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 11? 25, 2017)

by OpenXM Developing Team

@overfullrule=0pt

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1 はじめに

数式処理システム asirOpenXM 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:

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.


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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$

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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.


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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.


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5 基礎(標準函数)


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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)]); 

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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);

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5.0.3 base_flatten

base_flatten(S)

: It flattens a nested list S.

Example:

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

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5.0.4 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]]);

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5.0.5 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); 

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5.0.6 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]);

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5.0.7 base_permutation

base_permutation(L)

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

Example:

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

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5.0.8 base_position

base_position(A,S)

: It returns the position of A in S.

Example:

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

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5.0.9 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);

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5.0.10 base_prune

base_prune(A,S)

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

Example:

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

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5.0.11 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]]


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5.0.12 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.


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5.0.13 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.


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5.0.14 base_set_minus

base_set_minus(A,B)

:

Example:

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

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5.0.15 base_set_union

base_set_union(A,B)

:

Example:

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

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5.0.16 base_subsetq

base_subsetq(A,B)

:

Example:

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

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5.0.17 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]);

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5.0.18 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);

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5.0.19 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).

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6 数(標準数学函数)


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6.0.1 number_abs

number_abs(X)

:

Example:

 number_abs(-3);

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6.0.2 number_ceiling

number_ceiling(X)

:

Example:

 number_abs(1.5);

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6.0.3 number_eval

number_eval(X)

:

Example:

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

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6.0.4 number_factor

number_factor(X)

: It factors the given integer X.

Example:

 number_factor(20);

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6.0.5 number_float_to_rational

number_float_to_rational(X)

:

Example:

 number_float_to_rational(1.5234); 
             number_float_to_rational(1.5234 | prec=14); 

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6.0.6 number_floor

number_floor(X)

:

Example:

 number_floor(1.5);

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6.0.7 number_imaginary_part

number_imaginary_part(X)

:

Example:

 number_imaginary_part(1+2*@i);

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6.0.8 number_is_integer

number_is_integer(X)

:

Example:

 number_is_integer(2/3);

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6.0.9 number_real_part

number_real_part(X)

:

Example:

 number_real_part(1+2*@i);

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7 微積分(標準数学函数)


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8 級数(標準数学函数)


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9 特殊函数(標準数学函数)

まだ書いてない.


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10 行列(標準数学函数)


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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]]));

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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]]));

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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]]));

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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


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10.0.5 matrix_eigenavalues

matrix_eigenavalues(M)

: It returns the eigenvalues of the matrix M.

Example:

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

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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);

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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


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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);

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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


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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]);

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10.0.11 matrix_inverse

matrix_inverse(M)

: It returns the inverse of the matrix M.

Example:

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

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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]]));

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10.0.13 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]]);

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10.0.14 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


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10.0.15 matrix_matrix_to_list

matrix_matrix_to_list(M)

: It translates the matrix M to a list.

References:

matrix_list_to_matrix


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10.0.16 matrix_rank

matrix_rank(M)

: It returns the rank of the matrix M.

Example:

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

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10.0.17 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]);

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10.0.18 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]);

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10.0.19 matrix_transpose

matrix_transpose(M)

: It returns the transpose of the matrix M.

References:

matrix_list_to_matrix


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11 Graphic(標準数学函数)

まだ書いてない.


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12 表示(標準数学函数)


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12.0.1 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


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12.0.2 print_em

print_em(S)

: It outputs S by a font to emphasize it.

Example:

 print_em(x^2-1);

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12.0.3 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


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12.0.4 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));

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12.0.5 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));

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12.0.6 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


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12.0.7 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 , ( , )


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12.0.8 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


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12.0.9 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


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12.0.10 print_terminal_form

print_terminal_form(S)

: It transforms S to the terminal form???


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12.0.11 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


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12.0.12 print_tfb_form

print_tfb_form(S)

: It transforms S to the tfb format.

Example:

 print_tfb_form(x+1);

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12.0.13 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));

References:

print_tex_form , print_dvi_form


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12.0.14 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


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13 多項式(標準数学函数)


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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);

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13.0.2 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]);

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13.0.3 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,v=key1,homo=key2,grace=key3,strategy=key4)

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

Description:

If grobner_basis is "yes", 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_*


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13.0.4 poly_expand

poly_expand(F)

: This is an alias of poly_sort.

References:

poly_sort


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13.0.5 poly_factor

poly_factor(F)

: It factorizes the polynomial F.

Example:

 poly_factor(x^10-y^10);

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13.0.6 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);

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13.0.7 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_bais


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13.0.8 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.

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]);

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13.0.9 poly_hilbert_polynomial

poly_hilbert_polynomial(I)

: It returns the Hilbert polynomial of the ideal I.

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

: This function allows optional variables s, v

Description:

The optional variable v is a list of variables.

Example:

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

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13.0.10 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);

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13.0.11 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);

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13.0.12 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);

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13.0.13 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]);

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13.0.14 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] | v=[x,y],weight=[1,0]);

References:

poly_weight_to_omatrix , ( , W , V , ) , poly_grobner_basis , poly_gr_w , poly_in_w_


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13.0.15 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 , ( , W , V , ) , poly_grobner_basis , poly_gr_w


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13.0.16 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);

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13.0.17 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]);

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13.0.18 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]);

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13.0.19 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


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13.0.20 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);

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13.0.21 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);

References:

poly_weight_to_omatrix , ( , W , V , ) , ;


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13.0.22 poly_solve_linear

poly_solve_linear(Eqs,V)

: It solves the system of linear equations Eqs with respect to the set of variables V.

Example:

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

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13.0.23 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 trucate 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.

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13.0.24 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));

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13.0.25 poly_weight_to_omatrix

poly_weight_to_omatrix(W,V)

: 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);

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14 複体(標準数学函数)


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15 グラフィックライブラリ(2次元)


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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.


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16.0.1 glib_clear

glib_clear()

: Clear the screen.


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16.0.2 glib_flush

glib_flush()

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


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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);

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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.


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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));

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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);

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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


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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);

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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).


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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).


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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


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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);

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17 OpenXM-Contrib 一般函数


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17.1 函数一覧


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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

[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;


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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.


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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


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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.


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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


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19 Asir システム管理関数


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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"]);

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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.


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20.0.1 util_damepathq

util_damepathq(S)

: When S is a string by the ShiftJIS code and S contains dame-moji with respect to \, it returns [a non-zero number, the string].

Example:

  T = [0x5c,0xe4,0x5c,0x41,0x42]$
  T2=asciitostr(T)$
  util_damepathq(T2);

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20.0.2 util_file_exists

util_file_exists(Fname)

: It returns 1 when Fname exists. It returns 0 when Fname does not exist.


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20.0.3 util_filter

util_filter(Command,Input)

: It executes the filter program Command with the Input and returns the output of the filter as a string.

util_filter(Command,Input | env=key0)

: This function allows optional variables env

Example:

 util_filter("sort","cat\ndog\ncentipede\n");

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20.0.4 util_find_and_replace

util_find_and_replace(W,S,Wnew)

: It replaces W in S by Wnew. Arguments must be a list of ascii codes.


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20.0.5 util_find_start

util_find_start()

: It tries to find the gnome-open command or an installed browser in unix systems. It returns "open" on MacOS X and returns "start" on Windows.

util_find_start( | browser=key0)

: This function allows optional variables browser


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20.0.6 util_find_substr

util_find_substr(W,S)

: It returns the position of W in S. If W cannot be found, it returns -1. Arguments must be a list of ascii codes.


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20.0.7 util_index

util_index(V)

: It returns the name part and the index part of V.

Example:

 util_index(x_2_3)

References:

util_v


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20.0.8 util_load_file_as_a_string

util_load_file_as_a_string(F)

: It reads a file F as a string.


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20.0.9 util_part

util_part(S,P,Q)

: It returns from Pth element to Qth element of S.


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20.0.10 util_read_file_as_a_string

util_read_file_as_a_string(F)

: It reads a file F as a string.


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20.0.11 util_remove_cr

util_remove_cr(S)

: It removes cr/lf/tabs from S. Arguments must be a list of ascii codes.


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20.0.12 util_timing

util_timing(Q)

: Show the timing data to execute Q.

Example:

 util_timing( quote( fctr(x^50-y^50) ));

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20.0.13 util_v

util_v(V,L)

: It returns a variable indexed by L.

Example:

 util_v("x",[1,3]);

References:

util_index


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20.0.14 util_write_string_to_a_file

util_write_string_to_a_file(Fname,S)

: It writes a string S to a file Fname.


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21 その他のマニュアル

この節では asir-contrib のその他のマニュアルを紹介する.

それからまだ分類がおわっていない関数を解説する. これらの関数は将来は別の独立した節へ移す予定である.


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21.0.1 dsolv (Solving the initial ideal for holonomic systems)

../dsolv-html/dsolv-ja.html


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21.0.2 ok_diff (Okutani’s library for differential operators)

../ok_diff-html/ok_diff-ja.html


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21.0.3 ok_dmodule (Okutani’s library for D-modules)

../ok_dmodule-html/ok_dmodule-ja.html


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21.0.4 (Plucker relations)

../plucker-html/plucker-ja.html


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21.0.5 pfpcoh (Ohara’s library for homology/cohomology groups for p F q )

../pfpcoh-html/pfpcoh-ja.html


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21.0.6 (gnuplot ox server for graphics)

../gnuplot-html/gnuplot-ja.html


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21.0.7 mathematica (Mathematica (TM) ox server)

../mathematica-html/mathematica-ja.html


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21.0.8 om (om (java) ox server for translating CMO and OpenMath)

../om-html/om-ja.html


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21.0.9 phc (PHC ox server for solving systems of algebraic equations by the homotopy method)

../phc-html/phc-ja.html


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21.0.10 sm1 (Kan/sm1 ox server for the ring of differential operators)

../sm1-html/sm1-ja.html


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21.0.11 tigers (tigers ox server for toric universal Grobner bases)

../tigers-html/tigers-ja.html


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21.0.12 f_res (Comuting resultant)

../f_res-html/f_res-ja.html


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21.0.13 mt_graph (3D grapher)

../mk_graph-html/mk_graph-ja.html


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21.0.14 noro_mwl (Mordel Weil Lattice)

../noro_mwl-html/noro_mwl-ja.html


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21.0.15 nn_ndbf (local b-function)

../nn_ndbf-html/nn_ndbf-ja.html


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21.0.16 noro_pd (New Primary Ideal Decomposition)

../noro_pd-html/noro_pd-ja.html


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21.0.17 ns_twistedlog (twisted logarithmic cohomology group)

../ns_twistedlog-html/ns_twistedlog-ja.html


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21.0.18 nk_fb_gen_c (Fisher Bingham MLE)

../nk_fb_gen_c-html/nk_fb_gen_c-ja.html


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21.0.19 gtt_ekn (Two way contingency tables by HGM)

../gtt_ekn-html/gtt_ekn-ja.html


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21.0.20 noro_module_syz (syzygies for modules)

../noro_module_syz-html/noro_module_syz-ja.html


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21.0.21 n_wishartd (restriction of matrix 1F1)

../n_wishartd-html/n_wishartd-ja.html


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21.0.22 todo_parametrize

../todo_parametrize-html/todo_parametrize-ja.html

パッケージ todo_parametrize/todo_parametrize.rr をロードすることにより, 有理曲線のパラメータ表示を見付ける関数である, paramerize が利用できるようになる. 詳しくは See 概要 in Risa/Asir 代数曲線論用パッケージ説明書 を見よ (Web 版 Risa/Asir 代数曲線論用パッケージ説明書). このパッケージのマニュアルへの統合はまだできていない. このパッケージはまだ module 構造を利用していないので, 既存のライブラリと 名前の衝突の可能性がある.

[1205] load("todo_parametrize/todo_parametrize.rr");
1
[1425] parametrize(y^2-x^3);
[155*t^2+20*t+1,720*t^4+1044*t^3+580*t^2,155*t^4+20*t^3+t^2,(-x)/(y)]
[1426] parametrize(y^2+x^3);
[-t,1,t^3,(-x)/(y)]

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21.0.23 taji_alc

../taji_alc-html/taji_alc-ja.html

パッケージ taji_alc.rr をロードすることにより, 一変数代数的コホモロジ群に関連する関数をロードできる. (Web 版 Risa/Asir 一変数代数的局所コホモロジー類に関する Risa/Aisr パッケージ説明書).

import("taji_alc.rr");
taji_alc.laurent_expansion(x,(x-1)^3);

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21.0.24 Texinifo でないマニュアル, 論文等.

OpenXM documents. には texinfo で書かれていない asir-contrib のファイルや関数のマニュアル, および 関連論文へのリンクがある.

(yang, fj_curve, nk_mora 等)


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索引

??:   A   B   G   M   N   O   P   U  
?????  ?

A
asir_contrib_update 19.0.1 asir_contrib_update

B
base_cancel 5.0.1 base_cancel
base_choose 5.0.2 base_choose
base_flatten 5.0.3 base_flatten
base_intersection 5.0.4 base_intersection
base_makelist 5.0.5 base_makelist
base_memberq 5.0.6 base_memberq
base_permutation 5.0.7 base_permutation
base_position 5.0.8 base_position
base_product 5.0.9 base_product
base_prune 5.0.10 base_prune
base_rebuild_opt 5.0.11 base_rebuild_opt
base_replace 5.0.12 base_replace
base_replace_n 5.0.13 base_replace_n
base_set_minus 5.0.14 base_set_minus
base_set_union 5.0.15 base_set_union
base_subsetq 5.0.16 base_subsetq
base_subsets_of_size 5.0.17 base_subsets_of_size
base_sum 5.0.18 base_sum
base_var_list 5.0.19 base_var_list

G
glib_clear 16.0.1 glib_clear
glib_flush 16.0.2 glib_flush
glib_line 16.0.3 glib_line
glib_open 16.0.4 glib_open
glib_plot 16.0.5 glib_plot
glib_print 16.0.6 glib_print
glib_ps_form 16.0.7 glib_ps_form
glib_putpixel 16.0.8 glib_putpixel
glib_remove_last 16.0.9 glib_remove_last
glib_set_pixel_size 16.0.10 glib_set_pixel_size
glib_tops 16.0.11 glib_tops
glib_window 16.0.12 glib_window

M
matrix_adjugate 10.0.1 matrix_adjugate
matrix_clone 10.0.2 matrix_clone
matrix_det 10.0.3 matrix_det
matrix_diagonal_matrix 10.0.4 matrix_diagonal_matrix
matrix_eigenavalues 10.0.5 matrix_eigenavalues
matrix_gauge_transformation 10.0.6 matrix_gauge_transformation
matrix_identity_matrix 10.0.7 matrix_identity_matrix
matrix_ij 10.0.8 matrix_ij
matrix_image 10.0.9 matrix_image
matrix_inner_product 10.0.10 matrix_inner_product
matrix_inverse 10.0.11 matrix_inverse
matrix_inverse_singular 10.0.12 matrix_inverse_singular
matrix_kernel 10.0.13 matrix_kernel
matrix_list_to_matrix 10.0.14 matrix_list_to_matrix
matrix_matrix_to_list 10.0.15 matrix_matrix_to_list
matrix_rank 10.0.16 matrix_rank
matrix_solve_linear 10.0.17 matrix_solve_linear
matrix_submatrix 10.0.18 matrix_submatrix
matrix_transpose 10.0.19 matrix_transpose

N
number_abs 6.0.1 number_abs
number_ceiling 6.0.2 number_ceiling
number_eval 6.0.3 number_eval
number_factor 6.0.4 number_factor
number_float_to_rational 6.0.5 number_float_to_rational
number_floor 6.0.6 number_floor
number_imaginary_part 6.0.7 number_imaginary_part
number_is_integer 6.0.8 number_is_integer
number_real_part 6.0.9 number_real_part

O
oxshell.get_value 18.0.1 oxshell.get_value
oxshell.oxshell 18.0.2 oxshell.oxshell
oxshell.set_value 18.0.3 oxshell.set_value
ox_check_errors2 17.1.1 ox_check_errors2

P
poly_coefficient 13.0.1 poly_coefficient
poly_degree 13.0.2 poly_degree
poly_elimination_ideal 13.0.3 poly_elimination_ideal
poly_expand 13.0.4 poly_expand
poly_factor 13.0.5 poly_factor
poly_gcd 13.0.6 poly_gcd
poly_grobner_basis 13.0.8 poly_grobner_basis
poly_gr_w 13.0.7 poly_gr_w
poly_hilbert_polynomial 13.0.9 poly_hilbert_polynomial
poly_ideal_colon 13.0.10 poly_ideal_colon
poly_ideal_intersection 13.0.11 poly_ideal_intersection
poly_ideal_saturation 13.0.12 poly_ideal_saturation
poly_in 13.0.13 poly_in
poly_initial 13.0.16 poly_initial
poly_initial_coefficients 13.0.17 poly_initial_coefficients
poly_initial_term 13.0.18 poly_initial_term
poly_in_w 13.0.14 poly_in_w
poly_in_w_ 13.0.15 poly_in_w_
poly_ord_w 13.0.19 poly_ord_w
poly_prime_dec 13.0.20 poly_prime_dec
poly_r_omatrix 13.0.21 poly_r_omatrix
poly_solve_linear 13.0.22 poly_solve_linear
poly_sort 13.0.23 poly_sort
poly_toric_ideal 13.0.24 poly_toric_ideal
poly_weight_to_omatrix 13.0.25 poly_weight_to_omatrix
print_dvi_form 12.0.1 print_dvi_form
print_em 12.0.2 print_em
print_gif_form 12.0.3 print_gif_form
print_input_form 12.0.4 print_input_form
print_open_math_tfb_form 12.0.5 print_open_math_tfb_form
print_open_math_xml_form 12.0.6 print_open_math_xml_form
print_output 12.0.7 print_output
print_ox_rfc100_xml_form 12.0.8 print_ox_rfc100_xml_form
print_png_form 12.0.9 print_png_form
print_terminal_form 12.0.10 print_terminal_form
print_tex_form 12.0.11 print_tex_form
print_tfb_form 12.0.12 print_tfb_form
print_xdvi_form 12.0.13 print_xdvi_form
print_xv_form 12.0.14 print_xv_form

U
util_damepathq 20.0.1 util_damepathq
util_file_exists 20.0.2 util_file_exists
util_filter 20.0.3 util_filter
util_find_and_replace 20.0.4 util_find_and_replace
util_find_start 20.0.5 util_find_start
util_find_substr 20.0.6 util_find_substr
util_index 20.0.7 util_index
util_load_file_as_a_string 20.0.8 util_load_file_as_a_string
util_part 20.0.9 util_part
util_read_file_as_a_string 20.0.10 util_read_file_as_a_string
util_remove_cr 20.0.11 util_remove_cr
util_timing 20.0.12 util_timing
util_v 20.0.13 util_v
util_write_string_to_a_file 20.0.14 util_write_string_to_a_file

??:   A   B   G   M   N   O   P   U  

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