=================================================================== RCS file: /home/cvs/OpenXM/src/asir-contrib/packages/doc/mt_gkz/mt_gkz-en.texi,v retrieving revision 1.3 retrieving revision 1.7 diff -u -p -r1.3 -r1.7 --- OpenXM/src/asir-contrib/packages/doc/mt_gkz/mt_gkz-en.texi 2021/03/02 10:57:17 1.3 +++ OpenXM/src/asir-contrib/packages/doc/mt_gkz/mt_gkz-en.texi 2022/01/13 02:38:00 1.7 @@ -1,4 +1,4 @@ -%% $OpenXM: OpenXM/src/asir-contrib/packages/doc/mt_gkz/mt_gkz-en.texi,v 1.2 2021/01/20 08:17:54 takayama Exp $ +%% $OpenXM: OpenXM/src/asir-contrib/packages/doc/mt_gkz/mt_gkz-en.texi,v 1.6 2021/12/11 11:40:45 takayama Exp $ %% xetex mt_gkz-en.texi (.texi までつける. ) %% @math{tex形式の数式} %% 参考: http://www.fan.gr.jp/~ring/doc/texinfo/texinfo-ja_14.html#SEC183 @@ -19,7 +19,7 @@ @comment --- おまじない終り --- @comment --- GNU info ファイルの名前 --- -@setfilename mt_gkz_man +@setfilename asir-contrib-mt_gkz_man @comment --- タイトル --- @settitle GKZ hypergeometric system @@ -38,7 +38,7 @@ @title GKZ hypergeometric system @subtitle Pfaffian system (Pfaff equation), contiguity relations, cohomology intersection @subtitle Version 1.0 -@subtitle January 20, 2021 +@subtitle December 21, 2021 @author by S-J. Matsubara-Heo, N.Takayama @page @@ -123,6 +123,8 @@ systems, Linear Algebra and its Applications (LAA), 42 * mt_gkz.ff1:: * mt_gkz.ff2:: * mt_gkz.rvec_to_fvec:: +* mt_gkz.fvec_to_conn_mat:: +* mt_gkz.contiguity:: @end menu @node Pfaff equation for given cocycles,,, Pfaff equation @@ -134,6 +136,7 @@ systems, Linear Algebra and its Applications (LAA), 42 @subsection @code{mt_gkz.pfaff_eq} @comment --- 索引用キーワード @findex mt_gkz.pfaff_eq +@findex mt_gkz.use_hilbert_driven @table @t @item mt_gkz.pfaff_eq(@var{A},@var{Beta},@var{Ap},@var{Rvec},@var{DirX}) @@ -151,7 +154,7 @@ the parameter vector of the GKZ system. @item Ap See [MT2020]. @item Rvec -It is used to specify a basis of cocycles. See [MT2020] +It is used to specify a basis of cocycles as explained below. See also [MT2020]. @item DirX a list of dxi's. @end table @@ -168,6 +171,9 @@ the set of the cocycles standing for Av_1, Av_2, ..., (see [MT2020]) is supposed to be the basis to construct the Pfaffian system. +The exponents @math{(q_\ell, q)} of the integral representation +@math{\int \prod h_\ell^{-q_\ell} x^q {{dx} \over {x}}} +is shifted by Av_i@math{=:A_{v_i}} as @math{(q_\ell,q)+A_{v_i}}. Let a_1, a_2, ..., a_n be the column vectors of the matrix A and v be a column vector (x_1, x_2, ..., x_n)^T. Av is defined as a_1 x_1 + a_2 x_2 + ... + a_n x_n. @@ -225,6 +231,10 @@ Option @var{cg}. A constant matrix given by this optio for the Gauge transformation of the Pfaffian system. In other words, the basis of cocycles specified by @var{Rvec} is transformed by the constant matrix given by this option. +@item +By mt_gkz.use_hilbert_driven(Rank), the rank of the GKZ system is assumed to be +Rank. It makes the computation of Groebner basis by yang.rr faster. +This option is disabled by mt_gkz.use_hilbert_driven(0); @end itemize @comment --- @example〜@end example は実行例の表示 --- @@ -298,7 +308,7 @@ stopped in step_up at line 342 in file "./mt_gkz/saito @findex mt_gkz.ff @table @t -@item mt_gkz.ff(@var{Rvec0},@var{A},@var{Beta},@var{Ap}) +@item mt_gkz.ff(@var{Rvec0},@var{A},@var{Ap},@var{Beta}) @item mt_gkz.ff1(@var{Rvec0},@var{A},@var{Beta},@var{Ap}) @item mt_gkz.ff2(@var{Rvec0},@var{A},@var{Beta},@var{Ap},@var{BF},@var{C}) :: @code{ff} returns a differential operator whose action to 1 gives @@ -556,6 +566,55 @@ obtains connection matrices. @end table @comment おわり. +@comment --- contiguity +@comment --- section 名を正確に --- +@node mt_gkz.contiguity,,, Pfaff equation for given cocycles +@subsection @code{mt_gkz.contiguity} +@comment --- 索引用キーワード +@findex mt_gkz.contiguity + +@table @t +@item mt_gkz.contiguity(@var{A},@var{Beta},@var{Ap},@var{Rvec1},@var{Rvec2}) +:: It returns the coefficient matrix P that satisfies +@var{Rvec1} = P @var{Rvec2}. +@end table + +@comment --- 引数の簡単な説明 --- +@table @var +@item return +The coefficient matrix P that satisfies @var{Rvec1} = P @var{Rvec2}. +@item A Beta Ap Rvec1 Rvec2 +Same with @ref{mt_gkz.pfaff_eq}. +@end table + +@comment --- ここで関数の詳しい説明 --- +@comment --- @itemize〜@end itemize は箇条書き --- +@comment --- @bullet は黒点付き --- +@itemize @bullet +@item +It returns the contiguity relation between +@var{Rvec1} and @var{Rvec2} +@end itemize + +@comment --- @example〜@end example は実行例の表示 --- +Example: +@example +[1883] import("mt_gkz.rr"); +[3200] PP=mt_gkz.contiguity(A=[[1,1,0,0],[0,0,1,1],[0,1,0,1]], + Beta=[-g1,-g2,-c], + Ap = [[1,1,0,0],[0,0,1,1],[0,0,0,0]], + Rvec1 = [[1,0,0,0],[0,0,1,0]], + Rvec2 = [[0,0,1,0],[1,0,0,0]]); +@end example + +@comment --- 参照(リンク)を書く --- +@table @t +@item Refer to +@ref{mt_gkz.pfaff_eq} +@ref{mt_gkz.fvec_to_conn_mat} +@end table +@comment おわり. + @comment ---------- New Chapter --------------- @node b function,,, Top @chapter b function @@ -697,6 +756,7 @@ Example: * mt_gkz.get_check_fvec:: * mt_gkz.get_bf_step_up:: * mt_gkz.mytoric_ideal:: +* mt_gkz.cbase_by_euler:: @end menu @node some utility functions,,, utilities @@ -713,6 +773,7 @@ Example: @node mt_gkz.get_check_fvec,,, some utility functions @node mt_gkz.get_bf_step_up,,, some utility functions @node mt_gkz.mytoric_ideal,,, some utility functions +@node mt_gkz.cbase_by_euler,,, some utility functions @findex mt_gkz.reduce_by_toric @findex mt_gkz.tk_base_equal @@ -725,6 +786,7 @@ Example: @findex mt_gkz.get_check_fvec @findex mt_gkz.get_bf_step_up @findex mt_gkz.mytoric_ideal +@findex mt_gkz.cbase_by_euler @comment --- @example〜@end example は実行例の表示 --- We only show examples on these functions. As for details, please see @@ -789,6 +851,12 @@ the source code. [3192] mt_gkz.mytoric_ideal(0 | use_4ti2=0); // A slower method is used to obtain a generator set of the toric ideal // defined by the matrix A. 4ti2 is not needed. Default. +[3193] mt_gkz.cbase_by_euler(A=[[1,1,1,1],[0,1,3,4]]); +// Cohomology basis of the GKZ system defined by A for generic beta. +// Basis is given by a set of Euler operators. +// It is an implementation of the algorithm in http://dx.doi.org/10.1016/j.aim.2016.10.021 +// beta is set by random numbers. Option: no_prob=1 + @end example