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Annotation of OpenXM/doc/calc2000p/func1.tex, Revision 1.4

1.4     ! takayama    1: % $OpenXM: OpenXM/doc/calc2000p/func1.tex,v 1.3 2000/07/31 07:26:12 noro Exp $
1.1       noro        2: \documentclass[twocolumn]{article}
                      3: \pagestyle{empty}
                      4: %\markright{ {\tt http://www.openxm.org} }
                      5: \usepackage{color}
                      6: \usepackage{epsfig}
1.4     ! takayama    7: \title{\huge \color{blue} 1077 functions are available
1.1       noro        8: on our servers and libraries}
                      9: \author{} \date{}
                     10: \begin{document}
                     11: \maketitle
                     12:
                     13: \noindent
                     14: \fbox{\huge {\color{green}Operations on Integers}}
                     15:
                     16: \noindent
                     17: {\color{red} idiv},{\color{red} irem} (division with remainder),
                     18: {\color{red} ishift} (bit shifting),
                     19: {\color{red} iand},{\color{red} ior},{\color{red} ixor} (logical operations),
                     20: {\color{red} igcd},(GCD by various methods such as Euclid's algorithm and
                     21: the accelerated GCD algorithm),
                     22: {\color{red} fac} (factorial),
                     23: {\color{red} inv} (inverse modulo an integer),
1.3       noro       24: {\color{red} random} (random number generator by the Mersenne twister algorithm).
1.1       noro       25:
                     26:
                     27:
                     28: \medbreak
                     29:
                     30: \noindent
                     31: \fbox{\huge {\color{green}Ground Fields}}
                     32:
                     33: \noindent
                     34: Arithmetics on various fields: the rationals,
1.2       noro       35: ${\bf Q}(\alpha_1,\alpha_2,\ldots,\alpha_n)$
                     36: ($\alpha_i$ is algebraic over ${\bf Q}(\alpha_1,\ldots,\alpha_{i-1})$),
1.1       noro       37: $GF(p)$ ($p$ is a prime of arbitrary size), $GF(2^n)$.
                     38:
                     39: \medbreak
                     40:
                     41: \noindent
                     42: \fbox{\huge {\color{green}Operations on Polynomials}}
                     43:
                     44: \noindent
                     45: {\color{red} sdiv }, {\color{red} srem } (division with remainder),
                     46: {\color{red} ptozp } (removal of the integer content),
                     47: {\color{red} diff } (differentiation),
                     48: {\color{red} gcd } (GCD over the rationals),
                     49: {\color{red} res } (resultant),
                     50: {\color{red} subst } (substitution),
                     51: {\color{red} umul} (fast multiplication of dense univariate polynomials
                     52: by a hybrid method with Karatsuba and FFT+Chinese remainder),
                     53: {\color{red} urembymul\_precomp} (fast dense univariate polynomial
                     54: division with remainder by the fast multiplication and
                     55: the precomputed inverse of a divisor),
                     56:
                     57: \noindent
                     58: \fbox{\huge {\color{green}Polynomial Factorization}}
                     59: {\color{red} fctr } (factorization over the rationals),
                     60: {\color{red} fctr\_ff } (univariate factorization over finite fields),
                     61: {\color{red} af } (univariate factorization over algebraic number fields),
                     62: {\color{red} sp} (splitting field computation).
                     63:
                     64: \medbreak
                     65:
                     66: \noindent
                     67: \fbox{\huge {\color{green} Groebner basis}}
                     68:
                     69: \noindent
                     70: {\color{red} dp\_gr\_main } (Groebner basis computation of a polynomial ideal
                     71: over the rationals by the trace lifting),
                     72: {\color{red} dp\_gr\_mod\_main } (Groebner basis over small finite fields),
                     73: {\color{red} tolex } (Modular change of ordering for a zero-dimensional ideal),
                     74: {\color{red} tolex\_gsl } (Modular rational univariate representation
                     75: for a zero-dimensional ideal),
                     76: {\color{red} dp\_f4\_main } ($F_4$ over the rationals),
                     77: {\color{red} dp\_f4\_mod\_main } ($F_4$ over small finite fields).
                     78:
                     79: \medbreak
                     80: \noindent
                     81: \fbox{\huge {\color{green} Ideal Decomposition}}
                     82:
                     83: \noindent
                     84: {\color{red} primedec} (Prime decomposition of the radical),
                     85: {\color{red} primadec} (Primary decomposition of ideals by Shimoyama/Yokoyama algorithm).
                     86:
                     87: \medbreak
                     88:
                     89: \noindent
                     90: \fbox{\huge {\color{green} Quantifier Elimination}}
                     91:
                     92: \noindent
                     93: {\color{red} qe} (real quantifier elimination in a linear and
                     94: quadratic first-order formula),
                     95: {\color{red} simpl} (heuristic simplification of a first-order formula).
                     96:
1.3       noro       97: {\scriptsize
                     98: \begin{verbatim}
                     99: [0] MTP2 = ex([x11,x12,x13,x21,x22,x23,x31,x32,x33],
                    100: x11+x12+x13 @== a1 @&& x21+x22+x23 @== a2 @&& x31+x32+x33 @== a3
                    101: @&& x11+x21+x31 @== b1 @&& x12+x22+x32 @== b2 @&& x13+x23+x33 @== b3
                    102: @&& 0 @<= x11 @&& 0 @<= x12 @&& 0 @<= x13 @&& 0 @<= x21
                    103: @&& 0 @<= x22 @&& 0 @<= x23 @&& 0 @<= x31 @&& 0 @<= x32 @&& 0 @<= x33)$
                    104: [1] TSOL= a1+a2+a3@=b1+b2+b3 @&& a1@>=0 @&& a2@>=0 @&& a3@>=0
                    105: @&& b1@>=0 @&& b2@>=0 @&& b3@>=0$
                    106: [2] QE_MTP2 = qe(MTP2)$
                    107: [3] qe(all([a1,a2,a3,b1,b2,b3],QE_MTP2 @equiv TSOL));
                    108: @true
                    109: \end{verbatim}}
1.1       noro      110: \medbreak
                    111:
                    112: \noindent
                    113: \fbox{\huge {\color{green} Visualization of curves}}
                    114:
                    115: \noindent
                    116: {\color{red} plot} (plotting of a univariate function),
                    117: {\color{red} ifplot} (plotting zeros of a bivariate polynomial),
                    118: {\color{red} conplot} (contour plotting of a bivariate polynomial function).
                    119:
                    120: \medbreak
                    121:
                    122: \noindent
                    123: \fbox{\huge {\color{green} Miscellaneous functions}}
                    124:
                    125: \noindent
                    126: {\color{red} det} (determinant),
                    127: {\color{red} qsort} (sorting of an array by the quick sort algorithm),
1.2       noro      128: {\color{red} eval} (evaluation of a formula containing transcendental functions
                    129: such as
                    130: {\color{red} sin}, {\color{red} cos}, {\color{red} tan}, {\color{red} exp},
                    131: {\color{red} log})
1.1       noro      132: {\color{red} roots} (finding all roots of a univariate polynomial),
                    133: {\color{red} lll} (computation of an LLL-reduced basis of a lattice).
                    134:
                    135: \medbreak
                    136: \vfill
                    137: \noindent
                    138: \rightline{ {\color{red} {\tt http://www.openxm.org} }}
                    139:
                    140: \end{document}