OpenXM/Risa/Asir-Contrib

OpenXM/Risa/Asir-Contrib User’s Manual (日本語版)

Edition 1.3.2-3 for OpenXM/Asir2000

March 2017 (minor update on 5月 1, 2025)

@overfullrule=0pt

1 はじめに

数式処理システム asir は OpenXM プロトコル (Open message eXchange for Mathematics, http://www.openxm.org) をサポートしたサーバを コンポーネントとして利用できる. これらのサーバを呼ぶためのインタフェース関数はファイル ‘OpenXM/rc/asirrc’ をロードすることによりシステムに読み込まれる. Risa/Asir (OpenXM 配布版) では起動時に自動的にこのファイルが読まれる. Risa/Asir (OpenXM 配布版) は, このマニュアルでは OpenXM/Risa/Asir と呼ぶ. このマニュアルでは asir 用のこれらの関数 およびユーザ言語で書かれた数学関数およびユーティリティ関数を説明する.

HEAD branch に同期した最新版の asir-contrib マニュアルは http://www.math.kobe-u.ac.jp/OpenXM/Current/doc/index-doc-ja.html を参照.

OpenXM プロトコルの技術的詳細については, ‘$(OpenXM_HOME)/doc/OpenXM-specs’ にあるファイル ‘openxm-jp.tex’ を見て下さい.

それでは, あなたの計算機上で数学をお楽しみ下さい.

List of contributors:

• Maekawa, Masahide (Oct., 1999 – : CVS server)
• Noro, Masayuki (Jan., 1996 – : OpenXM Protocol OXRFC-100, asir2000)
• Ohara, Katsuyoshi (Jan., 1998 – : ox_math, oxc OXRFC-101)
• Takayama, Nobuki (Jan., 1996 – : OpenXM Protocol OXRFC-100, kan/sm1, asir-contrib)
• Tamura, Yasushi (Nov., 1998 – : OpenMath proxy, tfb)

• Fujimoto, Mitsushi (Windows)
• Iwane, Hidenao (Knapsack factorizer)
• Nakayama, Hiromasa (Gaussian elimination)
• Okutani, Yukio (Oct., 1999 – Feb., 2000 : matrix, diff, ...)
• Stillman, Mike (Macaulay 2 client and server)
• Tsai, Harrison (Macaulay 2 client and server)

この Contrib パッケージの 著作権については, OpenXM/Copyright を見て下さい.

有用だとおもいますが無保証です.

2 Asir/Contrib のロード方法.

‘OpenXM/rc/asirrc’ をロードすることにより Asir/Contrib の主な関数が利用可能となる. OpenXM/Risa/Asir では ASIR_CONFIG 環境変数により このファイルを起動時に読みこんでいる. ‘names.rr’ が Asir/Contrib のトップレベルのファイルである. このファイルよりその他のファイルが読み込まれている. 一部のパッケージは ‘names.rr’ からは読み込まれないので, 明示的に読み込む必要がある.

A sample of ‘asirrc’ to use Asir/Contrib.

load("gr")$
load("primdec")$
load("katsura")$
load("bfct")$
load("names.rr")$
load("oxrfc103.rr")$
User_asirrc=which(getenv("HOME")+"/.asirrc")$
if (type(User_asirrc)!=0) {
   if (!ctrl("quiet_mode")) print("Loading ~/.asirrc")$
   load(User_asirrc)$
else{ $
end$

3 Asir Contrib の函数名について

Asir Contrib には (1) 標準的な名前で定義された数学函数 (names.rr および (2) Asir 標準函数以外の有用なライブラリ函数 および (3) OpenXM サーバを asir から呼ぶための函数 が含まれている.

Asir Contrib の函数名はモジュール化されているかまたは次の形をしている: カテゴリ名_函数名

標準的な数学函数は実体へのラッパーである. たとえば sm1.hilbert は OpenXM サーバ sm1 の Hilbert 函数の計算函数を呼び出す函数である. 一方 poly_hilbert_polynomial は Asir Contrib の Hilbert 函数を計算するための (1) に属する標準的な函数名である. 標準函数 poly_hilbert_polynomial は, 現在 sm1.hilbert を呼び出して Hilbert 函数を計算しているが, これは将来変更されるかもしれない. たとえば, Asir 言語で記述された有用なライブラリ函数集 commutativeRing.rr が開発されて Hilbert 函数の計算函数 commutativeRing_hilbert_polynomial が含まれるようになったら, 標準函数 poly_hilbert_polynomial は, commutativeRing_hilbert_polynomial を呼び出して Hilbert 函数を計算するようになるかもしれない. したがって, ユーザプログラムは標準数学函数名を用いるのが望ましい.

標準数学函数名は, OpenXM project において, 全てのプロジェクトで共通の 仕様を持つように努力している. たとえば, kan/k0 も Asir Contrib と同様の標準数学函数名を持つ 予定である. 現在実験的に数学函数のカテゴリ complex 複体 (複素数でない) のマニュアルを kan/k0, asir/contrib で共通化を試みている.

以下の章は, 標準数学函数の解説をおこない, それから ライブラリ函数, それから, OpenXM サーバのインタフェースの説明を おこなう.

4 Windows 版 Asir-contrib

Windows でも不完全ながら asir-contrib が動作する. 現在, 外部コンポーネント sm1 および, 外部 コンポーネント を利用しない asir-contrib の関数が動作する. Cygwin 環境では外部コンポーネント sm1, phc が動作する. その他の外部コンポーネントは動作しない.

次の関数は Windows では動作しない. Windows での cygwin 環境では動作する場合がある.

• gnuplot.*
• om.*
• mathematica.*
• phc.*
• print_dvi_form
• print_gif_form
• print_open_math_xml_form
• print_png_form
• print_xdvi_form
• print_xv_form
• tigers_xv_form

5 基礎(標準函数)

5.0.1 base_cancel

base_cancel(S)

: It simplifies S by canceling the common factors of denominators and numerators.

Example:

 base_cancel([(x-1)/(x^2-1), (x-1)/(x^3-1)]); 

5.0.2 base_choose

base_choose(L,M)

: It returns the list of the order M subsets of L.

Example:

 base_choose([1,2,3],2);

5.0.3 base_f_definedp

base_f_definedp(Func)

: returns 1 if the function Func is defined.

5.0.4 base_flatten

base_flatten(S)

: It flattens a nested list S.

Example:

 base_flatten([[1,2,3],4]);

5.0.5 base_intersection

base_intersection(A,B)

: It returns the intersection of A and B as a set.

Example:

 base_intersection([1,2,3],[2,3,5,[6,5]]);

5.0.6 base_is_asir2018

base_is_asir2018()

: returns 1 if the system is asir2018.

5.0.7 base_is_equal

base_is_equal(L1,L2)

: returns 1 if the objects L1 and L2 are equal else return 0

5.0.8 base_ith

base_ith(A,I)

: It returns A[I].

Example:

 R=[[x,10],[y,20]]; map(base_ith,R,0);

5.0.9 base_makelist

base_makelist(Obj,K,B,T)

: base_makelist generate a list from Obj where K runs in [B,T]. Options are qt=1 (keep quote data), step (step size). When B is a list, T is ignored and K runs in B.

Example 0:

 base_makelist(k^2,k,1,10);

Example 1:

 map(print_input_form,base_makelist(quote(x^2),x,1,10 | qt=1, step=0.5))

Example 2:

 base_makelist(quote("the "+k),k,["cat","dog"],0); 

5.0.10 base_memberq

base_memberq(A,S)

: It returns 1 if A is a member of the set S else returns 0.

Example:

 base_memberq(2,[1,2,3]);

5.0.11 base_permutation

base_permutation(L)

: It outputs all permutations of L. BUG; it uses a slow algorithm.

Example:

 base_permutation([1,2,3,4]);

5.0.12 base_position

base_position(A,S)

: It returns the position of A in S.

Example:

 base_position("cat",["dog","cat","monkey"]);

5.0.13 base_preplace

base_preplace(S,Rule)

: It rewrites S by using the rule Rule. psubst is used instead of subst. The replacement is not performed for function arguments.

Example:

 base_preplace(exp(x)+x^2,[[x,a+1],[exp(x),b]]);

x is replaced by a+1 and exp(x) is replaced by b in exp(x)+x^2.

5.0.14 base_product

base_product(Obj,K,B,T)

: base_product returns the product of Obj where K runs in [B,T]. Options are qt=1 (keep quote data), step (step size). When B is a list, K runs in B and T is ignored.

Example 0:

 base_product(k^2,k,1,10);

Example 1:

 base_product(quote(x^2),x,1,10 | qt=1, step=0.5);

Example 2:

 base_product(quote(x^2),x,[a,b,c],0 | qt=1);

5.0.15 base_prune

base_prune(A,S)

: It returns a list in which A is removed from S.

Example:

 base_prune("cat",["dog","cat","monkey"]);

5.0.16 base_range

base_range(Start,End)

: It returns a list numbers [Start, Start+Step, Start+2*Step, ..., Start+n*Step] where Start+n*Step < End <= Start+(n+1)*Step Default value of step is 1.

base_range(Start,End | step=Step=key0)

: This function allows optional variables step=Step

Example:

 base_range(0,10);

5.0.17 base_rebuild_opt

base_rebuild_opt(Opt)

: It rebuilt the option list Opt

Example:

 base_rebuild_opt([[key1,1],[key2,3]] | remove_keys=["key2"]);

it returns [[key1,1]]

5.0.18 base_replace

base_replace(S,Rule)

: It rewrites S by using the rule Rule

Example:

 base_replace(x^2+y^2,[[x,a+1],[y,b]]);

x is replaced by a+1 and y is replaced by b in x^2+y^2.

5.0.19 base_replace_n

base_replace_n(S,Rule)

: It rewrites S by using the rule Rule. It is used only for specializing variables to numbers and faster than base_replace.

Example:

 base_replace_n(x^2+y^2,[[x,1/2],[y,2.0+3*@i]]);

x is replaced by 1/2 and y is replaced by 2.0+3*@i in x^2+y^2.

5.0.20 base_rest

base_rest(L)

: It returns cdr(L).

Example:

 R=[[x,10,30],[y,20,40]]; map(base_rest,R);

5.0.21 base_set_intersection

base_set_intersection(A,B)

:

Example:

 base_set_intersection([1,2,3],[3,4,5]);

5.0.22 base_set_minus

base_set_minus(A,B)

:

Example:

 base_set_minus([1,2,3],[3,4,5]);

5.0.23 base_set_union

base_set_union(A,B)

:

Example:

 base_set_union([1,2,3],[3,4,5]);

5.0.24 base_subsequenceq

base_subsequenceq(A,B)

: if A is a subsequence B, then it returns 1 else 0.

Example:

 base_subsequence([3,2,5],[1,2,3,4,5]);

5.0.25 base_subsetq

base_subsetq(A,B)

:

Example:

 base_subsetq([1,2],[1,2,3,4,5]);

5.0.26 base_subsets_of_size

base_subsets_of_size(K,S)

: It outputs all subsets of S of the size K. BUG; it uses a slow algorithm. Do not input a large S.

Example:

 base_subsets_of_size(2,[3,5,3,2]);

5.0.27 base_sum

base_sum(Obj,K,B,T)

: base_sum returns the sum of Obj where K runs in [B,T]. Options are qt=1 (keep quote data), step (step size). When B is a list, K runs in B and T is ignored. When K is 0, then Obj is assumed to be a list or vector and Obj[B]+...+Obj[T] is returned.

Example 0:

 base_sum(k^2,k,1,10);

Example 1:

 base_sum(quote(x^2),x,1,10 | qt=1, step=0.5);

Example 2:

 base_sum(quote(x^2),x,[a,b,c],0 | qt=1);

5.0.28 base_var_list

base_var_list(Name,B,T)

: base_var_list generate a list of variables Name+Index where Index runs on [B,T].

Example 0:

 base_var_list(x,0,10);

Example 1:

 base_var_list(x,1,4 | d = 1);
 Options are d=1 (add d before the name).

6 数(標準数学函数)

6.0.1 number_abs

number_abs(X)

:

Example:

 number_abs(-3);

6.0.2 number_ceiling

number_ceiling(X)

:

Example:

 number_abs(1.5);

6.0.3 number_eval

number_eval(X)

:

Example:

 number_eval([1/10^10,@pi,exp(1)]);

6.0.4 number_factor

number_factor(X)

: It factors the given integer X.

Example:

 number_factor(20);

6.0.5 number_float_to_rational

number_float_to_rational(X)

:

Example:

 number_float_to_rational(1.5234); 
             number_setprec(30); //About 30 digits after the decimal point.  It also set setprec

6.0.6 number_floor

number_floor(X)

:

Example:

 number_floor(1.5);

6.0.7 number_imaginary_part

number_imaginary_part(X)

:

Example:

 number_imaginary_part(1+2*@i);

6.0.8 number_is_integer

number_is_integer(X)

:

Example:

 number_is_integer(2/3);

6.0.9 number_real_part

number_real_part(X)

:

Example:

 number_real_part(1+2*@i);

6.0.10 number_setprec

number_setprec(X)

: When X is 0, it returns the current value of precision.

Example:

  number_setprec(30); 
   number_float_to_rational(F) returns 
   an approximation of F by a rational number with the accuracy
   about 30 digits after the decimal point.  
   It also calls setprec(30);

7 微積分(標準数学函数)

8 級数(標準数学函数)

9 特殊函数(標準数学函数)

まだ書いてない.

10 行列(標準数学函数)

10.0.1 matrix_adjugate

matrix_adjugate(M)

: It generates the adjugate matrix of the matrix M.

Example:

 matrix_adjugate(matrix_list_to_matrix([[a,b],[c,d]]));

10.0.2 matrix_clone

matrix_clone(M)

: It generates the clone of the matrix M.

Example:

 matrix_clone(matrix_list_to_matrix([[1,1],[0,1]]));

10.0.3 matrix_det

matrix_det(M)

: It returns the determinant of the matrix M.

Example:

 poly_factor(matrix_det([[1,x,x^2],[1,y,y^2],[1,z,z^2]]));

10.0.4 matrix_diagonal_matrix

matrix_diagonal_matrix(L)

: It returns the diagonal matrix with diagonal entries L.

Example:

 matrix_diagonal_matrix([1,2,3]);

References:

matrix_list_to_matrix

10.0.5 matrix_eigenavalues

matrix_eigenavalues(M)

: It returns the eigenvalues of the matrix M. if the option num=1, it returns the numerical approximate eigenvalues.

Example:

 matrix_eigenvalues([[x,1],[0,y]]);

10.0.6 matrix_gauge_transformation

matrix_gauge_transformation(M,T,V)

: It returns T^(-1) M T - T^(-1) dT/dV

Example:

 matrix_gauge_transformation([[0,x],[1,x]],[[x,0],[0,1]],x);

10.0.7 matrix_identity_matrix

matrix_identity_matrix(N)

: It returns the identity matrix of the size N.

Example:

 matrix_identity_matrix(5);

References:

matrix_diagonal_matrix

10.0.8 matrix_ij

matrix_ij(N,II,JJ)

: It returns the matrix for exchanging II-th row(col) and JJ-th row(col).

Example:

 matrix_ij(4,0,2);

10.0.9 matrix_image

matrix_image(M)

: It computes the image of M. Redundant vectors are removed.

Example:

 matrix_image([[1,2,3],[2,4,6],[1,0,0]]);

References:

matrix_kernel

10.0.10 matrix_inner_product

matrix_inner_product(A,B)

: It returns the inner product of two vectors A and B.

Example:

 matrix_inner_product([1,2],[x,y]);

10.0.11 matrix_inverse

matrix_inverse(M)

: It returns the inverse of the matrix M.

Example:

 matrix_inverse([[1,2],[0,1]]);

10.0.12 matrix_inverse_singular

matrix_inverse_singular(Mat)

: It returns a quasi-inverse matrix of Mat when it has 0-row and 0-column.

Example:

 matrix_inverse_singular(newmat(3,3,[[1,0,2],[0,0,0],[3,0,4]]));

10.0.13 matrix_is_zero

matrix_is_zero(A)

: If it is 0 matrix or 0 vector or list consisting of 0, then it returns 1 else it returns 0.

Example:

 matrix_is_zero(newmat(2,3));

10.0.14 matrix_kernel

matrix_kernel(M)

: It returns the basis of the kernel of the matrix M.

Example:

 matrix_kernel([[1,1,1,1],[0,1,3,4]]);

10.0.15 matrix_kronecker_product

matrix_kronecker_product(A,B)

: Kronecker product of the matrices A and B.

Example:

 matrix_kronecker_product([[a11,a12],[a21,a22]],[[b11,b12],[b21,b22]]);

10.0.16 matrix_list_to_matrix

matrix_list_to_matrix(M)

: It translates the list M to a matrix.

Example:

 print_xdvi_form(matrix_list_to_matrix([[1,1],[0,2]]));

References:

matrix_matrix_to_list

10.0.17 matrix_matrix_to_list

matrix_matrix_to_list(M)

: It translates the matrix M to a list.

References:

matrix_list_to_matrix

10.0.18 matrix_ones

matrix_ones(N)

: It returns the vector [1 1 ... 1] of length N. When one=m, it returns [m m ... m]. When size=[p,q] is given, N is ignored and returns p by q matrix with entries 1.

matrix_ones(N | one=m=key0,size=[p=key1,q]=key2)

: This function allows optional variables one=m, size=[p, q]

Example:

 vtol(matrix_ones(3));  returns the list [1,1,1]

10.0.19 matrix_poly_to_matrix

matrix_poly_to_matrix(Poly,Rule)

: Replace variables in the polynomial Poly by matrices in the Rule.

Example:

 matrix_poly_to_matrix(x^2-1,[[x,newmat(2,2,[[2,0],[0,3]])]]);

10.0.20 matrix_rank

matrix_rank(M)

: It returns the rank of the matrix M.

Example:

 matrix_rank([[1,1,1,1],[0,1,3,4]]);

10.0.21 matrix_rank_ff

matrix_rank_ff(Mat,P)

: It evaluates the rank of the matrix Mat by mod P. Entries may be rational numbers, and the inverse of the denominator D in F_P is properly computedd when P does not divide D, but the case P divides D does not raise an error.

10.0.22 matrix_row_matrix

matrix_row_matrix(L)

: It returns 1*n matrix [[L,L,...,L]] when L is a scalar. It returns 1*length(L) matrix [L].

matrix_row_matrix(L | size=n=key0)

: This function allows optional variables size=n

Example:

 matrix_row_matrix(1 | size=5);

10.0.23 matrix_solve_linear

matrix_solve_linear(M,X,B)

: It solves the system of linear equations M X = B

Example:

 matrix_solve_linear([[1,2],[0,1]],[x,y],[1,2]);

10.0.24 matrix_stack

matrix_stack(A,B)

: Stack the matrices A and B.

Example:

 matrix_stack([[a11,a12],[a21,a22]],[[b11,b12],[b21,b22]]);

10.0.25 matrix_submatrix

matrix_submatrix(M,Ind)

: It returns the submatrix of M defined by the index set Ind.

Example:

 matrix_submatrix([[0,1],[2,3],[4,5]],[1,2]);

10.0.26 matrix_transpose

matrix_transpose(M)

: It returns the transpose of the matrix M.

References:

matrix_list_to_matrix

11 Graphic(標準数学函数)

まだ書いてない.

12 表示(標準数学函数)

12.0.1 print_c_form

print_c_form(S)

: It transforms S to the C format or python format string.

Example 0:

 print_c_form(x^2+1);

Example 1:

 print_c_form(x^2+1 | mode=python);

Example 2:

 print_c_form(sin(x^2+1)/5 | mode=c);

12.0.2 print_dvi_form

print_dvi_form(S)

: It outputs S to a dvi file.

Example:

 print_dvi_form(x^2-1);

References:

print_xdvi_form , print_tex_form

12.0.3 print_em

print_em(S)

: It outputs S by a font to emphasize it.

Example:

 print_em(x^2-1);

12.0.4 print_format

print_format(S)

: It changes the list format of S. Options are list, sep. Defaults are list=["",""], sep=",".

Example 0:

 print_format([1,[x,y^2]]);

Example 1:

 print_format([1,[x,y^2]] | list=["(",")"], sep=" ");

Example 2:

 print_format(print_c_form([1,[x,y^2]]) );

12.0.5 print_gif_form

print_gif_form(S)

: It outputs S to a file of the gif format.

print_gif_form(S | table=key0)

: This function allows optional variables table

Example:

 print_gif_form(newmat(2,2,[[x^2,x],[y^2-1,x/(x-1)]]));

References:

print_tex_form

12.0.6 print_input_form

print_input_form(S)

: It transforms S to a string which can be parsed by asir.

Example:

 print_input_form(quote(x^3-1));

12.0.7 print_open_math_tfb_form

print_open_math_tfb_form(S)

: It transforms S to a tfb format of OpenMath XML.

Description:

It is experimental. You need to load taka_print_tfb.rr to call it.

Example:

 print_open_math_tfb_form(quote(f(x,1/(y+1))+2));

12.0.8 print_open_math_xml_form

print_open_math_xml_form(S)

: It transforms S to a string which is compliant to OpenMath(1999).

Example:

 print_open_math_xml_form(x^3-1);

References:

www.openmath.org

12.0.9 print_output

print_output(Obj)

: It outputs the object Obj to a file. If the optional variable file is set, then it outputs the Obj to the specified file, else it outputs it to "asir_output_tmp.txt". If the optional variable mode is set to "w", then the file is newly created. If the optional variable is not set, the Obj is appended to the file.

print_output(Obj | file=key0,mode=key1)

: This function allows optional variables file, mode

Example:

 print_output("Hello"|file="test.txt");

References:

glib_tops , ( , )

12.0.10 print_ox_rfc100_xml_form

print_ox_rfc100_xml_form(S)

: It transforms S to a string which is compliant to OpenXM RFC 100.

Example:

 print_ox_rfc100_xml_form(x^3-1);

References:

www.openxm.org

12.0.11 print_pdf_form

print_pdf_form(S)

: It transforms S to a pdf file and previews the file.

Example 0:

 print_pdf_form(newmat(2,2,[[x^2,x],[y^2-1,x/(x-1)]]));

Example 1:

 print_pdf_form(poly_factor(x^10-1));

Optinal variabes: nopreview=1 does not preview the PDF file.

References:

print_tex_form , print_xdvi_form

12.0.12 print_png_form

print_png_form(S)

: It transforms S to a file of the format png. dvipng should be installed.

Example:

 print_png_form(x^3-1);

References:

print_tex_form

12.0.13 print_terminal_form

print_terminal_form(S)

: It transforms S to the terminal form???

12.0.14 print_tex_form

print_tex_form(S)

: It transforms S to a string of the LaTeX format.

print_tex_form(S | table=key0,raw=key1)

: This function allows optional variables table, raw

Description:

The global variable Print_tex_form_fraction_format takes the values "auto", "frac", or "/". The global variable Print_tex_form_no_automatic_subscript takes the values 0 or 1. BUG; A large input S cannot be translated.

Example:

 print_tex_form(x*dx+1 | table=[["dx","\\partial_x"]]);

The optional variable table is used to give a translation table of asir symbols and tex symbols. when AMSTeX = 1, "begin pmatrix" and "end pmatrix" will be used to output matrix.

References:

print_xdvi_form

12.0.15 print_tfb_form

print_tfb_form(S)

: It transforms S to the tfb format.

Example:

 print_tfb_form(x+1);

12.0.16 print_xdvi_form

print_xdvi_form(S)

: It transforms S to a xdvi file and previews the file by xdvi.

Example 0:

 print_xdvi_form(newmat(2,2,[[x^2,x],[y^2-1,x/(x-1)]]));

Example 1:

 print_xdvi_form(print_tex_form(1/2) | texstr=1);

References:

print_tex_form , print_dvi_form

12.0.17 print_xv_form

print_xv_form(S)

: It transforms S to a gif file and previews the file by xv.

print_xv_form(S | input=key0,format=key1)

: This function allows optional variables input, format

Example 0:

 print_xv_form(newmat(2,2,[[x^2,x],[y^2-1,x/(x-1)]]));

Example 1:

 print_xv_form(x+y | format="png");

If the optional variable format="png" is set, png format will be used to generate an input for xv.

References:

print_tex_form , print_gif_form

13 多項式(標準数学函数)

13.0.1 poly_coefficient

poly_coefficient(F,Deg,V)

: It returns the coefficient of V^Deg in F. F may be rational or list or vector.

Example:

  F=[(x+y+z)^10/z^2,(x-y+z)^10/z^3]$
  poly_coefficient(F,10,x);

13.0.2 poly_coefficients_list

poly_coefficients_list(F,V)

: It returns the list of coefficients of F with respect to the variable list V. F may be rational or list or vector.

Example:

  F=[(x+y+c*z)^2/c^2,(x-y+c*z)^2/c^3]$
  poly_coefficients_list(F,[x,y,z]);

13.0.3 poly_coefficients_of_monomial_list

poly_coefficients_of_monomial_list(F,VV)

: It returns the list of coefficients of F with respect to a list of monomials VV.

Example:

poly_coefficients_of_monomial_list(2+3*x+4*z,[1,x,y,z]);
poly_coefficients_of_monomial_list((x+z)^3+5*y,[1,x,y,z,x^2*z]);
poly_coefficients_of_monomial_list([(x+y)^3,x+y],[x,x^2,x^3,x^2*y,x*y^2,y^3]);

References:

poly_construct_from_coefficients_of_monomial_list

13.0.4 poly_construct_from_coefficients_of_monomial_list

poly_construct_from_coefficients_of_monomial_list(L,VV)

: It returns the inner product of L and VV.

Example:

L=tk_poly_coefficients_of_monomial_list((x+y)^3,VV=[x,x^2,x^3,x^2*y,x*y^2,y^3]);
poly_construct_from_coefficients_of_monomial_list(L,VV);

References:

poly_coefficients_of_monomial_list

13.0.5 poly_dact

poly_dact(Op,F,XL)

: Act the differential operator Op to F. XL is a list of x variables.

Example:

 poly_dact( x*dx+y*dy+a, x^(-3)*y^(-2), [x,y]);

13.0.6 poly_decompose_by_weight

poly_decompose_by_weight(F,V,W)

: decompose F into homogeneous components with respect to the variable V with the weight W. The return value is [[Max_ord,Min_ord],[component of Max_ord, ..., component of Min_ord]];

Example:

 poly_decompose_by_weight(x^2*dx^2-x*(x*dx+y*dy+a),[x,y,dx,dy],[-1,-1,1,1]);

13.0.7 poly_degree

poly_degree(F)

: It returns the degree of F with respect to the given weight vector.

poly_degree(F | weight=key0,v=key1)

: This function allows optional variables weight, v

Description:

The weight is given by the optional variable weight w. It returns

Example:

 poly_degree(x^2+y^2-4 |weight=[100,1],v=[x,y]);

13.0.8 poly_denominator

poly_denominator(L)

: It returns the denominator of L. L may be a list.

Example:

  poly_denominator([1/(x^2-1),1/(x^3-1)]);

13.0.9 poly_diff2euler

poly_diff2euler(Op,XL)

: Express the differential operator Op by the euler operators. XL is a list of x variables. When XL=[x,y], dx,dy are differential operators and tx,ty are Euler operators (tx=x*dx, ty=y*dy). t stands for theta. When the return value is R, R[0]*R[1]=Op.

Example:

 poly_diff2euler(dx^2-a*x,[x]);

13.0.10 poly_dmul

poly_dmul(Op1,Op2,XL)

: Multiply Op1 and Op2 in the Weyl algebra (the ring of differential operators). XL is a list of x variables.

Example:

 poly_dmul( x*dx+y*dy+a*x, x*y*dx*dy, [x,y]);

13.0.11 poly_dvar

poly_dvar(V)

: Add d to the variable name V.

Example:

 poly_dvar([x1,x2,x3]);
          poly_dvar([x1,x2,x3] | d=t);

13.0.12 poly_elimination_ideal

poly_elimination_ideal(I,VV)

: It computes the intersection of the ideal I and the subring K[VV].

poly_elimination_ideal(I,VV | grobner_basis=key0,gb=key1,v=key2,homo=key3,grace=key4,strategy=key5)

: This function allows optional variables grobner_basis, gb, v, homo, grace, strategy

Description:

If grobner_basis is "yes" or gb=1, I is assumed to be a Grobner basis. The optional variable v is a list of variables which defines the ring of polynomials.

Example 0:

 poly_elimination_ideal([x^2+y^2-4,x*y-1],[x]);

Example 1:

 A = poly_grobner_basis([x^2+y^2-4,x*y-1]|order=2,v=[y,x]);
          poly_elimination_ideal(A,[x]|grobner_basis="yes");
 When strategy=1(default), 
   nd_gr is used when trace=0(defauult),
   nd_gr_trace is used when trace=1.

References:

gr , hgr , gr_mod , dp_*

13.0.13 poly_euler2diff

poly_euler2diff(Op,XL)

: Translate the differential operator Op expressed in terms of euler operators into the operators in terms of d. XL is a list of x variables. When XL=[x,y], dx,dy are differential operators and tx,ty are Euler operators (tx=x*dx, ty=y*dy). t stands for theta.

Example:

 poly_euler2diff(tx^2-x*(tx+1/2)^2,[x]);

13.0.14 poly_expand

poly_expand(F)

: This is an alias of poly_sort.

References:

poly_sort

13.0.15 poly_factor

poly_factor(F)

: It factorizes the polynomial F.

Example:

 poly_factor(x^10-y^10);

13.0.16 poly_gcd

poly_gcd(F,G)

: It computes the polynomial GCD of F and G.

Example:

 poly_gcd(x^10-y^10,x^25-y^25);

13.0.17 poly_gr_w

poly_gr_w(F,V,W)

: It returns the Grobner basis of F for the weight vector W. It is the second interface for poly_grobner_basis.

Example:

 poly_gr_w([x^2+y^2-1,x*y-1],[x,y],[1,0]);

References:

poly_in_w , poly_grobner_basis

13.0.18 poly_grobner_basis

poly_grobner_basis(I)

: It returns the Grobner basis of I.

poly_grobner_basis(I | order=key0,v=key1)

: This function allows optional variables order, v

Description:

The optional variable v is a list of variables which defines the ring of polynomials. Other Options; p (characteristic), homo, method (nd_gr_trace(default), nd_gr, nd_weyl_gr, nd_weyl_gr_trace, nd_f4, nd_f4_trace), order_matrix, order. See also asir manual. alias; poly_groebner_basis

Example:

 A = poly_grobner_basis([x^2+y^2-4,x*y-1]|order=2,v=[y,x],str=1);
          A->Generators;
          A->Ring->Variables;
          A->Ring->Order;
          B = poly_grobner_basis([x^2+y^2-4,x*y-1]|order=[[10,1]],v=[y,x]);
          C = poly_grobner_basis([x^2+y^2-4,x*y-1]|order=[block,[0,1],[0,1]],v=[y,x]);

13.0.19 poly_hilbert_polynomial

poly_hilbert_polynomial(I)

: It returns the Hilbert polynomial of the poly_init(I).

poly_hilbert_polynomial(I | s=key0,v=key1,sm1=key2)

: This function allows optional variables s, v, sm1

Description:

The optional variable v is a list of variables. sm1=1 forces to call sm1. [sum(H(k),{k,0,h), H(h)] where H(h) is the number of degree h monomials when h>>0. On asir2018, it returns [sum(H(k),{k,0,h), H(h),[H[0],H[1],...],F,d] where F/(1-h)^d is the Poincare series.

Example:

 poly_hilbert_polynomial([x1*y1,x1*y2,x2*y1,x2*y2]|s=k,v=[x1,x2,y1,y2]);

13.0.20 poly_ideal_colon

poly_ideal_colon(I,J,V)

: It computes the colon ideal of I by J V is the list of variables.

Example:

  B=[(x+y+z)^50,(x-y+z)^50]$
  V=[x,y,z]$
  B=poly_ideal_colon(B,[(x+y+z)^49,(x-y+z)^49],V);

13.0.21 poly_ideal_intersection

poly_ideal_intersection(I,J,V,Ord)

: It computes the intersection of the ideal I and J V is the list of variables. Ord is the order.

Example:

     A=[j*h*g*f*e*d*b,j*i*g*d*c*b,j*i*h*g*d*b,j*i*h*e*b,i*e*c*b,z]$
     B=[a*d-j*c,b*c,d*e-f*g*h]$
     V=[a,b,c,d,e,f,g,h,i,j,z]$
     poly_ideal_intersection(A,B,V,0);

13.0.22 poly_ideal_saturation

poly_ideal_saturation(I,J,V)

: It computes the saturation ideal of I by J. V is the list of variables.

Example:

  B=[(x+y+z)^50,(x-y+z)^50]$
  V=[x,y,z]$
  B=poly_ideal_saturation(B,[(x+y+z)^49,(x-y+z)^49],V);

13.0.23 poly_in

poly_in(I)

: It is an alias of poly_initial().

poly_in(I | order=key0,v=key1)

: This function allows optional variables order, v

Example:

 poly_in([x^2+y^2-4,x*y-1]|order=0,v=[x,y]);
          poly_in([x^2+y^2-4,x*y-1]|order=[1,0],v=[x,y]);

13.0.24 poly_in_w

poly_in_w(F,V,W)

: It returns the initial term or the initial ideal in_w(F) for the weight vector given by order. F is s single polynomial or a list of polynomials.

poly_in_w(F,V,W | gb=key0)

: This function allows optional variables gb

Example:

 poly_in_w([x^2+y^2-1,x*y-x] , [x,y], [1,0]);

References:

poly_weight_to_omatrix , poly_grobner_basis , poly_gr_w , poly_in_w_

13.0.25 poly_in_w_

poly_in_w_(F)

: It returns the initial term or the initial ideal in_w(F) for the weight vector given by order. F is s single polynomial or a list of polynomials. This is a new interface of poly_in_w with shorter args.

poly_in_w_(F | v=key0,weight=key1,gb=key2)

: This function allows optional variables v, weight, gb

Example:

 poly_in_w_([x^2+y^2-1,x*y-x] | v=[x,y],weight=[1,0]);

References:

poly_weight_to_omatrix , poly_grobner_basis , poly_gr_w

13.0.26 poly_initial

poly_initial(I)

: It returns the initial ideal of I with respect to the given order.

poly_initial(I | order=key0,v=key1)

: This function allows optional variables order, v

Description:

The optional variable v is a list of variables. This function computes

Example:

 poly_initial([x^2+y^2-4,x*y-1]|order=0,v=[x,y]);
    poly_initial([x^2+y^2-4,x*y-1]|order=0,v=[x,y],gb=1);
    poly_in([x^2+y^2-4,x*y-1]|order=[1,0],v=[x,y]);

13.0.27 poly_initial_coefficients

poly_initial_coefficients(I)

: It computes the coefficients of the initial ideal of I with respect to the given order.

poly_initial_coefficients(I | order=key0,v=key1)

: This function allows optional variables order, v

Description:

The optional variable v is a list of variables. The order is specified by the optional variable order

Example:

 poly_initial_coefficients([x^2+y^2-4,x*y-1]|order=0,v=[x,y]);

13.0.28 poly_initial_term

poly_initial_term(F)

: It returns the initial term of a polynomial F with respect to the given weight vector.

poly_initial_term(F | weight=key0,order=key1,v=key2)

: This function allows optional variables weight, order, v

Description:

The weight is given by the optional variable weight w. It returns

Example:

 poly_initial_term( x^2+y^2-4 |weight=[100,1],v=[x,y]);

13.0.29 poly_is_linear

poly_is_linear(F,V)

:

Example:

 poly_is_linear([x+t*y-1],[x,y]);

13.0.30 poly_lcm

poly_lcm(L)

: It returns the LCM of L[0], L[1], ...

Example:

  poly_lcm([x^2-1,x^3-1]);

13.0.31 poly_numerator

poly_numerator(L)

: It returns the numerator of L. L may be a list.

Example:

  poly_numerator([1/(x^2-1),1/(x^3-1)]);

13.0.32 poly_ord_w

poly_ord_w(F,V,W)

: It returns the order with respect to W of F.

Example:

 poly_ord_w(x^2+y^2-1,[x,y],[1,3]);

References:

poly_in_w

13.0.33 poly_pop_ord

poly_pop_ord()

: Restore the order saved by poly_push_ord.

13.0.34 poly_prime_dec

poly_prime_dec(I,V)

: It computes the prime ideal decomposition of the radical of I. V is a list of variables.

Example:

 B=[x00*x11-x01*x10,x01*x12-x02*x11,x02*x13-x03*x12,x03*x14-x04*x13,
          -x11*x20+x21*x10,-x21*x12+x22*x11,-x22*x13+x23*x12,-x23*x14+x24*x13];
          V=[x00,x01,x02,x03,x04,x10,x11,x12,x13,x14,x20,x21,x22,x23,x24];
          poly_prime_dec(B,V | radical=1);

13.0.35 poly_push_ord

poly_push_ord(Ord)

: Save the current value of dp_ord and set dp_ord(Ord).

13.0.36 poly_r_omatrix

poly_r_omatrix(N)

: It gives a weight matrix, which is used to compute a Grobner basis in K(x)<dx>, |x|=|dx|=N.

Example:

 poly_r_omatrix(3);
 When the option lex is given, the last lex variables are
 compared firstly by the lexicographic order, e.g.,
 poly_r_omatrix(4 | lex=2) is compared by the matrix
  0 0 0 0   0 0 0 1
  0 0 0 0   0 0 1 0
  0 0 0 0   1 1 0 0
  .... 

References:

poly_weight_to_omatrix

13.0.37 poly_replace_factor

poly_replace_factor(F,Rule)

: It factorizes F and replaces factors by the Rule.

Example:

 poly_replace_factor(2*x/((x-y)^3*y), [[x-y,s]]);
   It returns 2*x/(s^3*y).

13.0.38 poly_solve_linear

poly_solve_linear(Eqs,V)

: It solves the system of linear equations Eqs with respect to the set of variables V. When the option p=P is given, it solves the system by mod P. When the option reverse=1 is given, the lex order of reverse(V) is used.

Example:

 poly_solve_linear([2*x+3*y-z-2, x+y+z-1], [x,y,z]);
          poly_solve_linear([2*x+3*y-z-2, x+y+z-1], [x,y,z] | p=13);

13.0.39 poly_sort

poly_sort(F)

: It expands F with a given variables v=V and a given weight w=W. It returns a quote object. If truncate option is set, the expansion is truncated at the given degree.

poly_sort(F | v=key0,w=key1,truncate=key2)

: This function allows optional variables v, w, truncate

Example:

 poly_sort((x-y-a)^3 | v=[x,y], w=[-1,-1])  
    returns a series expansion in terms of x and y.

13.0.40 poly_subsetq

poly_subsetq(II,JJ,V)

: If the ideal II is contained in the ideal JJ, it returns 1, else 0.

Example:

 poly_subsetq([x^2-1,(x-1)*(y-2)],[x-1,y-2],[x,y]);

Optinal variabes: gb=1 (if JJ is already a GB). verbose=1 Note that when gb=1, the order must not be changed since the GB of JJ was computed. Otherwise, this function does not give correct answer or stucks. If gb=1 is not given, dp_ord(0) is executed in this function.

13.0.41 poly_toric_ideal

poly_toric_ideal(A,V)

: It returns generators of the affine toric ideal defined by the matrix(list) A. V is the list of variables.

Example:

 poly_toric_ideal([[1,1,1,1],[0,1,2,3]],base_var_list(x,0,3));

Optinal variabes: nk_toric=1 (disable 4ti2)

13.0.42 poly_w_marking

poly_w_marking(Id,V,W)

: The monomials x^a in Id is rewritten to x^a*t_w^(<a,w>+b). <a,w> is the inner product and b is an integer to avoid negative powers of t_w. Return value is [w-marked polynomial, b]

Example:

     poly_w_marking(x*dx^2+y*dy+a,[x,y,dx,dy],[-1,-1,1,1]);
       [t_w*x*dx^2+y*dy+a,0]

Optinal variabes: specify a name of homogenization variable by the option hvar. The default is t_w.

13.0.43 poly_weight_to_omatrix

poly_weight_to_omatrix(W,V)

: [obsoleted] It translates the weight vector W into a matrix, which is used to set the order in asir Grobner basis functions. V is the list of variables.

Example:

 M=poly_weight_to_omatrix([2,1,0],[x,y,z]);
          nd_gr([x^3+z^3-1,x*y*z-1,y^2+z^2-1,[x,y,z],0,M);

13.0.44 poly_weight_to_ord_matrix

poly_weight_to_ord_matrix(W)

: Weight vector W is transformed to a matrix defined order for dp_ord, nd_gr, ... It is a new version of poly_weight_to_omatrix(W,V) [obsoleted]

Example:

     Mat=poly_weight_to_ord_matrix([1,1,1,1,0,1,1,1,1,0]);
     Mat=poly_weight_to_ord_matrix([]|tie_breaker=[lex,0,1,2,3,5,6,7,8,4,9]);

Optinal variabes: tie_breaker=[lex,n1,n2,n3,...] defines the lexcographic order x_n1,x_n2, x_n3, ... when variables are x_*

13.0.45 poly_weyl_subsetq

poly_weyl_subsetq(II,JJ,V)

: If the ideal II in the Weyl algebra is contained in the ideal JJ, it returns 1, else 0.

Example:

 poly_weyl_subsetq([x*dx^2],[x*dx-1],[x,dx]);

Optinal variabes: gb=1 (if JJ is already a GB). verbose=1. Note that when gb=1, the order must not be changed since the GB of JJ was computed. Otherwise, this function does not give correct answer or stucks. If gb=1 is not given, dp_ord(0) is executed in this function.

14 複体(標準数学函数)

15 グラフィックライブラリ(2次元)

16 Graphic Library (2 dimensional)

ライブラリ glib は, Risa/Asir の グラフィック基本関数 (draw_obj) に対する, 昔の BASIC のような単純なインタフェースを提供する.

16.0.1 glib_clear

glib_clear()

: Clear the screen.

16.0.2 glib_flush

glib_flush()

: ; Flush the output. (Cfep only. It also set initGL to 1.).

16.0.3 glib_line

glib_line(X0,Y0,X1,Y1)

: It draws the line [X0,Y0]– [X1,Y1] with color and shape

glib_line(X0,Y0,X1,Y1 | color=key0,shape=key1)

: This function allows optional variables color, shape

Example:

 glib_line(0,0,5,3/2 | color=0xff00ff);
           glib_line(0,0,10,0 | shape=arrow);

16.0.4 glib_open

glib_open()

: It starts the ox_plot server and opens a canvas. The canvas size is set to Glib_canvas_x X Glib_canvas_y (the default value is 400). This function is automatically called when the user calls glib functions.

16.0.5 glib_plot

glib_plot(F)

: It plots an object F on the glib canvas.

Example 0:

 glib_plot([[0,1],[0.1,0.9],[0.2,0.7],[0.3,0.5],[0.4,0.8]]);

Example 1:

 glib_plot(tan(x));

16.0.6 glib_print

glib_print(X,Y,Text)

: It put a string Text at [X,Y] on the glib canvas.

glib_print(X,Y,Text | color=key0)

: This function allows optional variables color

Example:

 glib_print(100,100,"Hello Worlds" | color=0xff0000);

16.0.7 glib_ps_form

glib_ps_form(S)

: It returns the PS code generated by executing S (experimental).

Example 0:

 glib_ps_form(quote( glib_line(0,0,100,100) ));

Example 1:

 glib_ps_form(quote([glib_line(0,0,100,100),glib_line(100,0,0,100)]));

References:

glib_tops

16.0.8 glib_putpixel

glib_putpixel(X,Y)

: It puts a pixel at [X,Y] with color

glib_putpixel(X,Y | color=key0)

: This function allows optional variables color

Example:

 glib_putpixel(1,2 | color=0xffff00);

16.0.9 glib_remove_last

glib_remove_last()

: Remove the last object. glib_flush() should also be called to remove the last object. (cfep only).

16.0.10 glib_set_pixel_size

glib_set_pixel_size(P)

: Set the size of putpixel to P. 1.0 is the default. (cfep only).

16.0.11 glib_tops

glib_tops()

: If Glib_ps is set to 1, it returns a postscript program to draw the picture on the canvas.

References:

print_output

16.0.12 glib_window

glib_window(Xmin,Ymin,Xmax,Ymax)

: It generates a window with the left top corner [Xmin,Ymin] and the right bottom corner [Xmax,Ymax]. If the global variable Glib_math_coordinate is set to 1, mathematical coordinate system will be employed, i.e., the left top corner will have the coordinate [Xmin,Ymax].

Example:

 glib_window(-1,-1,10,10);

17 OpenXM-Contrib 一般函数

17.1 函数一覧

17.1.1 ox_check_errors2

ox_check_errors2(p)

:: 識別番号 p のサーバのスタック上にあるエラーオブジェクトをリストで戻す.

return

リスト

p

数

• 識別番号 p のサーバのスタック上にあるエラーオブジェクトをリストで戻す.
• エラーオブジェクトのポップはしない.

[219] P=sm1.start();
0
[220] sm1.sm1(P," 0 get ");
0
[221] ox_check_errors2(P);
[error([7,4294967295,executeString: Usage:get])]
Error on the server of the process number = 1
To clean the stack of the ox server,
type in ox_pops(P,N) (P: process number, N: the number of data you need to pop)
out of the debug mode.
If you like to automatically clean data on the server stack,
set XM_debug=0;

18 OXshell の関数

OXshell はシステムのコマンドを ox server より実行する仕組みである. 詳しくは OpenXM/src/kan96xx/Doc/oxshell.oxw および OpenXM/doc/Papers/rims-2003-12-16-ja.tex を見よ.

18.0.1 oxshell.get_value

oxshell.get_value(NAME,V)

: It get the value of the variable NAME on the server ox_shell.

Example:

 oxshell.set_value("abc","Hello world!");
           oxshell.oxshell(["cp", "stringIn://abc", "stringOut://result"]);
           oxshell.get_value("result");
   What we do is a file $TMP/abc* is generated with the contents Hello world! and copied to $TMP/result*
   The contents of the file is stored in the variable result on ox_sm1.

References:

oxshell.oxshell , oxshell.set_value

18.0.2 oxshell.oxshell

oxshell.oxshell(L)

: It executes command L on a ox_shell server. L must be an array. The result is the outputs to stdout and stderr. A temporary file will be generated under $TMP. cf. oxshell.keep_tmp()

Example:

 oxshell.oxshell(["ls"]);

References:

ox_shell , oxshell.set_value , oxshell.get_value , oxshell , of , sm1.

18.0.3 oxshell.set_value

oxshell.set_value(NAME,V)

: It set the value V to the variable Name on the server ox_shell.

Example:

 oxshell.set_value("abc","Hello world!");
           oxshell.oxshell(["cat", "stringIn://abc"]);

References:

oxshell.oxshell , oxshell.get_value

19 Asir システム管理関数

19.0.1 asir_contrib_update

asir_contrib_update()

: It updates the asir-contrib library and/or some other files to the HEAD branch. The usage will be shown by asir_contrib_update() without the option update. Options are update, clean, url, install_dir, zip_files, tmp. Default values update=0, clean=0, url="http://www.math.kobe-u.ac.jp/OpenXM/Current", install_dir=%APPDATA%/OpenXM (win) or install_dir=$OpenXM_tmp/OpenXM (others) zip_files=["lib-asir-contrib.zip"]

Example:

 
   asir_contrib_update();
   asir_contrib_update(|update=1);    update the library 
   asir_contrib_update(|update=3);    update the library and the documents
   asir_contrib_update(|clean=1);
   asir_contrib_update(|zip_files=["lib-asir-contrib.zip","doc-asir2000.zip","doc-asir-contrib.zip","doc-other-docs.zip"]);

20 便利な関数

システムの資源にアクセスするためおよび文字列処理の便利な関数を集めてある.