version 1.2, 2009/12/02 04:38:47 |
version 1.7, 2017/03/30 06:16:36 |
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%comment $OpenXM: OpenXM/src/asir-contrib/packages/doc/nn_ndbf/nn_ndbf.texi,v 1.1 2009/12/02 02:55:29 noro Exp $ |
%comment $OpenXM: OpenXM/src/asir-contrib/packages/doc/nn_ndbf/nn_ndbf.texi,v 1.6 2010/06/19 09:43:45 noro Exp $ |
%comment --- おまじない --- |
%comment --- 障 --- |
\input ../../../../asir-doc/texinfo |
\input ../../../../asir-doc/texinfo |
@iftex |
@iftex |
@catcode`@#=6 |
@catcode`@#=6 |
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@overfullrule=0pt |
@overfullrule=0pt |
@c -*-texinfo-*- |
@c -*-texinfo-*- |
@comment %**start of header |
@comment %**start of header |
@comment --- おまじない終り --- |
@comment --- 障腟 --- |
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@comment --- GNU info ファイルの名前 --- |
@comment --- GNU info <ゃ --- |
@setfilename asir-contrib-nn_ndbf |
@setfilename asir-contrib-nn_ndbf |
@comment @documentlanguage ja |
@comment @documentlanguage ja |
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@comment --- タイトル --- |
@comment --- 帥ゃ --- |
@settitle nn_ndbf |
@settitle nn_ndbf |
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@comment %**end of header |
@comment %**end of header |
@comment %@setchapternewpage odd |
@comment %@setchapternewpage odd |
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@comment --- おまじない --- |
@comment --- 障 --- |
@ifinfo |
@ifinfo |
@macro fref{name} |
@macro fref{name} |
@ref{\name\,,@code{\name\}} |
@ref{\name\,,@code{\name\}} |
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@end iftex |
@end iftex |
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@titlepage |
@titlepage |
@comment --- おまじない終り --- |
@comment --- 障腟 --- |
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@comment --- タイトル, バージョン, 著者名, 著作権表示 --- |
@comment --- 帥ゃ, 若吾с, , 篏罔茵腓 --- |
@title nn_ndbf |
@title nn_ndbf |
@subtitle nn_ndbf User's Manual |
@subtitle nn_ndbf User's Manual |
@subtitle Edition 1.0 |
@subtitle Edition 1.0 |
Line 50 Copyright @copyright{} Masayuki Noro and Kenta Nishiya |
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Line 50 Copyright @copyright{} Masayuki Noro and Kenta Nishiya |
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2009. All rights reserved. |
2009. All rights reserved. |
@end titlepage |
@end titlepage |
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@comment --- おまじない --- |
@comment --- 障 --- |
@synindex vr fn |
@synindex vr fn |
@comment --- おまじない終り --- |
@comment --- 障腟 --- |
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@comment --- @node は GNU info, HTML 用 --- |
@comment --- @node GNU info, HTML --- |
@comment --- @node の引数は node-name, next, previous, up --- |
@comment --- @node 綣違 node-name, next, previous, up --- |
@node Top,, (dir), (dir) |
@node Top,, (dir), (dir) |
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@comment --- @menu は GNU info, HTML 用 --- |
@comment --- @menu GNU info, HTML --- |
@comment --- chapter 名を正確に並べる --- |
@comment --- chapter 罩g∈筝鴻 --- |
@menu |
@menu |
\JP * 新 b 関数パッケージ nn_ndbf.rr:: |
\JP * b ∽違宴若 nn_ndbf.rr:: |
\EG * New b-function package nn_ndbf.rr:: |
\EG * New b-function package nn_ndbf.rr:: |
* Index:: |
* Index:: |
@end menu |
@end menu |
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@comment --- chapter の開始 --- |
@comment --- chapter 紮 --- |
@comment --- 親 chapter 名を正確に --- |
@comment --- 荀 chapter 罩g∈ --- |
\JP @node 新 b 関数パッケージ nn_ndbf.rr,,, Top |
\JP @node b ∽違宴若 nn_ndbf.rr,,, Top |
\EG @node New b-function package nn_ndbf.rr,,, Top |
\EG @node New b-function package nn_ndbf.rr,,, Top |
\JP @chapter New b-function package nn_ndbf.rr |
\JP @chapter New b-function package nn_ndbf.rr |
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@comment --- section 名を正確に並べる --- |
@comment --- section 罩g∈筝鴻 --- |
@menu |
@menu |
\JP * b 関数計算:: |
\JP * b ∽域膊:: |
\EG * Computation of b-function:: |
\EG * Computation of b-function:: |
\JP * Annihilator イデアル計算:: |
\JP * Annihilator ゃ≪荐膊:: |
\EG * Computation of annihilator ideal:: |
\EG * Computation of annihilator ideal:: |
@end menu |
@end menu |
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\BJP |
\BJP |
このマニュアルでは, asir-contrib パッケージに収録されている, |
ャ≪с, asir-contrib 宴若吾蚊, |
新 b 関数パッケージ @samp{nn_ndbf.rr} について解説する. |
b ∽違宴若 @samp{nn_ndbf.rr} ゃ茹h. |
このパッケージを使うには, まず @samp{nn_ndbf.rr} をロードする. |
宴若吾篏帥, 障 @samp{nn_ndbf.rr} 若. |
\E |
\E |
\BEG |
\BEG |
In this manual we explain about a new b-function package @samp{nn_ndbf.rr} |
In this manual we explain about a new b-function package @samp{nn_ndbf.rr} |
in asir-contrib. To use this package one has to load @samp{nn_ndbf.rr}. |
in asir-contrib. To use this package one has to load @samp{nn_ndbf.rr}. |
\E |
\E |
@example |
@example |
[1518] load("nn_ndbf.rr"); |
[...] load("nn_ndbf.rr"); |
@end example |
@end example |
\BJP |
\BJP |
このパッケージの函数を呼び出すには, 全て @code{ndbf.} を先頭につける. |
宴若吾醇違若喝冴, @code{ndbf.} ゃ. |
このマニュアルでは, 関連する組込み関数についても解説する. |
ャ≪с, ∫c腟莨若翠∽違ゃ茹h. |
\E |
\E |
\BEG |
\BEG |
A prefix @code{ndbf.} is necessary to call the functions in this package. |
A prefix @code{ndbf.} is necessary to call the functions in this package. |
In this manual we also explain about some related built-in functions. |
In this manual we also explain about some related built-in functions. |
\E |
\E |
@comment --- section の開始 --- |
@comment --- section 紮 --- |
@comment --- 書体指定について --- |
@comment --- 娯絎ゃ --- |
@comment --- @code{} はタイプライタ体表示 --- |
@comment --- @code{} 帥ゃゃ推茵腓 --- |
@comment --- @var{} は斜字体表示 --- |
@comment --- @var{} 絖篏茵腓 --- |
@comment --- @b{} はボールド表示 --- |
@comment --- @b{} 若茵腓 --- |
@comment --- @samp{} はファイル名などの表示 --- |
@comment --- @samp{} <ゃ茵腓 --- |
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\JP @node b 関数計算,,, 新 b 関数パッケージ nn_ndbf.rr |
\JP @node b ∽域膊,,, b ∽違宴若 nn_ndbf.rr |
\EG @node Computation of b-function,,, New b-function package nn_ndbf.rr |
\EG @node Computation of b-function,,, New b-function package nn_ndbf.rr |
\JP @section b 関数計算 |
\JP @section b ∽域膊 |
\EG @section Computation of b-function |
\EG @section Computation of b-function |
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@menu |
@menu |
* ndbf.bfunction:: |
* ndbf.bfunction:: |
* ndbf.bf_local:: |
* ndbf.bf_local:: |
* ndbf.bf_strat:: |
* ndbf.bf_strat:: |
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* ndbf.action_on_gfs:: |
@end menu |
@end menu |
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\JP @node ndbf.bfunction,,, b 関数計算 |
\JP @node ndbf.bfunction,,, b ∽域膊 |
\EG @node ndbf.bfunction,,, Computation of b-function |
\EG @node ndbf.bfunction,,, Computation of b-function |
@subsection @code{ndbf.bfunction} |
@subsection @code{ndbf.bfunction} |
@findex ndbf.bfunction |
@findex ndbf.bfunction |
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@table @t |
@table @t |
\JP @item ndbf.bfunction(@var{f}[|weight=@var{w},heruristic=@var{yesno},vord=@var{v}]) :: 多項式 @var{f} の大域 b 関数を計算する. |
\JP @item ndbf.bfunction(@var{f}[|weight=@var{w},heruristic=@var{yesno},vord=@var{v},op=@var{yesno}]) :: 紊綣 @var{f} 紊у b ∽違荐膊. |
\EG @item ndbf.bfunction(@var{f}[|weight=@var{w},heruristic=@var{yesno},vord=@var{v}]) :: computes the global b-function of a polynomial @var{f} |
\EG @item ndbf.bfunction(@var{f}[|weight=@var{w},heruristic=@var{yesno},vord=@var{v},op=@var{yesno}]) :: computes the global b-function of a polynomial @var{f} |
@end table |
@end table |
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@table @var |
@table @var |
@item return |
@item return |
\JP 多項式 |
\JP 紊綣 |
\EG a polynomial |
\EG a polynomial |
@item f |
@item f |
\JP 多項式 |
\JP 紊綣 |
\EG a polynomial |
\EG a polynomial |
@item w |
@item w |
\JP @code{[@var{v1,w1,...,vn,wn}]} なるリスト |
\JP @code{[@var{v1,w1,...,vn,wn}]} 鴻 |
\EG a list @code{[@var{v1,w1,...,vn,wn}]} |
\EG a list @code{[@var{v1,w1,...,vn,wn}]} |
@item yesno |
@item yesno |
\JP 0 または 1 |
\JP 0 障 1 |
\EG 0 or 1 |
\EG 0 or 1 |
@item v |
@item v |
\JP 変数のリスト |
\JP 紊違鴻 |
\EG a list of variables |
\EG a list of variables |
@end table |
@end table |
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@itemize @bullet |
@itemize @bullet |
\BJP |
\BJP |
@item |
@item |
この関数は asir-contrib のパッケージ @samp{nn_ndbf.rr} で定義されている. |
∽違 asir-contrib 宴若 @samp{nn_ndbf.rr} у臂. |
@item |
@item |
多項式 @var{f} の大域 b 関数 (global b-function) を計算する. 出力は |
紊綣 @var{f} 紊у b ∽ (global b-function) 荐膊. |
変数 @var{s} の多項式である. |
с紊у b ∽違帥阪, 激с @code{op=1} 絎 |
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翫, 紊у b ∽ @var{b}, |
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緇篏膣 @var{P} 腟 @var{[b,P]} 菴. |
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@var{Pf^(s+1)=b(s)f^s} 羣. |
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緇篏膣 @var{v1,...,vn,dv1,...,dvn} 紊綣 |
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茵憗. 茵憗, 緇茵 d ゃ紊違 |
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筝絎宴, 篆医綣違紊違茵腓冴 |
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, 紊綣鋓違綏臀閫h頫憗茹c綽荀. |
@item |
@item |
オプション @code{weight=[@var{v1,w1,...,vn,wn}]} が指定された場合, |
激с @code{weight=[@var{v1,w1,...,vn,wn}]} 絎翫, |
変数リスト @var{(v1,...,vn)} に対して weight @var{(w1,...,wn)} |
紊違鴻 @var{(v1,...,vn)} 絲障 weight @var{(w1,...,wn)} |
を設定して計算が行われる. このオプションは, @var{f} が @var{(w1,...,wn)} |
荐絎荐膊茵. 激с潟, @var{f} @var{(w1,...,wn)} |
に関して weighted homogeneous の場合に有効に働く. |
≪ weighted homogeneous 翫鴻. |
@item |
@item |
オプション @code{heuristic=1} が指定された場合, あるイデアルのグレブナー |
激с @code{heuristic=1} 絎翫, ゃ≪違 |
基底を別の項順序に変換してから消去計算を行う. この方法により全体の計算が |
阪ャ綺紊羔サ荐膊茵. 号篏荐膊 |
高速化する場合がある. |
蕭翫. |
@item |
@item |
デフォルトでは, 内部で用いられる変数順序は自動的に決定されるが, |
с, х紊育綺羆阪, |
オプション @code{vord=@var{v}} が指定された場合その変数順序が使われる. |
激с @code{vord=@var{v}} 絎翫紊育綺篏帥. |
\E |
\E |
\BEG |
\BEG |
@item |
@item |
This function is defined in an asir-contrib package @samp{nn_ndbf.rr}. |
This function is defined in an asir-contrib package @samp{nn_ndbf.rr}. |
@item |
@item |
This function computes the global b-function of a polynomial @var{f}. |
This function computes the global b-function of a polynomial @var{f}. |
The output is a polynomial in @var{s}. |
By default only the global b-function is returned. |
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If an option @code{op=1} is given, |
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a pair @var{[b,P]} of the global b-function and |
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a differential operator satisfying @var{Pf^(s+1)=b(s)f^s}. |
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The operator P is represented as a commutative polynomial |
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of variables @var{v1,...,vn,dv1,...,dvn}. The d-variables |
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are treated as commutative indeterminates in this representation and |
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the polynomial should be regarded as a canonical representation with each polynomial coefficient |
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placed at the left of d-variables. |
@item |
@item |
If an option @code{weight=[@var{v1,w1,...,vn,wn}]} is given, |
If an option @code{weight=[@var{v1,w1,...,vn,wn}]} is given, |
the computation is done with a weight @var{(w1,...,wn)} for @var{(v1,...,vn)}. |
the computation is done with a weight @var{(w1,...,wn)} for @var{(v1,...,vn)}. |
Line 186 If an option @code{vord=@var{v}} is given, a variable |
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Line 202 If an option @code{vord=@var{v}} is given, a variable |
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\E |
\E |
@end itemize |
@end itemize |
@example |
@example |
[1519] load("nn_ndbf.rr"); |
[...] load("nn_ndbf.rr"); |
[1602] ndbf.bfunction(x^3-y^2*z^2); |
[...] ndbf.bfunction(x^3-y^2*z^2); |
-11664*s^7-93312*s^6-316872*s^5-592272*s^4-658233*s^3-435060*s^2 |
-11664*s^7-93312*s^6-316872*s^5-592272*s^4-658233*s^3-435060*s^2 |
-158375*s-24500 |
-158375*s-24500 |
[1603] F=256*u1^3-128*u3^2*u1^2+(144*u3*u2^2+16*u3^4)*u1-27*u2^4 |
[...] ndbf.bfunction(x^3-y^2*z^2|op=1); |
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[-11664*s^7-93312*s^6-316872*s^5-592272*s^4-658233*s^3-435060*s^2 |
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-158375*s-24500,(108*z^3*x*dz^3+756*z^2*x*dz^2+1080*z*x*dz+216*x)*dx^4 |
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... |
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+(729/8*z^3*dz^5+9477/8*z^2*dz^4+5103/2*z*dz^3+2025/2*dz^2)*dy^2] |
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[...] F=256*u1^3-128*u3^2*u1^2+(144*u3*u2^2+16*u3^4)*u1-27*u2^4 |
-4*u3^3*u2^2$ |
-4*u3^3*u2^2$ |
[1604] ndbf.bfunction(F|weight=[u3,2,u2,3,u1,4]); |
[...] ndbf.bfunction(F|weight=[u3,2,u2,3,u1,4]); |
576*s^6+3456*s^5+8588*s^4+11312*s^3+8329*s^2+3250*s+525 |
576*s^6+3456*s^5+8588*s^4+11312*s^3+8329*s^2+3250*s+525 |
@end example |
@end example |
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\JP @node ndbf.bf_local,,, b 関数計算 |
\JP @node ndbf.bf_local,,, b ∽域膊 |
\EG @node ndbf.bf_local,,, Computation of b-function |
\EG @node ndbf.bf_local,,, Computation of b-function |
@subsection @code{ndbf.bf_local} |
@subsection @code{ndbf.bf_local} |
@findex ndbf.bf_local |
@findex ndbf.bf_local |
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@table @t |
@table @t |
\JP @item ndbf.bf_local(@var{f},@var{p}[|weight=@var{w},heruristic=@var{yesno},vord=@var{v},op=@var{yesno}]) :: 多項式 @var{f} の点 @var{p} における局所 b 関数を計算する. |
\JP @item ndbf.bf_local(@var{f},@var{p}[|weight=@var{w},heruristic=@var{yesno},vord=@var{v},op=@var{yesno}]) :: 紊綣 @var{f} @var{p} 絮 b ∽違荐膊. |
\EG @item ndbf.bf_local(@var{f},@var{p}[|weight=@var{w},heruristic=@var{yesno},vord=@var{v},op=@var{yesno}]) :: computes the local b-function of a polynomial @var{f} at @var{p}. |
\EG @item ndbf.bf_local(@var{f},@var{p}[|weight=@var{w},heruristic=@var{yesno},vord=@var{v},op=@var{yesno}]) :: computes the local b-function of a polynomial @var{f} at @var{p}. |
@end table |
@end table |
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@table @var |
@table @var |
@item return |
@item return |
\JP リスト |
\JP 鴻 |
\EG a list |
\EG a list |
@item f |
@item f |
\JP 多項式 |
\JP 紊綣 |
\EG a polynomail |
\EG a polynomail |
@item p |
@item p |
\JP @code{[@var{v1,a1,...,vn,an}]} なるリスト |
\JP @code{[@var{v1,a1,...,vn,an}]} 鴻 |
\EG a list @code{[@var{v1,a1,...,vn,an}]} |
\EG a list @code{[@var{v1,a1,...,vn,an}]} |
@item w |
@item w |
\JP @code{[@var{v1,w1,...,vn,wn}]} なるリスト |
\JP @code{[@var{v1,w1,...,vn,wn}]} 鴻 |
\EG a list @code{[@var{v1,w1,...,vn,wn}]} |
\EG a list @code{[@var{v1,w1,...,vn,wn}]} |
@item yesno |
@item yesno |
\JP 0 または 1 |
\JP 0 障 1 |
\EG 0 or 1 |
\EG 0 or 1 |
@item v |
@item v |
\JP 変数のリスト |
\JP 紊違鴻 |
\EG a list of variables |
\EG a list of variables |
@end table |
@end table |
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\BJP |
\BJP |
@itemize @bullet |
@itemize @bullet |
@item |
@item |
この関数は asir-contrib のパッケージ @samp{nn_ndbf.rr} で定義されている. |
∽違 asir-contrib 宴若 @samp{nn_ndbf.rr} у臂. |
@item |
@item |
多項式 @var{f} の @var{(v1,...,vn)=(a1,...,an)} における |
紊綣 @var{f} @var{(v1,...,vn)=(a1,...,an)} |
局所 b 関数 (local b-function) を計算する. 出力は局所 $b$ 関数の因子, 重複度のペアのリストである. |
絮 b ∽ (local b-function) 荐膊. 阪絮 $b$ ∽違絖, 茲綺≪鴻с. |
@item |
@item |
デフォルトでは局所 b 関数のみが出力されるが, オプション @code{op=1} が指定 |
с絮 b ∽違帥阪, 激с @code{op=1} 絎 |
された場合, 局所 b 関数 @var{b} と, それを実現する微分作用素 @var{P} の |
翫, 絮 b ∽ @var{b}, 緇篏膣演罸 $a(x)$ |
ペア @var{[b,P]} を返す. これらは |
緇篏膣 @var{P} 腟 @var{[b,a(x),P]} 菴. |
@var{Pf^(s+1)=b(s)f^s} |
@var{a(x)Pf^(s+1)=b(s)f^s} 羣. |
を満たす. 微分作用素は @var{v1,...,vn,dv1,...,dvn} の可換多項式として |
緇篏膣 @var{v1,...,vn,dv1,...,dvn} 紊綣 |
表現されている. この表現においては, 微分を表す d のついた変数も単なる |
茵憗. 茵憗, 緇茵 d ゃ紊違 |
不定元として扱われているため, 係数多項式環の変数の前に表示されることも |
筝絎宴, 篆医綣違紊違茵腓冴 |
ありうるが, 多項式係数を左に置く正規表現として理解する必要がある. |
, 紊綣鋓違綏臀閫h頫憗茹c綽荀. |
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@item |
@item |
オプション @code{weight=[@var{v1,w1,...,vn,wn}]} が指定された場合, |
激с @code{weight=[@var{v1,w1,...,vn,wn}]} 絎翫, |
変数リスト @var{(v1,...,vn)} に対して weight @var{(w1,...,wn)} |
紊違鴻 @var{(v1,...,vn)} 絲障 weight @var{(w1,...,wn)} |
を設定して計算が行われる. このオプションは, @var{f} が @var{(w1,...,wn)} |
荐絎荐膊茵. 激с潟, @var{f} @var{(w1,...,wn)} |
に関して weighted homogeneous の場合に有効に働く. |
≪ weighted homogeneous 翫鴻. |
@item |
@item |
オプション @code{heuristic=1} が指定された場合, あるイデアルのグレブナー |
激с @code{heuristic=1} 絎翫, ゃ≪違 |
基底を別の項順序に変換してから消去計算を行う. この方法により全体の計算が |
阪ャ綺紊羔サ荐膊茵. 号篏荐膊 |
高速化する場合がある. |
蕭翫. |
@item |
@item |
デフォルトでは, 内部で用いられる変数順序は自動的に決定されるが, |
с, х紊育綺羆阪, |
オプション @code{vord=@var{v}} が指定された場合その変数順序が使われる. |
激с @code{vord=@var{v}} 絎翫紊育綺篏帥. |
@end itemize |
@end itemize |
\E |
\E |
|
|
Line 267 This function is defined in an asir-contrib package @s |
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Line 288 This function is defined in an asir-contrib package @s |
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This function computes the local b-function of a polynomial @var{f} at a point @var{(v1,...,vn)=(a1,...,an)}. |
This function computes the local b-function of a polynomial @var{f} at a point @var{(v1,...,vn)=(a1,...,an)}. |
The output is a list of pairs of each factor of the local b-function and its multiplicity. |
The output is a list of pairs of each factor of the local b-function and its multiplicity. |
@item |
@item |
By default only the local b-function is returned. If an option @code{op=1} is given, |
By default only the local b-function is returned. |
a pair @var{[b,P]} of the local b-function and a differential operator satisfying |
If an option @code{op=1} is given, |
@var{Pf^(s+1)=b(s)f^s}. The operator P is represented as a commutative polynomial |
a triple @var{[b,a,P]} of the local b-function, a polynomial and |
of variables @var{v1,...,vn,dv1,...,dvn}. Although the d-variables |
a differential operator satisfying @var{Pf^(s+1)=ab(s)f^s}. |
|
The operator P is represented as a commutative polynomial |
|
of variables @var{v1,...,vn,dv1,...,dvn}. The d-variables |
are treated as commutative indeterminates in this representation, |
are treated as commutative indeterminates in this representation, |
it should be regarded as a canonical representation with each polynomial coefficient |
the polynomial should be regarded as a canonical representation with each polynomial coefficient |
placed at the left of d-variables. |
placed at the left of d-variables. |
@item |
@item |
If an option @code{weight=[@var{v1,w1,...,vn,wn}]} is given, |
If an option @code{weight=[@var{v1,w1,...,vn,wn}]} is given, |
Line 290 If an option @code{vord=@var{v}} is given, a variable |
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Line 313 If an option @code{vord=@var{v}} is given, a variable |
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\E |
\E |
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@example |
@example |
[1527] load("nn_ndbf.rr"); |
[...] load("nn_ndbf.rr"); |
[1610] ndbf.bf_local(y*((x+1)*x^3-y^2),[x,-1,y,0]); |
[...] ndbf.bf_local(y*((x+1)*x^3-y^2),[x,-1,y,0]); |
[[-s-1,2]] |
[[-s-1,2]] |
[1611] ndbf.bf_local(y*((x+1)*x^3-y^2),[x,-1,y,0]|op=1); |
[...] ndbf.bf_local(y*((x+1)*x^3-y^2),[x,-1,y,0]|op=1); |
[[[-s-1,2]],12*x^3+36*y^2*x-36*y^2,(32*y*x^2+56*y*x)*dx^2 |
[[[-s-1,2]],12*x^3+36*y^2*x-36*y^2,(32*y*x^2+56*y*x)*dx^2 |
+((-8*x^3-2*x^2+(128*y^2-6)*x+112*y^2)*dy+288*y*x+(-240*s-128)*y)*dx |
+((-8*x^3-2*x^2+(128*y^2-6)*x+112*y^2)*dy+288*y*x+(-240*s-128)*y)*dx |
+(32*y*x^2-6*y*x+128*y^3-9*y)*dy^2+(32*x^2+6*s*x+640*y^2+39*s+30)*dy |
+(32*y*x^2-6*y*x+128*y^3-9*y)*dy^2+(32*x^2+6*s*x+640*y^2+39*s+30)*dy |
+(-1152*s^2-3840*s-2688)*y] |
+(-1152*s^2-3840*s-2688)*y] |
@end example |
@end example |
|
|
\JP @node ndbf.bf_strat,,, b 関数計算 |
\JP @node ndbf.bf_strat,,, b ∽域膊 |
\EG @node ndbf.bf_strat,,, Computation of b-function |
\EG @node ndbf.bf_strat,,, Computation of b-function |
@subsection @code{ndbf.bf_strat} |
@subsection @code{ndbf.bf_strat} |
@findex ndbf.bf_strat |
@findex ndbf.bf_strat |
|
|
@table @t |
@table @t |
@item ndbf.bf_strat(@var{f}[|weight=@var{w},heruristic=@var{h},vord=@var{v}]) |
@item ndbf.bf_strat(@var{f}[|weight=@var{w},heruristic=@var{h},vord=@var{v}]) |
\JP :: 多項式 @var{f} の, 局所 b 関数に付随する滑層分割 (stratification) を計算する. |
\JP :: 紊綣 @var{f} , 絮 b ∽違篁羯絮ゅ (stratification) 荐膊. |
\EG :: computes a stratification associated with local b-function of a polynomial @var{f}. |
\EG :: computes a stratification associated with local b-function of a polynomial @var{f}. |
@end table |
@end table |
|
|
@table @var |
@table @var |
@item return |
@item return |
\JP リスト |
\JP 鴻 |
\EG a list |
\EG a list |
@item f |
@item f |
\JP 多項式 |
\JP 紊綣 |
\EG a polynomial |
\EG a polynomial |
@item w |
@item w |
\JP @code{[@var{v1,w1,...,vn,wn}]} なるリスト |
\JP @code{[@var{v1,w1,...,vn,wn}]} 鴻 |
\EG a list @code{[@var{v1,w1,...,vn,wn}]} |
\EG a list @code{[@var{v1,w1,...,vn,wn}]} |
@item h |
@item h |
\JP 0 または 1 |
\JP 0 障 1 |
\EG 0 or 1 |
\EG 0 or 1 |
@item v |
@item v |
\JP 変数のリスト |
\JP 紊違鴻 |
\EG li ist of variables |
\EG li ist of variables |
@end table |
@end table |
|
|
\BJP |
\BJP |
@itemize @bullet |
@itemize @bullet |
@item |
@item |
この関数は asir-contrib のパッケージ @samp{nn_ndbf.rr} で定義されている. |
∽違 asir-contrib 宴若 @samp{nn_ndbf.rr} у臂. |
@item |
@item |
多項式 @var{f} の大域 b 関数 (global b-function) を計算する. 出力は |
紊綣 @var{f} 紊у b ∽ (global b-function) 荐膊. 阪 |
変数 @var{s} の多項式である. |
紊 @var{s} 紊綣с. |
@item |
@item |
オプション @code{weight=[@var{v1,w1,...,vn,wn}]} が指定された場合, |
激с @code{weight=[@var{v1,w1,...,vn,wn}]} 絎翫, |
変数リスト @var{(v1,...,vn)} に対して weight @var{(w1,...,wn)} |
紊違鴻 @var{(v1,...,vn)} 絲障 weight @var{(w1,...,wn)} |
を設定して計算が行われる. このオプションは, @var{f} が @var{(w1,...,wn)} |
荐絎荐膊茵. 激с潟, @var{f} @var{(w1,...,wn)} |
に関して weighted homogeneous の場合に有効に働く. |
≪ weighted homogeneous 翫鴻. |
@item |
@item |
オプション @code{heuristic=1} が指定された場合, あるイデアルのグレブナー |
激с @code{heuristic=1} 絎翫, ゃ≪違 |
基底を別の項順序に変換してから消去計算を行う. この方法により全体の計算が |
阪ャ綺紊羔サ荐膊茵. 号篏荐膊 |
高速化する場合がある. |
蕭翫. |
@item |
@item |
デフォルトでは, 内部で用いられる変数順序は自動的に決定されるが, |
с, х紊育綺羆阪, |
オプション @code{vord=@var{v}} が指定された場合その変数順序が使われる. |
激с @code{vord=@var{v}} 絎翫紊育綺篏帥. |
@end itemize |
@end itemize |
\E |
\E |
|
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Line 376 If an option @code{vord=@var{v}} is given, a variable |
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Line 399 If an option @code{vord=@var{v}} is given, a variable |
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\E |
\E |
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@example |
@example |
[1537] load("nn_ndbf.rr"); |
[...] load("nn_ndbf.rr"); |
[1620] F=256*u1^3-128*u3^2*u1^2+(144*u3*u2^2+16*u3^4)*u1-27*u2^4 |
[...] F=256*u1^3-128*u3^2*u1^2+(144*u3*u2^2+16*u3^4)*u1-27*u2^4 |
-4*u3^3*u2^2$ |
-4*u3^3*u2^2$ |
[1621] ndbf.bf_strat(F); |
[...] ndbf.bf_strat(F); |
[[u3^2,-u1,-u2],[-1],[[-s-1,2],[16*s^2+32*s+15,1],[36*s^2+72*s+35,1]]] |
[[[u3^2,-u1,-u2],[-1],[[-s-1,2],[16*s^2+32*s+15,1],[36*s^2+72*s+35,1]]], |
[[-4*u1+u3^2,-u2],[96*u1^2+40*u3^2*u1-9*u3*u2^2,...],[[-s-1,2]]] |
[[-4*u1+u3^2,-u2],[96*u1^2+40*u3^2*u1-9*u3*u2^2,...],[[-s-1,2]]], |
[[...],[-u3*u2,u2*u1,...],[[-s-1,1],...]]] |
[[-2048*u1^3-...],[-u3*u2,u2*u1,...],[[-s-1,1],...]]], |
[[-256*u1^3+128*u3^2*u1^2+...],[...],[[-s-1,1]]] |
[[-256*u1^3+128*u3^2*u1^2+...],[...],[[-s-1,1]]], |
[[],[-256*u1^3+128*u3^2*u1^2+...],[]] |
[[],[-256*u1^3+128*u3^2*u1^2+...],[]]] |
@end example |
@end example |
|
|
\JP @node Annihilator 計算,,, 新 b 関数パッケージ nn_ndbf.rr |
\JP @node ndbf.action_on_gfs,,, b ∽域膊 |
|
\EG @node ndbf.action_on_gfs,,, Computation of b-function |
|
@subsection @code{ndbf.action_on_gfs} |
|
@findex ndbf.action_on_gfs |
|
|
|
@table @t |
|
@item ndbf.action_on_gfs(@var{op},@var{v},@var{gfs}) |
|
\JP :: 緇篏膣 @var{op} @var{gf^(s+a)} 吾篏荐膊. |
|
\EG :: computes the action of an operatior @var{op} on @var{gf^(s+a)} |
|
@end table |
|
|
|
@table @var |
|
@item return |
|
\JP 鴻 |
|
\EG a list |
|
@item op |
|
\JP 緇篏膣 |
|
\EG a differential operator |
|
@item gfs |
|
\JP @var{[g,f,s+a]} 鴻 |
|
\EG a list @var{[g,f,s+a]} |
|
@item v |
|
\JP @var{f} 紊違鴻 (@var{v=[v1,...,vn]}) |
|
\EG list of variables of @var{f} (@var{v=[v1,...,vn]}) |
|
@end table |
|
|
|
\BJP |
|
@itemize @bullet |
|
@item 緇篏膣 @var{op} @var{gf^(s+a)} 篏腟荐膊. |
|
@item @var{g} @var{v1,...,vn} 紊違紊綣с. |
|
@item @var{op} @var{[v1,...,vn,dv1,...,dvn]} 紊違紊綣ц;憗. |
|
@item ュ鴻 @var{[g,f,s+a]} @var{gf^(s+a)} 茵. |
|
@item 腟 @var{[h,f,s+c]} 鴻, @var{hf^(s+b)} |
|
潟. @var{c} 贋違с. |
|
@var{op} b-∽ @var{b(s)} 筝篏膣, |
|
@var{a=1} 絲障 @var{c=0} , @var{h=b(s)} (global case) 障 |
|
@var{h=d(v)b(s)} (local case) с. |
|
@end itemize |
|
\E |
|
|
|
\BEG |
|
@itemize @bullet |
|
@item This function computes the action of a differential operator |
|
@var{op} on @var{gf^(s+a)}. |
|
@item @var{g} is a polynomial with variables @var{v1,...,vn}. |
|
@item @var{op} is represented by a polynonmial with @var{[v1,...,vn,dv1,...,dvn]}. |
|
@item The input list @var{[g,f,s+a]} represents @var{gf^(s+a)}. |
|
@item The result is a list @var{[h,f,s+c]} and it means @var{hf^(s+c)}, |
|
where @var{c} is an integer. If @var{op} is an operator giving b-function |
|
@var{b(s)}, |
|
then @var{c=0} for @var{a=1} and @var{h=b(s)} (global case) |
|
or @var{h=b(s)d(v)} (local case). |
|
@end itemize |
|
\E |
|
|
|
@example |
|
[...] load("nn_ndbf.rr"); |
|
[...] F=x^5-y^2*z^2$ |
|
[...] B=ndbf.bfunction(F|op=1)$ |
|
[...] ndbf.action_on_gfs(B[1],[x,y,z],[1,F,s+1]); |
|
[-62500000000*s^13-...-2985505717194*s-245434132944,x^5-z^2*y^2,s] |
|
[...] L=ndbf.bf_local(F,[x,0,y,0,z,1]|op=1)$ |
|
[...] ndbf.action_on_gfs(L[2],[x,y,z],[1,F,s+1]); |
|
[(-100000*s^5-500000*s^4-990000*s^3-970000*s^2-470090*s-90090)*z^2, |
|
x^5-z^2*y^2,s] |
|
@end example |
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|
|
\JP @node Annihilator ゃ≪荐膊,,, b ∽違宴若 nn_ndbf.rr |
\EG @node Computation of annihilator ideal,,, New b-function package nn_ndbf.rr |
\EG @node Computation of annihilator ideal,,, New b-function package nn_ndbf.rr |
\JP @section Annihilator 計算 |
\JP @section Annihilator ゃ≪荐膊 |
\EG @section Computation of annihilator ideal |
\EG @section Computation of annihilator ideal |
|
|
@menu |
@menu |
* ndbf.ann:: |
* ndbf.ann:: |
@end menu |
@end menu |
|
|
\JP @node ndbf.ann,,, Annihilator イデアル計算 |
\JP @node ndbf.ann,,, Annihilator ゃ≪荐膊 |
\EG @node ndbf.ann,,, Computation of annihilator ideal |
\EG @node ndbf.ann,,, Computation of annihilator ideal |
@subsection @code{ndbf.ann} |
@subsection @code{ndbf.ann} |
@findex ndbf.ann |
@findex ndbf.ann |
|
|
@table @t |
@table @t |
\JP @item ndbf.ann(@var{f}[|weight=@var{w}]) :: 多項式 @var{f} に対し @var{f^s} の annihilator ideal を計算する. |
\JP @item ndbf.ann(@var{f}[|weight=@var{w}]) :: 紊綣 @var{f} 絲障 @var{f^s} annihilator ideal 荐膊. |
\EG @item ndbf.ann(@var{f}[|weight=@var{w}]) :: computes the annihilator ideal of @var{f^s} for a polynomial @var{f}. |
\EG @item ndbf.ann(@var{f}[|weight=@var{w}]) :: computes the annihilator ideal of @var{f^s} for a polynomial @var{f}. |
@end table |
@end table |
|
|
@table @var |
@table @var |
@item return |
@item return |
\JP 微分作用素のリスト |
\JP 緇篏膣鴻 |
\EG a list of differential operators |
\EG a list of differential operators |
@item f |
@item f |
\JP 多項式 |
\JP 紊綣 |
\EG a polynomial |
\EG a polynomial |
@item w |
@item w |
\JP @code{[@var{v1,w1,...,vn,wn}]} なるリスト |
\JP @code{[@var{v1,w1,...,vn,wn}]} 鴻 |
\EG a list @code{[@var{v0,w1,...,vn,wn}]} |
\EG a list @code{[@var{v0,w1,...,vn,wn}]} |
@end table |
@end table |
|
|
\BJP |
\BJP |
@itemize @bullet |
@itemize @bullet |
@item |
@item |
この関数は asir-contrib のパッケージ @samp{nn_ndbf.rr} で定義されている. |
∽違 asir-contrib 宴若 @samp{nn_ndbf.rr} у臂. |
@item |
@item |
多項式 @var{f} に対し, @var{f^s} の annihilator ideal を計算する. |
紊綣 @var{f} 絲障, @var{f^s} annihilator ideal 荐膊. |
出力は, @var{s} を係数に含む微分作用素のリストである. 微分作用素の |
阪, @var{s} 篆違緇篏膣鴻с. 緇篏膣 |
表現方法は, @code{ndbf.bf_local} と同様である. |
茵憝号, @code{ndbf.bf_local} 罕с. |
@item |
@item |
オプション @code{weight=[@var{v1,w1,...,vn,wn}]} が指定された場合, |
激с @code{weight=[@var{v1,w1,...,vn,wn}]} 絎翫, |
変数リスト @var{(v1,...,vn)} に対して weight @var{(w1,...,wn)} |
紊違鴻 @var{(v1,...,vn)} 絲障 weight @var{(w1,...,wn)} |
を設定して計算が行われる. このオプションは, @var{f} が @var{(w1,...,wn)} |
荐絎荐膊茵. 激с潟, @var{f} @var{(w1,...,wn)} |
に関して weighted homogeneous の場合に有効に働く. |
≪ weighted homogeneous 翫鴻. |
@end itemize |
@end itemize |
\E |
\E |
\BEG |
\BEG |
Line 449 This option is useful when @var{f} is weighted homogen |
|
Line 539 This option is useful when @var{f} is weighted homogen |
|
\E |
\E |
|
|
@example |
@example |
[1542] load("nn_ndbf.rr"); |
[...] load("nn_ndbf.rr"); |
[1625] ndbf.ann(x*y*z*(x^3-y^2*z^2)); |
[...] ndbf.ann(x*y*z*(x^3-y^2*z^2)); |
[(-x^4*dy^2+3*z^4*x*dz^2+12*z^3*x*dz+6*z^2*x)*dx+4*z*x^3*dz*dy^2 |
[(-x^4*dy^2+3*z^4*x*dz^2+12*z^3*x*dz+6*z^2*x)*dx+4*z*x^3*dz*dy^2 |
-z^5*dz^3-6*z^4*dz^2-6*z^3*dz, |
-z^5*dz^3-6*z^4*dz^2-6*z^3*dz, |
(x^4*dy-3*z^3*y*x*dz-6*z^2*y*x)*dx-4*z*x^3*dz*dy+z^4*y*dz^2+3*z^3*y*dz, |
(x^4*dy-3*z^3*y*x*dz-6*z^2*y*x)*dx-4*z*x^3*dz*dy+z^4*y*dz^2+3*z^3*y*dz, |
Line 458 This option is useful when @var{f} is weighted homogen |
|
Line 548 This option is useful when @var{f} is weighted homogen |
|
-y*dy+z*dz] |
-y*dy+z*dz] |
@end example |
@end example |
|
|
@comment --- おまじない --- |
@comment --- 障 --- |
@node Index,,, Top |
@node Index,,, Top |
@unnumbered Index |
@unnumbered Index |
@printindex fn |
@printindex fn |
Line 469 This option is useful when @var{f} is weighted homogen |
|
Line 559 This option is useful when @var{f} is weighted homogen |
|
@summarycontents |
@summarycontents |
@contents |
@contents |
@bye |
@bye |
@comment --- おまじない終り --- |
@comment --- 障腟 --- |
|
|