kbd{down}\hfill\break
! 989: %
! 990: \li{relative {\tt nfalgtobasis}}{rnfalgtobasis$(\var{rnf},x)$}
! 991: \li{relative {\tt nfbasistoalg}}{rnfbasistoalg$(\var{rnf},x)$}
! 992: \li{relative {\tt idealhnf}}{rnfidealhnf$(\var{rnf},x)$}
! 993: \li{relative {\tt idealmul}}{rnfidealmul$(\var{rnf},x,y)$}
! 994: \li{relative {\tt idealtwoelt}}{rnfidealtwoelt$(\var{rnf},x)$}
! 995: %
! 996: \subsec{Projective $\ZZ_K$-modules, maximal order}
! 997: \li{relative {\tt polred}}{rnfpolred$(\var{nf},g)$}
! 998: \li{relative {\tt polredabs}}{rnfpolredabs$(\var{nf},g)$}
! 999: \li{characteristic poly.\ of $a$ mod $g$}{rnfcharpoly$(\var{nf},g,a,\{v\})$}
! 1000: \li{relative Dedekind criterion, prime $pr$}{rnfdedekind$(\var{nf},g,pr)$}
! 1001: \li{discriminant of relative extension}{rnfdisc$(\var{nf},g)$}
! 1002: \li{pseudo-basis of $\ZZ_L$}{rnfpseudobasis$(\var{nf},g)$}
! 1003: \li{relative HNF basis of $order$}{rnfhnfbasis$(\var{bnf},order)$}
! 1004: \li{reduced basis for $order$}{rnflllgram$(\var{nf},g,order)$}
! 1005: \li{determinant of pseudo-matrix $A$}{rnfdet$(\var{nf},A)$}
! 1006: \li{Steinitz class of $order$}{rnfsteinitz$(\var{nf},order)$}
! 1007: \li{is \var{order} a free $\ZZ_K$-module?}{rnfisfree$(\var{bnf},\var{order})$}
! 1008: \li{true basis of \var{order}, if it is free}{rnfbasis$(\var{bnf},\var{order})$}
! 1009: %
! 1010: \subsec{Norms}
! 1011: \li{absolute norm of ideal $x$}{rnfidealnormabs$(\var{rnf},x)$}
! 1012: \li{relative norm of ideal $x$}{rnfidealnormrel$(\var{rnf},x)$}
! 1013: \li{solutions of $N_{K/\QQ}(y)=x\in \ZZ$}{bnfisintnorm$(\var{bnf},x)$}
! 1014: \li{is $x\in\QQ$ a norm from $K$?}{bnfisnorm$(\var{bnf},x,\fl)$}
! 1015: \li{is $x\in K$ a norm from $L$?}{rnfisnorm$(\var{bnf},ext,x,\fl)$}
! 1016: \vfill
! 1017: \copyrightnotice
! 1018: \bye
! 1019: % Local variables:
! 1020: % compile-command: "tex PARIRefCard"
! 1021: % End:
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