version 1.1, 2008/01/23 02:36:14 |
version 1.2, 2009/02/22 05:40:48 |
|
|
@end iftex |
@end iftex |
@overfullrule=0pt |
@overfullrule=0pt |
@c -*-texinfo-*- |
@c -*-texinfo-*- |
@comment $OpenXM$ |
@comment $OpenXM: OpenXM/src/asir-contrib/packages/doc/taji_alc/taji_alc-ja.texi,v 1.1 2008/01/23 02:36:14 takayama Exp $ |
@comment %**start of header |
@comment %**start of header |
@comment --- $B$*$^$8$J$$=*$j(B --- |
@comment --- おまじない終り --- |
|
|
@comment --- GNU info $B%U%!%$%k$NL>A0(B --- |
@comment --- GNU info ファイルの名前 --- euc code で記述すること. |
@setfilename asir-contrib-taji_alc_ja |
@setfilename asir-contrib-taji_alc_ja |
|
|
@comment --- $B%?%$%H%k(B --- |
@comment --- タイトル --- |
@settitle 1$BJQ?tBe?tE*6I=j%3%[%b%m%8!<N`MQ%Q%C%1!<%8(B taji_alc |
@settitle 1変数代数的局所コホモロジー類用パッケージ taji_alc |
|
|
@comment %**end of header |
@comment %**end of header |
@comment %@setchapternewpage odd |
@comment %@setchapternewpage odd |
|
|
@comment --- $B$*$^$8$J$$(B --- |
@comment --- おまじない --- |
@ifinfo |
@ifinfo |
@macro fref{name} |
@macro fref{name} |
@ref{\name\,,@code{\name\}} |
@ref{\name\,,@code{\name\}} |
|
|
@end iftex |
@end iftex |
|
|
@titlepage |
@titlepage |
@comment --- $B$*$^$8$J$$=*$j(B --- |
@comment --- おまじない終り --- |
|
|
@comment --- $B%?%$%H%k(B, $B%P!<%8%g%s(B, $BCx<TL>(B, $BCx:n8"I=<((B --- |
@comment --- タイトル, バージョン, 著者名, 著作権表示 --- |
@title 1$BJQ?tBe?tE*6I=j%3%[%b%m%8!<N`MQ%Q%C%1!<%8(B taji_alc |
@title 1変数代数的局所コホモロジー類用パッケージ taji_alc |
@subtitle 1$BJQ?tBe?tE*6I=j%3%[%b%m%8!<N`MQ%Q%C%1!<%8(B taji_alc |
@subtitle 1変数代数的局所コホモロジー類用パッケージ taji_alc |
@subtitle 1.0 $BHG(B |
@subtitle 1.0 版 |
@subtitle 2007 $BG/(B 11 $B7n(B |
@subtitle 2007 年 11 月 |
|
|
@author $B>1;JBnL4(B, $BEDEg?50l(B |
@author 庄司卓夢, 田島慎一 |
@page |
@page |
@vskip 0pt plus 1filll |
@vskip 0pt plus 1filll |
Copyright @copyright{} Takumu Shoji, Shinichi Tajima. |
Copyright @copyright{} Takumu Shoji, Shinichi Tajima. |
2007. All rights reserved. Licensed by GPL. |
2007. All rights reserved. Licensed by GPL. |
@end titlepage |
@end titlepage |
|
|
@comment --- $B$*$^$8$J$$(B --- |
@comment --- おまじない --- |
@synindex vr fn |
@synindex vr fn |
@comment --- $B$*$^$8$J$$=*$j(B --- |
@comment --- おまじない終り --- |
|
|
@comment --- @node $B$O(B GNU info, HTML $BMQ(B --- |
@comment --- @node は GNU info, HTML 用 --- |
@comment --- @node $B$N0z?t$O(B node-name, next, previous, up --- |
@comment --- @node の引数は node-name, next, previous, up --- |
@node Top,, (dir), (dir) |
@node Top,, (dir), (dir) |
|
|
@comment --- @menu $B$O(B GNU info, HTML $BMQ(B --- |
@comment --- @menu は GNU info, HTML 用 --- |
@comment --- chapter $BL>$r@53N$KJB$Y$k(B --- |
@comment --- chapter 名を正確に並べる --- |
@comment --- $B$3$NJ8=q$G$O(B chapter XYZ, Chapter Index $B$,$"$k(B. |
@comment --- この文書では chapter XYZ, Chapter Index がある. |
@comment --- Chapter XYZ $B$K$O(B section XYZ$B$K$D$$$F(B, section XYZ$B$K4X$9$k4X?t$,$"$k(B. |
@comment --- Chapter XYZ には section XYZについて, section XYZに関する関数がある. |
@menu |
@menu |
* 1$BJQ?tBe?tE*6I=j%3%[%b%m%8!<N`(B:: |
* 1変数代数的局所コホモロジー類:: |
* Index:: |
* Index:: |
@end menu |
@end menu |
|
|
@comment --- chapter $B$N3+;O(B --- |
@comment --- chapter の開始 --- |
@comment --- $B?F(B chapter $BL>$r@53N$K(B. $B?F$,$J$$>l9g$O(B Top --- |
@comment --- 親 chapter 名を正確に. 親がない場合は Top --- |
@node 1$BJQ?tBe?tE*6I=j%3%[%b%m%8!<N`(B,,, Top |
@node 1変数代数的局所コホモロジー類,,, Top |
@chapter 1$BJQ?tBe?tE*6I=j%3%[%b%m%8!<N`(B |
@chapter 1変数代数的局所コホモロジー類 |
|
|
@comment --- section $BL>$r@53N$KJB$Y$k(B. --- |
@comment --- section 名を正確に並べる. --- |
@menu |
@menu |
* 1$BJQ?tBe?tE*6I=j%3%[%b%m%8!<N`MQ$N%Q%C%1!<%8(B taji_alc $B$K$D$$$F(B:: |
* 1変数代数的局所コホモロジー類用のパッケージ taji_alc について:: |
* 1$BJQ?tBe?tE*6I=j%3%[%b%m%8!<N`MQ$N4X?t(B:: |
* 1変数代数的局所コホモロジー類用の関数:: |
@end menu |
@end menu |
|
|
@comment --- section ``XYZ$B$K$D$$$F(B'' $B$N3+;O(B --- section XYZ$B$K$D$$$F$N?F$O(B chapter XYZ |
@comment --- section ``XYZについて'' の開始 --- section XYZについての親は chapter XYZ |
@node 1$BJQ?tBe?tE*6I=j%3%[%b%m%8!<N`MQ$N%Q%C%1!<%8(B taji_alc $B$K$D$$$F(B,,, 1$BJQ?tBe?tE*6I=j%3%[%b%m%8!<N`(B |
@node 1変数代数的局所コホモロジー類用のパッケージ taji_alc について,,, 1変数代数的局所コホモロジー類 |
@section 1$BJQ?tBe?tE*6I=j%3%[%b%m%8!<N`MQ$N%Q%C%1!<%8(B taji_alc $B$K$D$$$F(B |
@section 1変数代数的局所コホモロジー類用のパッケージ taji_alc について |
|
|
@comment --- $B=qBN;XDj$K$D$$$F(B --- |
@comment --- 書体指定について --- |
@comment --- @code{} $B$O%?%$%W%i%$%?BNI=<((B --- |
@comment --- @code{} はタイプライタ体表示 --- |
@comment --- @var{} $B$O<P;zBNI=<((B --- |
@comment --- @var{} は斜字体表示 --- |
@comment --- @b{} $B$O%\!<%k%II=<((B --- |
@comment --- @b{} はボールド表示 --- |
@comment --- @samp{} $B$O%U%!%$%kL>$J$I$NI=<((B --- |
@comment --- @samp{} はファイル名などの表示 --- |
|
|
$B$3$N@bL@=q$G$O(B |
この説明書では |
1$BJQ?tBe?tE*6I=j%3%[%b%m%8!<N`MQ$N%Q%C%1!<%8(B taji_alc $B$K$D$$$F@bL@$9$k(B. |
1変数代数的局所コホモロジー類用のパッケージ taji_alc について説明する. |
$B?t3XE*2r@b$dGX7J$K$D$$$F$O(B, $B2r@b5-;v(B |
数学的解説や背景については, 解説記事 |
``1$BJQ?tBe?tE*6I=j%3%[%b%m%8!<N`MQ$KBP$9$k(B Risa/Asir $BMQ%Q%C%1!<%8(B taji_alc'' |
``1変数代数的局所コホモロジー類用に対する Risa/Asir 用パッケージ taji_alc'' |
(Risa/Asir Journal (2007)) |
(Risa/Asir Journal (2007)) |
$B$*$h$S$=$N;29MJ88%$r;2>H(B. |
およびその参考文献を参照. |
|
|
|
|
@comment --- section ``$B<B83E*4X?t(B'' $B$N3+;O(B --- |
@comment --- section ``実験的関数'' の開始 --- |
@node 1$BJQ?tBe?tE*6I=j%3%[%b%m%8!<N`MQ$N4X?t(B,,, 1$BJQ?tBe?tE*6I=j%3%[%b%m%8!<N`(B |
@node 1変数代数的局所コホモロジー類用の関数,,, 1変数代数的局所コホモロジー類 |
@section 1$BJQ?tBe?tE*6I=j%3%[%b%m%8!<N`MQ$N4X?t(B |
@section 1変数代数的局所コホモロジー類用の関数 |
|
|
@comment --- section ``$B<B83E*4X?t(B'' $B$N(B subsection xyz_abc |
@comment --- section ``実験的関数'' の subsection xyz_abc |
@comment --- subsection xyz_pqr xyz_stu $B$,$"$k(B. |
@comment --- subsection xyz_pqr xyz_stu がある. |
@menu |
@menu |
* taji_alc.cpfd:: |
* taji_alc.cpfd:: |
* taji_alc.snoether:: |
* taji_alc.snoether:: |
Line 112 Copyright @copyright{} Takumu Shoji, Shinichi Tajima. |
|
Line 112 Copyright @copyright{} Takumu Shoji, Shinichi Tajima. |
|
* taji_alc.inv:: |
* taji_alc.inv:: |
@end menu |
@end menu |
|
|
$BK\%;%/%7%g%s$N4X?t$r8F$S=P$9$K$O(B, |
本セクションの関数を呼び出すには, |
@example |
@example |
import("taji_alc.rr")$ |
import("taji_alc.rr")$ |
@end example |
@end example |
$B$r<B9T$7$F%W%m%0%i%`$r%m!<%I$9$k(B. |
を実行してプログラムをロードする. |
|
|
|
|
@comment **************************************************************** |
@comment **************************************************************** |
@node taji_alc.cpfd,,, 1$BJQ?tBe?tE*6I=j%3%[%b%m%8!<N`MQ$N4X?t(B |
@node taji_alc.cpfd,,, 1変数代数的局所コホモロジー類用の関数 |
@subsection @code{taji_alc.cpfd} |
@subsection @code{taji_alc.cpfd} |
@findex taji_alc.cpfd |
@findex taji_alc.cpfd |
|
|
@table @t |
@table @t |
@item taji_alc.cpfd(@var{num},@var{den}) |
@item taji_alc.cpfd(@var{num},@var{den}) |
:: $BM-M}4X?t(B@var{num}/@var{den}$B$NItJ,J,?tJ,2r$r5a$a$k(B. |
:: 有理関数@var{num}/@var{den}の部分分数分解を求める. |
@end table |
@end table |
|
|
@table @var |
@table @var |
@item return |
@item return |
@var{switch}$B$,(B0$B$+(B1$B$J$i$P(B, [[[$BJ,;R(B,[$BJ,Jl$N0x;R(B,$B=EJ#EY(B]],...],...] $B$J$k%j%9%H(B. |
@var{switch}が0か1ならば, [[[分子,[分母の因子,重複度]],...],...] なるリスト. |
|
|
@var{switch}$B$,(B10$B$+(B11$B$J$i$P(B, [[$BJ,;R(B,[$BJ,Jl$N0x;R(B,$B=EJ#EY(B]],...] $B$J$k%j%9%H(B. |
@var{switch}が10か11ならば, [[分子,[分母の因子,重複度]],...] なるリスト. |
|
|
@item num |
@item num |
($BM-M}4X?t$NJ,;R$N(B) $BB?9`<0(B |
(有理関数の分子の) 多項式 |
@item den |
@item den |
($BM-M}4X?t$NJ,Jl$N(B) $BB?9`<0(B |
(有理関数の分母の) 多項式 |
|
|
$B$^$?$O(B ($BM-M}4X?t$NJ,Jl$r(BQ$B>e$G4{LsJ,2r$7$?(B) [[$B0x;R(B,$B=EJ#EY(B],...] $B$J$k%j%9%H(B |
または (有理関数の分母をQ上で既約分解した) [[因子,重複度],...] なるリスト |
@item switch |
@item switch |
$B%*%W%7%g%s;XDj(B |
オプション指定 |
|
|
case 0 : complete$B$JItJ,J,?tJ,2r$rJV$9(B. ($BJ,;R$OM-M}?t78?tB?9`<0(B) |
case 0 : completeな部分分数分解を返す. (分子は有理数係数多項式) |
|
|
case 1 : complete$B$JItJ,J,?tJ,2r$rJV$9(B. ($BJ,;R$O@0?t78?t2=%j%9%H(B) |
case 1 : completeな部分分数分解を返す. (分子は整数係数化リスト) |
|
|
case 10 : $BJ,Jl$rQQE83+$7$J$$ItJ,J,?tJ,2r$rJV$9(B. ($BJ,;R$OM-M}?t78?tB?9`<0(B) |
case 10 : 分母を冪展開しない部分分数分解を返す. (分子は有理数係数多項式) |
|
|
case 11 : $BJ,Jl$rQQE83+$7$J$$ItJ,J,?tJ,2r$rJV$9(B. ($BJ,;R$O@0?t78?t2=%j%9%H(B) |
case 11 : 分母を冪展開しない部分分数分解を返す. (分子は整数係数化リスト) |
|
|
default : case 0 |
default : case 0 |
@end table |
@end table |
|
|
@itemize @bullet |
@itemize @bullet |
@item taji_alc.cpfd()$B$O(B, proper$B$JM-M}4X?t$rBP>]$H$9$k(B. |
@item taji_alc.cpfd()は, properな有理関数を対象とする. |
$BF~NOCM$,(Bproper$B$G$J$$>l9g$G$b@5>o$KF0:n$9$k$,(B, $BB?9`<0$H$7$F=P$F$/$kItJ,$OI=<($7$J$$(B. |
入力値がproperでない場合でも正常に動作するが, 多項式として出てくる部分は表示しない. |
@item $BItJ,J,?tJ,2r$O(B, $BQQE83+$r$9$k(Bcomplete$B$J%?%$%W$H(B, $BQQE83+$r$7$J$$%?%$%W$N(B2$B$D$N%?%$%W$,$"$k(B. |
@item 部分分数分解は, 冪展開をするcompleteなタイプと, 冪展開をしないタイプの2つのタイプがある. |
taji_alc.cpfd()$B$G:NMQ$7$F$$$k%"%k%4%j%:%`$G$O(B, $BA0<T$,@h$K5a$^$k(B. |
taji_alc.cpfd()で採用しているアルゴリズムでは, 前者が先に求まる. |
$B8e<T$O(B, $BA0<T$N%G!<%?$r%[!<%J!<K!$GB-$7>e$2$F5a$a$k(B. |
後者は, 前者のデータをホーナー法で足し上げて求める. |
@item @var{den}$B$O(B, $B%j%9%H$G$NF~NO$,K>$^$7$$(B. |
@item @var{den}は, リストでの入力が望ましい. |
($BB?9`<0$GF~NO$9$k$H(B, $B4JLs2=$N=hM}$,@8$8$k$?$a=E$/$J$k(B.) |
(多項式で入力すると, 簡約化の処理が生じるため重くなる.) |
$B$?$@$7$=$N>l9g$K$O(B, $B4{Ls%A%'%C%/(B, $BM-M}<0$NLsJ,(B, $B@0?t78?t2=$O9T$o$J$$$N$GCm0U$9$k(B. |
ただしその場合には, 既約チェック, 有理式の約分, 整数係数化は行わないので注意する. |
$BF~NOCM$O%f!<%6B&$,@UG$$r$b$D(B. |
入力値はユーザ側が責任をもつ. |
@end itemize |
@end itemize |
|
|
@example |
@example |
Line 179 taji_alc.cpfd()$B$G:NMQ$7$F$$$k%"%k%4%j%:%`$G$O(B, |
|
Line 179 taji_alc.cpfd()$B$G:NMQ$7$F$$$k%"%k%4%j%:%`$G$O(B, |
|
@end example |
@end example |
|
|
@table @t |
@table @t |
@item $B;2>H(B |
@item 参照 |
@end table |
@end table |
|
|
@comment --- ChangeLog $B$r=q$/(B. $BF05!(B. $B%=!<%9%3!<%I$N0LCV(B. $BJQ99F|;~(B $B$J$I(B CVS$B%5!<%P$r8+$k$?$a(B |
@comment --- ChangeLog を書く. 動機. ソースコードの位置. 変更日時 など CVSサーバを見るため |
@comment --- openxm $B$N30It$+$i$N4sM?$b=R$Y$k(B. Credit. |
@comment --- openxm の外部からの寄与も述べる. Credit. |
@noindent |
@noindent |
ChangeLog |
ChangeLog |
@itemize @bullet |
@itemize @bullet |
|
|
|
|
@page |
@page |
@comment **************************************************************** |
@comment **************************************************************** |
@node taji_alc.snoether,,, 1$BJQ?tBe?tE*6I=j%3%[%b%m%8!<N`MQ$N4X?t(B |
@node taji_alc.snoether,,, 1変数代数的局所コホモロジー類用の関数 |
@subsection @code{taji_alc.snoether} |
@subsection @code{taji_alc.snoether} |
@findex taji_alc.snoether |
@findex taji_alc.snoether |
|
|
@table @t |
@table @t |
@item taji_alc.snoether(@var{num},@var{den}) |
@item taji_alc.snoether(@var{num},@var{den}) |
:: $BM-M}4X?t(B@var{num}/@var{den}$B$,Dj$a$kBe?tE*6I=j%3%[%b%m%8!<N`$N%M!<%?!<:nMQAG$r5a$a$k(B. |
:: 有理関数@var{num}/@var{den}が定める代数的局所コホモロジー類のネーター作用素を求める. |
@end table |
@end table |
|
|
@table @var |
@table @var |
@item return |
@item return |
[[$B0x;R(B,$B%M!<%?!<:nMQAG(B],...] $B$J$k%j%9%H(B. |
[[因子,ネーター作用素],...] なるリスト. |
|
|
$B%M!<%?!<:nMQAG$O(B, $B78?t$r9b3,$NItJ,$+$i9_=g$KJB$Y$?%j%9%H(B |
ネーター作用素は, 係数を高階の部分から降順に並べたリスト |
|
|
@item num |
@item num |
($BM-M}4X?t$NJ,;R$N(B)$BB?9`<0(B |
(有理関数の分子の)多項式 |
@item den |
@item den |
($BM-M}4X?t$NJ,Jl$N(B)$BB?9`<0(B |
(有理関数の分母の)多項式 |
|
|
$B$^$?$O(B ($BM-M}4X?t$NJ,Jl$r(BQ$B>e$G4{LsJ,2r$7$?(B) [[$B0x;R(B,$B=EJ#EY(B],...] $B$J$k%j%9%H(B. |
または (有理関数の分母をQ上で既約分解した) [[因子,重複度],...] なるリスト. |
@item switch |
@item switch |
$B%*%W%7%g%s;XDj(B |
オプション指定 |
|
|
case 0 : $B%M!<%?!<:nMQAG$r(B [$BM-M}?t78?tB?9`<0(B,...] $B$J$k%j%9%H$GJV$9(B. |
case 0 : ネーター作用素を [有理数係数多項式,...] なるリストで返す. |
|
|
case 1 : $B%M!<%?!<:nMQAG$r(B [$B@0?t78?t2=%j%9%H(B,...] $B$J$k%j%9%H$GJV$9(B. |
case 1 : ネーター作用素を [整数係数化リスト,...] なるリストで返す. |
|
|
case 10 : $B%M!<%?!<:nMQAG$r(B [[$B@0?t78?tB?9`<0(B,...],$B@0?t(B] $B$J$k%j%9%H$GJV$9(B. |
case 10 : ネーター作用素を [[整数係数多項式,...],整数] なるリストで返す. |
|
|
case 20 : $B%M!<%?!<:nMQAG$r(B [[$B@0?t78?t2=%j%9%H(B,...],$B@0?t(B] $B$J$k%j%9%H$GJV$9(B. |
case 20 : ネーター作用素を [[整数係数化リスト,...],整数] なるリストで返す. |
|
|
default : case 0 |
default : case 0 |
@end table |
@end table |
|
|
@itemize @bullet |
@itemize @bullet |
@item taji_alc.snoether()$B$O(B, @var{den}$B$r(BQ$B>e$G4{LsJ,2r$7(B, |
@item taji_alc.snoether()は, @var{den}をQ上で既約分解し, |
$B3F0x;R$KBP1~$9$k%M!<%?!<:nMQAG$rJV$9(B. |
各因子に対応するネーター作用素を返す. |
@item @var{den}$B$O(B, $B%j%9%H$G$NF~NO$,K>$^$7$$(B. |
@item @var{den}は, リストでの入力が望ましい. |
($BB?9`<0$GF~NO$9$k$H(B, $B4JLs2=$N=hM}$,@8$8$k$?$a=E$/$J$k(B.) |
(多項式で入力すると, 簡約化の処理が生じるため重くなる.) |
$B$?$@$7$=$N>l9g$K$O(B, $B4{Ls%A%'%C%/(B, $BM-M}<0$NLsJ,(B, $B@0?t78?t2=$O9T$o$J$$$N$GCm0U$9$k(B. |
ただしその場合には, 既約チェック, 有理式の約分, 整数係数化は行わないので注意する. |
$BF~NOCM$O%f!<%6B&$,@UG$$r$b$D(B. |
入力値はユーザ側が責任をもつ. |
@item $BLa$jCM$N7?$O(B@var{switch}$B$GA*Br$G$-$k(B. |
@item 戻り値の型は@var{switch}で選択できる. |
|
|
case 10$B$O(B, $B%M!<%?!<:nMQAG$N3F78?tA4BN$rDLJ,$7(B, $B$=$NJ,JlItJ,$H3,>h$N@Q$r%j%9%H$GJ,$1$?I=8=$G$"$k(B. |
case 10は, ネーター作用素の各係数全体を通分し, その分母部分と階乗の積をリストで分けた表現である. |
$B$o$+$j$d$9$$$,(B, $BDLJ,CM$H78?tItJ,$H$GLsJ,$G$-$kItJ,$,$"$k(B($BFC$K9b3,$NItJ,$KB?$$(B)$B$N$G(B, $B>iD9@-$r$b$C$F$$$k(B. |
わかりやすいが, 通分値と係数部分とで約分できる部分がある(特に高階の部分に多い)ので, 冗長性をもっている. |
|
|
case 20$B$O(B, $B3,>h$NItJ,$GA4BN$r$/$/$j(B($B%j%9%H$GJ,$1(B), $B%M!<%?!<:nMQAG$N3F78?t$r8DJL$KDLJ,$7%j%9%H2=$9$k(B. |
case 20は, 階乗の部分で全体をくくり(リストで分け), ネーター作用素の各係数を個別に通分しリスト化する. |
$B3,>h$NItJ,$H78?tItJ,$H$GLsJ,$G$-$kItJ,$,$"$k(B($BFC$KDc3,$NItJ,$KB?$$(B)$B$N$G(B, |
階乗の部分と係数部分とで約分できる部分がある(特に低階の部分に多い)ので, |
$B>iD9$H8@$($J$/$b$J$$(B(case 10$B$h$j$O$^$7(B)$B$,(B, $B?t3XE*$J9=B$$,e:No$K8+$($kI=8=$G$"$k(B. |
冗長と言えなくもない(case 10よりはまし)が, 数学的な構造が綺麗に見える表現である. |
@end itemize |
@end itemize |
|
|
@example |
@example |
Line 266 case 20$B$O(B, $B3,>h$NItJ,$GA4BN$r$/$/$j(B($B%j% |
|
Line 266 case 20$B$O(B, $B3,>h$NItJ,$GA4BN$r$/$/$j(B($B%j% |
|
@end example |
@end example |
|
|
@table @t |
@table @t |
@item $B;2>H(B |
@item 参照 |
@end table |
@end table |
|
|
@comment --- ChangeLog $B$r=q$/(B. $BF05!(B. $B%=!<%9%3!<%I$N0LCV(B. $BJQ99F|;~(B $B$J$I(B CVS$B%5!<%P$r8+$k$?$a(B |
@comment --- ChangeLog を書く. 動機. ソースコードの位置. 変更日時 など CVSサーバを見るため |
@comment --- openxm $B$N30It$+$i$N4sM?$b=R$Y$k(B. Credit. |
@comment --- openxm の外部からの寄与も述べる. Credit. |
@noindent |
@noindent |
ChangeLog |
ChangeLog |
@itemize @bullet |
@itemize @bullet |
|
|
|
|
@page |
@page |
@comment **************************************************************** |
@comment **************************************************************** |
@node taji_alc.laurent_expansion,,, 1$BJQ?tBe?tE*6I=j%3%[%b%m%8!<N`MQ$N4X?t(B |
@node taji_alc.laurent_expansion,,, 1変数代数的局所コホモロジー類用の関数 |
@subsection @code{taji_alc.laurent_expansion} |
@subsection @code{taji_alc.laurent_expansion} |
@findex taji_alc.laurent_expansion |
@findex taji_alc.laurent_expansion |
|
|
@table @t |
@table @t |
@item taji_alc.laurent_expansion(@var{num},@var{den}) |
@item taji_alc.laurent_expansion(@var{num},@var{den}) |
:: $BM-M}4X?t(B@var{num}/@var{den}$B$N6K$K$*$1$k%m!<%i%sE83+$N<gMWIt$N78?t$r5a$a$k(B. |
:: 有理関数@var{num}/@var{den}の極におけるローラン展開の主要部の係数を求める. |
@end table |
@end table |
|
|
@table @var |
@table @var |
@item return |
@item return |
[[$B0x;R(B,$B%m!<%i%sE83+$N78?t(B],...] $B$J$k%j%9%H(B. |
[[因子,ローラン展開の係数],...] なるリスト. |
|
|
$B%m!<%i%sE83+$N78?t$O(B, $B9b0L$N78?t$+$i=g$KJB$Y$?%j%9%H(B. |
ローラン展開の係数は, 高位の係数から順に並べたリスト. |
|
|
@item num |
@item num |
($BM-M}4X?t$NJ,;R$N(B)$BB?9`<0(B |
(有理関数の分子の)多項式 |
@item den |
@item den |
($BM-M}4X?t$NJ,Jl$N(B)$BB?9`<0(B |
(有理関数の分母の)多項式 |
|
|
$B$^$?$O(B ($BM-M}4X?t$NJ,Jl$r(BQ$B>e$G4{LsJ,2r$7$?(B) [[$B0x;R(B,$B=EJ#EY(B],...] $B$J$k%j%9%H(B |
または (有理関数の分母をQ上で既約分解した) [[因子,重複度],...] なるリスト |
|
|
@item switch |
@item switch |
$B%*%W%7%g%s;XDj(B |
オプション指定 |
|
|
case 0 : $B%m!<%i%sE83+$N78?t$r(B [$BM-M}?t78?tB?9`<0(B,...] $B$J$k%j%9%H$GJV$9(B. |
case 0 : ローラン展開の係数を [有理数係数多項式,...] なるリストで返す. |
|
|
case 1 : $B%m!<%i%sE83+$N78?t$r(B [$B@0?t78?t2=%j%9%H(B,...] $B$J$k%j%9%H$GJV$9(B. |
case 1 : ローラン展開の係数を [整数係数化リスト,...] なるリストで返す. |
|
|
case 10 : $B%m!<%i%sE83+$N78?t$r(B [[$B@0?t78?tB?9`<0(B,...],$B@0?t(B] $B$J$k%j%9%H$GJV$9(B. |
case 10 : ローラン展開の係数を [[整数係数多項式,...],整数] なるリストで返す. |
|
|
case 20 : $B%m!<%i%sE83+$N78?t$r(B [[$B@0?t78?t2=%j%9%H(B,...],$B@0?t(B] $B$J$k%j%9%H$GJV$9(B. |
case 20 : ローラン展開の係数を [[整数係数化リスト,...],整数] なるリストで返す. |
|
|
default : case 0 |
default : case 0 |
@end table |
@end table |
|
|
@itemize @bullet |
@itemize @bullet |
@item taji_alc.laurent_expansion()$B$O(B, taji_alc.snoether()$B$r;H$C$F(B, $B%m!<%i%sE83+$N78?t$r5a$a$k(B. |
@item taji_alc.laurent_expansion()は, taji_alc.snoether()を使って, ローラン展開の係数を求める. |
@item taji_alc.laurent_expansion()$B$G$O(B, |
@item taji_alc.laurent_expansion()では, |
C$B>e$N(B1$BE@$KCmL\$9$k$N$G$O$J$/(B, Q$B>e$G$N4{Ls0x;R<+BN$KCmL\$7$F%m!<%i%sE83+$N78?t$r5a$a$k(B. |
C上の1点に注目するのではなく, Q上での既約因子自体に注目してローラン展開の係数を求める. |
$BLa$jCM$N78?t%j%9%H$N3F@.J,$O(B, $B$=$N0x;R$NA4$F$NNmE@$,6&DL$KK~$?$9%m!<%i%sE83+$N78?tB?9`<0$G$"$k(B. |
戻り値の係数リストの各成分は, その因子の全ての零点が共通に満たすローラン展開の係数多項式である. |
$B=>$C$F(B, 1$BE@$4$H$N%m!<%i%sE83+$N78?t$r$5$i$K5a$a$?$$>l9g$K$O(B, |
従って, 1点ごとのローラン展開の係数をさらに求めたい場合には, |
$B5a$a$?%m!<%i%sE83+$N78?tB?9`<0$K0x;R$NNmE@(B($BB($AFC0[E@(B)$B$NCM$rBeF~$9$kI,MW$,$"$k(B. |
求めたローラン展開の係数多項式に因子の零点(即ち特異点)の値を代入する必要がある. |
@end itemize |
@end itemize |
|
|
@example |
@example |
Line 334 C$B>e$N(B1$BE@$KCmL\$9$k$N$G$O$J$/(B, Q$B>e$G$N4{ |
|
Line 334 C$B>e$N(B1$BE@$KCmL\$9$k$N$G$O$J$/(B, Q$B>e$G$N4{ |
|
@end example |
@end example |
|
|
@table @t |
@table @t |
@item $B;2>H(B |
@item 参照 |
@ref{taji_alc.snoether} |
@ref{taji_alc.snoether} |
@end table |
@end table |
|
|
@comment --- ChangeLog $B$r=q$/(B. $BF05!(B. $B%=!<%9%3!<%I$N0LCV(B. $BJQ99F|;~(B $B$J$I(B CVS$B%5!<%P$r8+$k$?$a(B |
@comment --- ChangeLog を書く. 動機. ソースコードの位置. 変更日時 など CVSサーバを見るため |
@comment --- openxm $B$N30It$+$i$N4sM?$b=R$Y$k(B. Credit. |
@comment --- openxm の外部からの寄与も述べる. Credit. |
@noindent |
@noindent |
ChangeLog |
ChangeLog |
@itemize @bullet |
@itemize @bullet |
|
|
|
|
@page |
@page |
@comment **************************************************************** |
@comment **************************************************************** |
@node taji_alc.residue,,, 1$BJQ?tBe?tE*6I=j%3%[%b%m%8!<N`MQ$N4X?t(B |
@node taji_alc.residue,,, 1変数代数的局所コホモロジー類用の関数 |
@subsection @code{taji_alc.residue} |
@subsection @code{taji_alc.residue} |
@findex taji_alc.residue |
@findex taji_alc.residue |
|
|
@table @t |
@table @t |
@item taji_alc.residue(@var{num},@var{den}) |
@item taji_alc.residue(@var{num},@var{den}) |
:: $BM-M}4X?t(B@var{num}/@var{den}$B$N6K$K$*$1$kN1?t$r5a$a$k(B. |
:: 有理関数@var{num}/@var{den}の極における留数を求める. |
@end table |
@end table |
|
|
@table @var |
@table @var |
@item return |
@item return |
[[$B0x;R(B,$BN1?t(B],...] $B$J$k%j%9%H(B |
[[因子,留数],...] なるリスト |
|
|
@item num |
@item num |
($BM-M}4X?t$NJ,;R$N(B) $BB?9`<0(B |
(有理関数の分子の) 多項式 |
@item den |
@item den |
($BM-M}4X?t$NJ,Jl$N(B) $BB?9`<0(B |
(有理関数の分母の) 多項式 |
|
|
$B$^$?$O(B ($BM-M}4X?t$NJ,Jl$r(BQ$B>e$G4{LsJ,2r$7$?(B) [[$B0x;R(B,$B=EJ#EY(B],...] $B$J$k%j%9%H(B |
または (有理関数の分母をQ上で既約分解した) [[因子,重複度],...] なるリスト |
@item switch |
@item switch |
$B%*%W%7%g%s;XDj(B |
オプション指定 |
|
|
case 0 : $BN1?t$rM-M}?t78?tB?9`<0$GJV$9(B. |
case 0 : 留数を有理数係数多項式で返す. |
|
|
case 1 : $BN1?t$r@0?t78?t2=%j%9%H$GJV$9(B. |
case 1 : 留数を整数係数化リストで返す. |
|
|
default : case 0 |
default : case 0 |
|
|
@item pole |
@item pole |
$B%*%W%7%g%s;XDj(B |
オプション指定 |
|
|
[$B0x;R(B,...] $B$J$k%*%W%7%g%s%j%9%H(B |
[因子,...] なるオプションリスト |
@end table |
@end table |
|
|
@itemize @bullet |
@itemize @bullet |
@item taji_alc.residue()$B$O(B, @var{den}$B$r(BQ$B>e$G4{LsJ,2r$7(B, $B3F0x;R$NNmE@(B($BB($AM-M}4X?t$N6K(B)$B$K$*$1$kN1?t$rJV$9(B. |
@item taji_alc.residue()は, @var{den}をQ上で既約分解し, 各因子の零点(即ち有理関数の極)における留数を返す. |
@item $B%*%W%7%g%s$G(B@var{pole}$B$r;XDj$9$l$P$=$N0x;R$N$_$NN1?t$rJV$9(B. $B;XDj$,ITE,Ev$@$H(B0$B$rJV$9(B. |
@item オプションで@var{pole}を指定すればその因子のみの留数を返す. 指定が不適当だと0を返す. |
@item taji_alc.residue()$B$G:NMQ$7$F$$$k%"%k%4%j%:%`$G$O(B, |
@item taji_alc.residue()で採用しているアルゴリズムでは, |
C$B>e$N(B1$BE@$KCmL\$9$k$N$G$O$J$/(B, Q$B>e$G$N4{Ls0x;R<+BN$KCmL\$7$FN1?t$r5a$a$k(B. |
C上の1点に注目するのではなく, Q上での既約因子自体に注目して留数を求める. |
$BLa$jCM$NN1?t$O(B, $B$=$N0x;R$NA4$F$NNmE@$,6&DL$KK~$?$9N1?tB?9`<0$G$"$k(B. |
戻り値の留数は, その因子の全ての零点が共通に満たす留数多項式である. |
$B=>$C$F(B, 1$BE@$4$H$NN1?tCM$r$5$i$K5a$a$?$$>l9g$K$O(B, |
従って, 1点ごとの留数値をさらに求めたい場合には, |
$B5a$a$?N1?tB?9`<0$K0x;R$NNmE@(B($BB($AFC0[E@(B)$B$NCM$rBeF~$9$kI,MW$,$"$k(B. |
求めた留数多項式に因子の零点(即ち特異点)の値を代入する必要がある. |
|
|
@example |
@example |
[219] taji_alc.residue(1,x^4+1); |
[219] taji_alc.residue(1,x^4+1); |
[[x^4+1,-1/4*x]] |
[[x^4+1,-1/4*x]] |
@end example |
@end example |
|
|
$B$3$NNc$G8@$&$H(B, $B5a$a$?N1?tB?9`<0(B-1/4*x$B$K(B, x^4+1$B$N(B(4$B$D$"$k(B)$BNmE@$r$=$l$>$lBeF~$7$?$b$N$,8DJL$NN1?tCM$G$"$k(B. |
この例で言うと, 求めた留数多項式-1/4*xに, x^4+1の(4つある)零点をそれぞれ代入したものが個別の留数値である. |
@item @var{den}$B$O(B, $B%j%9%H$G$NF~NO$,K>$^$7$$(B. |
@item @var{den}は, リストでの入力が望ましい. |
($BB?9`<0$GF~NO$9$k$H(B, $B4JLs2=$N=hM}$,@8$8$k$?$a=E$/$J$k(B.) |
(多項式で入力すると, 簡約化の処理が生じるため重くなる.) |
$B$?$@$7$=$N>l9g$K$O(B, $B4{Ls%A%'%C%/(B, $BM-M}<0$NLsJ,(B, $B@0?t78?t2=$O9T$o$J$$$N$GCm0U$9$k(B. |
ただしその場合には, 既約チェック, 有理式の約分, 整数係数化は行わないので注意する. |
$BF~NOCM$O%f!<%6B&$,@UG$$r$b$D(B. |
入力値はユーザ側が責任をもつ. |
@end itemize |
@end itemize |
|
|
@example |
@example |
|
|
@end example |
@end example |
|
|
@table @t |
@table @t |
@item $B;2>H(B |
@item 参照 |
@end table |
@end table |
|
|
@comment --- ChangeLog $B$r=q$/(B. $BF05!(B. $B%=!<%9%3!<%I$N0LCV(B. $BJQ99F|;~(B $B$J$I(B CVS$B%5!<%P$r8+$k$?$a(B |
@comment --- ChangeLog を書く. 動機. ソースコードの位置. 変更日時 など CVSサーバを見るため |
@comment --- openxm $B$N30It$+$i$N4sM?$b=R$Y$k(B. Credit. |
@comment --- openxm の外部からの寄与も述べる. Credit. |
@noindent |
@noindent |
ChangeLog |
ChangeLog |
@itemize @bullet |
@itemize @bullet |
|
|
|
|
@page |
@page |
@comment **************************************************************** |
@comment **************************************************************** |
@node taji_alc.invpow,,, 1$BJQ?tBe?tE*6I=j%3%[%b%m%8!<N`MQ$N4X?t(B |
@node taji_alc.invpow,,, 1変数代数的局所コホモロジー類用の関数 |
@subsection @code{taji_alc.invpow} |
@subsection @code{taji_alc.invpow} |
@findex taji_alc.invpow |
@findex taji_alc.invpow |
|
|
@table @t |
@table @t |
@item taji_alc.invpow(@var{poly},@var{f},@var{m}) |
@item taji_alc.invpow(@var{poly},@var{f},@var{m}) |
:: $B>jM>BN(BQ[x]/<@var{f}>$B>e$G$N(B@var{poly}$B$N5U85$N(B@var{m}$B>h$r5a$a$k(B. |
:: 剰余体Q[x]/<@var{f}>上での@var{poly}の逆元の@var{m}乗を求める. |
@end table |
@end table |
|
|
@table @var |
@table @var |
@item return |
@item return |
$B5UQQ(B |
逆冪 |
|
|
@item poly |
@item poly |
$BB?9`<0(B |
多項式 |
@item f |
@item f |
Q$B>e$G4{Ls$JB?9`<0(B |
Q上で既約な多項式 |
@item m |
@item m |
$B<+A3?t(B |
自然数 |
@item switch |
@item switch |
$B%*%W%7%g%s;XDj(B |
オプション指定 |
|
|
case 0 : $B5UQQ$rM-M}?t78?tB?9`<0$GJV$9(B. |
case 0 : 逆冪を有理数係数多項式で返す. |
|
|
case 1 : $B5UQQ$r@0?t78?t2=%j%9%H$GJV$9(B. |
case 1 : 逆冪を整数係数化リストで返す. |
|
|
default : case 0 |
default : case 0 |
@end table |
@end table |
|
|
@itemize @bullet |
@itemize @bullet |
@item @var{poly}$B$H(B@var{f}$B$O8_$$$KAG$G$J$1$l$P$J$i$J$$(B. |
@item @var{poly}と@var{f}は互いに素でなければならない. |
@item $B%"%k%4%j%:%`$N9|3J$O7+$jJV$7(B2$B>hK!$G$"$k(B. $B$=$3$K:G>.B?9`<0$NM}O@$r1~MQ$7$F9bB.2=$7$F$$$k(B. |
@item アルゴリズムの骨格は繰り返し2乗法である. そこに最小多項式の理論を応用して高速化している. |
@end itemize |
@end itemize |
|
|
@example |
@example |
Line 479 default : case 0 |
|
Line 479 default : case 0 |
|
@end example |
@end example |
|
|
@table @t |
@table @t |
@item $B;2>H(B |
@item 参照 |
@end table |
@end table |
|
|
@comment --- ChangeLog $B$r=q$/(B. $BF05!(B. $B%=!<%9%3!<%I$N0LCV(B. $BJQ99F|;~(B $B$J$I(B CVS$B%5!<%P$r8+$k$?$a(B |
@comment --- ChangeLog を書く. 動機. ソースコードの位置. 変更日時 など CVSサーバを見るため |
@comment --- openxm $B$N30It$+$i$N4sM?$b=R$Y$k(B. Credit. |
@comment --- openxm の外部からの寄与も述べる. Credit. |
@noindent |
@noindent |
ChangeLog |
ChangeLog |
@itemize @bullet |
@itemize @bullet |
|
|
|
|
@page |
@page |
@comment **************************************************************** |
@comment **************************************************************** |
@node taji_alc.rem_formula,,, 1$BJQ?tBe?tE*6I=j%3%[%b%m%8!<N`MQ$N4X?t(B |
@node taji_alc.rem_formula,,, 1変数代数的局所コホモロジー類用の関数 |
@subsection @code{taji_alc.rem_formula} |
@subsection @code{taji_alc.rem_formula} |
@findex taji_alc.rem_formula |
@findex taji_alc.rem_formula |
|
|
@table @t |
@table @t |
@item taji_alc.rem_formula(@var{polylist}) |
@item taji_alc.rem_formula(@var{polylist}) |
:: $BB?9`<0(Bf(x)$B$rM?$($?$H$-$N>jM>8x<0$r5a$a$k(B. |
:: 多項式f(x)を与えたときの剰余公式を求める. |
@end table |
@end table |
|
|
@table @var |
@table @var |
@item return |
@item return |
@var{switch} $B$*$h$S(B $B@bL@J8$r;2>H(B |
@var{switch} および 説明文を参照 |
|
|
@item polylist |
@item polylist |
f(x)$B$r(BQ$B>e$G4{LsJ,2r$7$?(B [[$B0x;R(B,$B=EJ#EY(B,$BNmE@$N5-9f(B],...] $B$J$k%j%9%H(B |
f(x)をQ上で既約分解した [[因子,重複度,零点の記号],...] なるリスト |
|
|
@item switch |
@item switch |
$B%*%W%7%g%s;XDj(B |
オプション指定 |
|
|
case 0 : x$B$NQQ$G@0M}$7(B, $B%j%9%H$GJV$9(B. |
case 0 : xの冪で整理し, リストで返す. |
|
|
case 10 : f(x)$B$NQQ$G@0M}$7(B, $B%j%9%H$GJV$9(B. ($B0l0x;R$N>l9g$N$_BP1~(B) |
case 10 : f(x)の冪で整理し, リストで返す. (一因子の場合のみ対応) |
|
|
case 20 : x$B$NQQ$G@0M}$7(B, symbolic$B$JI=8=$GJV$9(B. |
case 20 : xの冪で整理し, symbolicな表現で返す. |
|
|
default : case 0 |
default : case 0 |
@end table |
@end table |
|
|
@itemize @bullet |
@itemize @bullet |
@item $B%"%k%4%j%:%`$O(B, $B%(%k%_!<%H$NJd4V>jM>$rMQ$$$F$$$k(B. |
@item アルゴリズムは, エルミートの補間剰余を用いている. |
@item $B>jM>8x<0$NI=8=J}K!$O$$$/$D$+9M$($i$l$k$?$a(B, @var{switch}$B$GA*Br<0$H$7$?(B. |
@item 剰余公式の表現方法はいくつか考えられるため, @var{switch}で選択式とした. |
@item @var{switch}=0 $B$NLa$jCM$N8+J}$r=R$Y$k(B. $BNc$H$7$F(B, f(x)=f1(x)^m1*f2(x)^m2$B$r9M$($k(B. |
@item @var{switch}=0 の戻り値の見方を述べる. 例として, f(x)=f1(x)^m1*f2(x)^m2を考える. |
$BF~NO$O(B [[f1(x),m1,z1],[f2(x),m2,z2]] $B$H$J$k(B. $B$=$N$H$-La$jCM$O(B, |
入力は [[f1(x),m1,z1],[f2(x),m2,z2]] となる. そのとき戻り値は, |
|
|
[r_{f1}(x,z1),r_{f2}(x,z2)] |
[r_{f1}(x,z1),r_{f2}(x,z2)] |
|
|
$B$J$k%j%9%H$GJV$5$l$k(B. $B$3$l$O(B, $B>jM>8x<0$,(B |
なるリストで返される. これは, 剰余公式が |
|
|
@tex |
@tex |
$r(x)=r_{f1}(x,z1)+r_{f2}(x,z2)$ |
$r(x)=r_{f1}(x,z1)+r_{f2}(x,z2)$ |
@end tex |
@end tex |
|
|
$B$J$k7A$GM?$($i$l$k$3$H$r0UL#$7$F$$$k(B. |
なる形で与えられることを意味している. |
$B3F@.J,$N(Br_{fi}(x,zi)$B$O(B, |
各成分のr_{fi}(x,zi)は, |
|
|
[p^(mi-1)(zi)$B$N78?t$H$J$k(Bx$B$H(Bzi$B$NB?9`<0(B,...,p^(0)(zi)$B$N78?t$H$J$k(Bx$B$H(Bzi$B$NB?9`<0(B] |
[p^(mi-1)(zi)の係数となるxとziの多項式,...,p^(0)(zi)の係数となるxとziの多項式] |
|
|
$B$J$k%j%9%H$G$"$k(B. |
なるリストである. |
@item @var{switch}=10 $B$NLa$jCM$N8+J}$r=R$Y$k(B. $BNc$H$7$F(B, f(x)=f1(x)^m$B$r9M$($k(B. |
@item @var{switch}=10 の戻り値の見方を述べる. 例として, f(x)=f1(x)^mを考える. |
$BF~NO$O(B [[f1(x),m,z]] $B$H$J$k(B. $B$=$N$H$-La$jCM$O(B, |
入力は [[f1(x),m,z]] となる. そのとき戻り値は, |
|
|
[r_(m-1)(x,z),...,r_0(x,z)] |
[r_(m-1)(x,z),...,r_0(x,z)] |
|
|
$B$J$k%j%9%H$GJV$5$l$k(B. $B3F@.J,$O(B, $B>jM>8x<0$r(B |
なるリストで返される. 各成分は, 剰余公式を |
|
|
@tex |
@tex |
$r(x)=r_{m-1}(x,z)f_1(x)^{m-1}+\cdots+r_0(x,z)$ |
$r(x)=r_{m-1}(x,z)f_1(x)^{m-1}+\cdots+r_0(x,z)$ |
@end tex |
@end tex |
|
|
$B$N$h$&$K(Bf1(x)$B$NQQ$GE83+$7$?$H$-$N3F78?t$r0UL#$7$F$$$k(B. |
のようにf1(x)の冪で展開したときの各係数を意味している. |
$B3F@.J,$N(Br_{i}(x,z)$B$O(B, |
各成分のr_{i}(x,z)は, |
|
|
[p^(m-1)(z)$B$N78?t$H$J$k(Bx$B$H(Bz$B$NB?9`<0(B,...,p^(0)(z)$B$N78?t$H$J$k(Bx$B$H(Bz$B$NB?9`<0(B] |
[p^(m-1)(z)の係数となるxとzの多項式,...,p^(0)(z)の係数となるxとzの多項式] |
|
|
$B$J$k%j%9%H$G$"$k(B. |
なるリストである. |
@item @var{switch}=20 $B$NLa$jCM$N8+J}$r=R$Y$k(B. |
@item @var{switch}=20 の戻り値の見方を述べる. |
symbolic$B$J=PNO$N(Bp^(m)(z)$B$O(B, p(x)$B$N(Bm$B3,$NF34X?t$K(Bz$B$rBeF~$7$?CM$H$$$&0UL#$G$"$k(B. |
symbolicな出力のp^(m)(z)は, p(x)のm階の導関数にzを代入した値という意味である. |
@item $BLa$jCM$O(B, $BM?$($?0x;R$NA4$F$NNmE@$rBeF~$7$?$b$N$NOB$H$7$F8+$k(B. |
@item 戻り値は, 与えた因子の全ての零点を代入したものの和として見る. |
$B$3$l$O0x;R$,(B2$B<!0J>e$NB?9`<0$N>l9g$K4X78$7$F$/$k(B. $BNc$($P(B, |
これは因子が2次以上の多項式の場合に関係してくる. 例えば, |
|
|
@example |
@example |
[228] taji_alc.rem_formula([[x^2+1,1,z]]); |
[228] taji_alc.rem_formula([[x^2+1,1,z]]); |
[[-1/2*z*x+1/2]] |
[[-1/2*z*x+1/2]] |
@end example |
@end example |
|
|
$B$N@5$7$$8+J}$O(B, x^2+1$B$NNmE@$r(Ba1,a2$B$H$*$$$?$H$-$K(B, z$B$K(Ba1$B$H(Ba2$B$rBeF~$7$?(B, |
の正しい見方は, x^2+1の零点をa1,a2とおいたときに, zにa1とa2を代入した, |
|
|
r(x)=(-1/2*a1*x+1/2)+(-1/2*a2*x+1/2) |
r(x)=(-1/2*a1*x+1/2)+(-1/2*a2*x+1/2) |
$B$G$"$k(B. $B$7$+$7=PNO$G$O(B, $BNmE@$NOB$NItJ,$rJX59>e>JN,$7$FJV$9(B. |
である. しかし出力では, 零点の和の部分を便宜上省略して返す. |
@end itemize |
@end itemize |
|
|
@example |
@example |
Line 609 z^2-174/529*z-108/529)*x^2+(-105/529*z^2+54/529*z+70/5 |
|
Line 609 z^2-174/529*z-108/529)*x^2+(-105/529*z^2+54/529*z+70/5 |
|
@end example |
@end example |
|
|
@table @t |
@table @t |
@item $B;2>H(B |
@item 参照 |
@end table |
@end table |
|
|
@comment --- ChangeLog $B$r=q$/(B. $BF05!(B. $B%=!<%9%3!<%I$N0LCV(B. $BJQ99F|;~(B $B$J$I(B CVS$B%5!<%P$r8+$k$?$a(B |
@comment --- ChangeLog を書く. 動機. ソースコードの位置. 変更日時 など CVSサーバを見るため |
@comment --- openxm $B$N30It$+$i$N4sM?$b=R$Y$k(B. Credit. |
@comment --- openxm の外部からの寄与も述べる. Credit. |
@noindent |
@noindent |
ChangeLog |
ChangeLog |
@itemize @bullet |
@itemize @bullet |
|
|
|
|
@page |
@page |
@comment **************************************************************** |
@comment **************************************************************** |
@node taji_alc.solve_ode_cp,,, 1$BJQ?tBe?tE*6I=j%3%[%b%m%8!<N`MQ$N4X?t(B |
@node taji_alc.solve_ode_cp,,, 1変数代数的局所コホモロジー類用の関数 |
@subsection @code{taji_alc.solve_ode_cp} |
@subsection @code{taji_alc.solve_ode_cp} |
@findex taji_alc.solve_ode_cp |
@findex taji_alc.solve_ode_cp |
|
|
@table @t |
@table @t |
@item taji_alc.solve_ode_cp(@var{poly},@var{var},@var{exppoly}) |
@item taji_alc.solve_ode_cp(@var{poly},@var{var},@var{exppoly}) |
:: $BM-M}?t78?t$N@~7A>oHyJ,J}Dx<0$N%3!<%7!<LdBj(B |
:: 有理数係数の線形常微分方程式のコーシー問題 |
|
|
@tex |
@tex |
$Pu(z)=f(z)$, $u^{(0)}(0)=c_0,\ldots,u^{(n-1)}(0)=c_{n-1}$ |
$Pu(z)=f(z)$, $u^{(0)}(0)=c_0,\ldots,u^{(n-1)}(0)=c_{n-1}$ |
@end tex |
@end tex |
|
|
$B$N2r$r5a$a$k(B. |
の解を求める. |
|
|
$B$?$@$7(B, P$B$O(Bn$B3,$NM-M}?t78?t$N@~7A>oHyJ,:nMQAG(B, f(z)$B$O;X?tB?9`<0$H$9$k(B. |
ただし, Pはn階の有理数係数の線形常微分作用素, f(z)は指数多項式とする. |
@end table |
@end table |
|
|
@table @var |
@table @var |
@item return |
@item return |
2$BDL$j$NI=8=$,$"$k(B. |
2通りの表現がある. |
|
|
$B!&I=8=(B1 ($B%3!<%7!<%G!<%?$G@0M}$7$?7A(B) |
・表現1 (コーシーデータで整理した形) |
|
|
$B%3!<%7!<LdBj$N0lHL2r(Bu(z)$B$O(B, |
コーシー問題の一般解u(z)は, |
|
|
@tex$u(z)=c_0u_0(z)+\cdots+c_{n-1}u_{n-1}(z)+v(z)$@end tex |
@tex$u(z)=c_0u_0(z)+\cdots+c_{n-1}u_{n-1}(z)+v(z)$@end tex |
|
|
$B$J$k@~7A7k9g$N7A$GM?$($i$l$k(B. |
なる線形結合の形で与えられる. |
@tex$u_0(z),\ldots,u_{n-1}(z)$@end tex |
@tex$u_0(z),\ldots,u_{n-1}(z)$@end tex |
$B$r%3!<%7!<LdBj$N4pK\2r(B, |
をコーシー問題の基本解, |
@tex$v(z)$@end tex |
@tex$v(z)$@end tex |
$B$r%3!<%7!<LdBj$NFC<l2r$H$$$$(B, |
をコーシー問題の特殊解といい, |
|
|
[u_0(z),...,u_(n-1)(z),v(z)] |
[u_0(z),...,u_(n-1)(z),v(z)] |
|
|
$B$J$k%j%9%H$GJV$9(B. |
なるリストで返す. |
$B4pK\2r$HFC<l2r$O(B, $B;X?tB?9`<0%j%9%H$G$"$k(B. |
基本解と特殊解は, 指数多項式リストである. |
|
|
$B!&I=8=(B2 ($B;X?t4X?t$G@0M}$7$?7A(B) |
・表現2 (指数関数で整理した形) |
|
|
@var{data}$B$K%3!<%7!<%G!<%?$rM?$($k$H(B, |
@var{data}にコーシーデータを与えると, |
$B%3!<%7!<LdBj$N0lHL2r(Bu(z)$B$N(B |
コーシー問題の一般解u(z)の |
@tex$c_0,\ldots,c_{n-1}$@end tex |
@tex$c_0,\ldots,c_{n-1}$@end tex |
$B$N$H$3$m$K%G!<%?$rBeF~$7(B, |
のところにデータを代入し, |
$B$=$l$r;X?t4X?t$G@0M}$7D>$7$?;X?tB?9`<0%j%9%H$rJV$9(B. |
それを指数関数で整理し直した指数多項式リストを返す. |
|
|
@item poly |
@item poly |
$BB?9`<0(B (P$B$NFC@-B?9`<0(B) |
多項式 (Pの特性多項式) |
|
|
$B$^$?$O(B (P$B$NFC@-B?9`<0$r(BQ$B>e$G4{LsJ,2r$7$?(B) [[$B0x;R(B,$B=EJ#EY(B],...] $B$J$k%j%9%H(B |
または (Pの特性多項式をQ上で既約分解した) [[因子,重複度],...] なるリスト |
|
|
@item var |
@item var |
$BITDj85(B ($B4X?t$NFHN)JQ?t(B) |
不定元 (関数の独立変数) |
|
|
@item exppoly |
@item exppoly |
$B@F<!7A$N$H$-(B0, $BHs@F<!7A$N$H$-(Bf(z)$B$N;X?tB?9`<0%j%9%H(B. |
斉次形のとき0, 非斉次形のときf(z)の指数多項式リスト. |
|
|
@item switch |
@item switch |
$B%*%W%7%g%s;XDj(B |
オプション指定 |
|
|
case 0 : $B;X?tB?9`<0%j%9%H$N(B2$BHVL\$N@.J,$rM-M}?t78?tB?9`<0$GJV$9(B. |
case 0 : 指数多項式リストの2番目の成分を有理数係数多項式で返す. |
|
|
case 1 : $B;X?tB?9`<0%j%9%H$N(B2$BHVL\$N@.J,$r@0?t78?t2=%j%9%H$GJV$9(B. |
case 1 : 指数多項式リストの2番目の成分を整数係数化リストで返す. |
|
|
default : case 0 |
default : case 0 |
@item data |
@item data |
$B%*%W%7%g%s;XDj(B |
オプション指定 |
|
|
$B%3!<%7!<%G!<%?$r(B [c_0,...,c_(n-1)] $B$N=g$KJB$Y$?%j%9%H(B. |
コーシーデータを [c_0,...,c_(n-1)] の順に並べたリスト. |
@end table |
@end table |
|
|
@itemize @bullet |
@itemize @bullet |
@item $B2rK!$O%(%k%_!<%H$NJ}K!(B($BN1?t7W;;$K5"Ce$5$;$kJ}K!(B)$B$r:NMQ$7$F$$$k(B. |
@item 解法はエルミートの方法(留数計算に帰着させる方法)を採用している. |
@item $BJQ?t$O(B2$B<oN`I,MW(B($BFC@-B?9`<0$NJQ?t$H4X?t$NFHN)JQ?t(B). |
@item 変数は2種類必要(特性多項式の変数と関数の独立変数). |
@var{poly}$B$NITDj85$H(B@var{var}$B$NITDj85$,>WFM$7$J$$$h$&Cm0U(B. |
@var{poly}の不定元と@var{var}の不定元が衝突しないよう注意. |
@item $BLa$jCM$NFC<l2r(B |
@item 戻り値の特殊解 |
@tex |
@tex |
$v(z)$ |
$v(z)$ |
@end tex |
@end tex |
$B$O(B, $B%3!<%7!<>r7o(B |
は, コーシー条件 |
@tex |
@tex |
$v(0)=0,\ldots,v^{(n-1)}(0)=0$ |
$v(0)=0,\ldots,v^{(n-1)}(0)=0$ |
@end tex |
@end tex |
$B$rK~$?$9%3!<%7!<LdBj$NFC<l2r$G$"$k(B. |
を満たすコーシー問題の特殊解である. |
@end itemize |
@end itemize |
|
|
@example |
@example |
Line 729 $v(0)=0,\ldots,v^{(n-1)}(0)=0$ |
|
Line 729 $v(0)=0,\ldots,v^{(n-1)}(0)=0$ |
|
@end example |
@end example |
|
|
@table @t |
@table @t |
@item $B;2>H(B |
@item 参照 |
@end table |
@end table |
|
|
@comment --- ChangeLog $B$r=q$/(B. $BF05!(B. $B%=!<%9%3!<%I$N0LCV(B. $BJQ99F|;~(B $B$J$I(B CVS$B%5!<%P$r8+$k$?$a(B |
@comment --- ChangeLog を書く. 動機. ソースコードの位置. 変更日時 など CVSサーバを見るため |
@comment --- openxm $B$N30It$+$i$N4sM?$b=R$Y$k(B. Credit. |
@comment --- openxm の外部からの寄与も述べる. Credit. |
@noindent |
@noindent |
ChangeLog |
ChangeLog |
@itemize @bullet |
@itemize @bullet |
|
|
|
|
@page |
@page |
@comment **************************************************************** |
@comment **************************************************************** |
@node taji_alc.solve_ode_cp_ps,,, 1$BJQ?tBe?tE*6I=j%3%[%b%m%8!<N`MQ$N4X?t(B |
@node taji_alc.solve_ode_cp_ps,,, 1変数代数的局所コホモロジー類用の関数 |
@subsection @code{taji_alc.solve_ode_cp_ps} |
@subsection @code{taji_alc.solve_ode_cp_ps} |
@findex taji_alc.solve_ode_cp_ps |
@findex taji_alc.solve_ode_cp_ps |
|
|
@table @t |
@table @t |
@item taji_alc.solve_ode_cp_ps(@var{poly},@var{var},@var{exppoly}) |
@item taji_alc.solve_ode_cp_ps(@var{poly},@var{var},@var{exppoly}) |
:: $BM-M}?t78?t$N@~7A>oHyJ,J}Dx<0$N%3!<%7!<LdBj(B |
:: 有理数係数の線形常微分方程式のコーシー問題 |
|
|
@tex |
@tex |
$Pu(z)=f(z)$, $u^{(0)}(0)=c_0,\ldots,u^{(n-1)}(0)=c_{n-1}$ |
$Pu(z)=f(z)$, $u^{(0)}(0)=c_0,\ldots,u^{(n-1)}(0)=c_{n-1}$ |
@end tex |
@end tex |
|
|
$B$NFC<l2r$r5a$a$k(B. |
の特殊解を求める. |
|
|
$B$?$@$7(B, $BHs@F<!7A$N$_$rBP>]$H$7$F$$$k$N$G(B, |
ただし, 非斉次形のみを対象としているので, |
@tex |
@tex |
$f(z)\neq0$ |
$f(z)\neq0$ |
@end tex |
@end tex |
$B$H$9$k(B. |
とする. |
|
|
@end table |
@end table |
|
|
@table @var |
@table @var |
@item return |
@item return |
$B;X?tB?9`<0%j%9%H(B |
指数多項式リスト |
|
|
@item poly |
@item poly |
$BB?9`<0(B (P$B$NFC@-B?9`<0(B) |
多項式 (Pの特性多項式) |
|
|
$B$^$?$O(B (P$B$NFC@-B?9`<0$r(BQ$B>e$G4{LsJ,2r$7$?(B) [[$B0x;R(B,$B=EJ#EY(B],...] $B$J$k%j%9%H(B |
または (Pの特性多項式をQ上で既約分解した) [[因子,重複度],...] なるリスト |
|
|
@item var |
@item var |
$BITDj85(B ($B4X?t$NFHN)JQ?t(B) |
不定元 (関数の独立変数) |
|
|
@item exppoly |
@item exppoly |
f(z)$B$N;X?tB?9`<0%j%9%H(B |
f(z)の指数多項式リスト |
|
|
@item switch |
@item switch |
$B%*%W%7%g%s;XDj(B |
オプション指定 |
|
|
case 0 : $B;X?tB?9`<0%j%9%H$N(B2$BHVL\$N@.J,$rM-M}?t78?tB?9`<0$GJV$9(B. |
case 0 : 指数多項式リストの2番目の成分を有理数係数多項式で返す. |
|
|
case 1 : $B;X?tB?9`<0%j%9%H$N(B2$BHVL\$N@.J,$r@0?t78?t2=%j%9%H$GJV$9(B. |
case 1 : 指数多項式リストの2番目の成分を整数係数化リストで返す. |
|
|
default : case 0 |
default : case 0 |
|
|
@item switch2 |
@item switch2 |
$B%*%W%7%g%s;XDj(B |
オプション指定 |
|
|
case 0 : $B%3!<%7!<LdBj$NFC<l2r$rJV$9(B. |
case 0 : コーシー問題の特殊解を返す. |
|
|
case 1 : $B4JC1$J7A$NFC<l2r$rJV$9(B. |
case 1 : 簡単な形の特殊解を返す. |
|
|
default : case 0 |
default : case 0 |
@end table |
@end table |
|
|
@itemize @bullet |
@itemize @bullet |
@item $BJQ?t$O(B2$B<oN`I,MW(B($BFC@-B?9`<0$NJQ?t$H4X?t$NFHN)JQ?t(B). |
@item 変数は2種類必要(特性多項式の変数と関数の独立変数). |
@var{poly}$B$NITDj85$H(B@var{var}$B$NITDj85$,>WFM$7$J$$$h$&Cm0U(B. |
@var{poly}の不定元と@var{var}の不定元が衝突しないよう注意. |
@end itemize |
@end itemize |
|
|
@example |
@example |
Line 824 x-3,[4232*z^2-4278*z-4295,97336]]] |
|
Line 824 x-3,[4232*z^2-4278*z-4295,97336]]] |
|
@end example |
@end example |
|
|
@table @t |
@table @t |
@item $B;2>H(B |
@item 参照 |
@end table |
@end table |
|
|
@comment --- ChangeLog $B$r=q$/(B. $BF05!(B. $B%=!<%9%3!<%I$N0LCV(B. $BJQ99F|;~(B $B$J$I(B CVS$B%5!<%P$r8+$k$?$a(B |
@comment --- ChangeLog を書く. 動機. ソースコードの位置. 変更日時 など CVSサーバを見るため |
@comment --- openxm $B$N30It$+$i$N4sM?$b=R$Y$k(B. Credit. |
@comment --- openxm の外部からの寄与も述べる. Credit. |
@noindent |
@noindent |
ChangeLog |
ChangeLog |
@itemize @bullet |
@itemize @bullet |
|
|
|
|
@page |
@page |
@comment **************************************************************** |
@comment **************************************************************** |
@node taji_alc.fbt,,, 1$BJQ?tBe?tE*6I=j%3%[%b%m%8!<N`MQ$N4X?t(B |
@node taji_alc.fbt,,, 1変数代数的局所コホモロジー類用の関数 |
@subsection @code{taji_alc.fbt} |
@subsection @code{taji_alc.fbt} |
@findex taji_alc.fbt |
@findex taji_alc.fbt |
|
|
@table @t |
@table @t |
@item taji_alc.fbt(@var{num},@var{den},@var{var}) |
@item taji_alc.fbt(@var{num},@var{den},@var{var}) |
:: $BM-M}4X?t(B@var{num}/@var{den}$B$,Dj$a$kBe?tE*6I=j%3%[%b%m%8!<N`$N%U!<%j%(!&%\%l%kJQ49$r9T$&(B. |
:: 有理関数@var{num}/@var{den}が定める代数的局所コホモロジー類のフーリエ・ボレル変換を行う. |
@end table |
@end table |
|
|
@table @var |
@table @var |
@item return |
@item return |
[$B;X?tB?9`<0%j%9%H(B,...] $B$J$k%j%9%H(B |
[指数多項式リスト,...] なるリスト |
|
|
@item num |
@item num |
($BM-M}4X?t$NJ,;R$N(B) $BB?9`<0(B |
(有理関数の分子の) 多項式 |
@item den |
@item den |
($BM-M}4X?t$NJ,Jl$N(B) $BB?9`<0(B |
(有理関数の分母の) 多項式 |
|
|
$B$^$?$O(B ($BM-M}4X?t$NJ,Jl$r(BQ$B>e$G4{LsJ,2r$7$?(B) [[$B0x;R(B,$B=EJ#EY(B],...] $B$J$k%j%9%H(B |
または (有理関数の分母をQ上で既約分解した) [[因子,重複度],...] なるリスト |
@item var |
@item var |
$BITDj85(B ($BA|$NFHN)JQ?t(B) |
不定元 (像の独立変数) |
@item switch |
@item switch |
$B%*%W%7%g%s;XDj(B |
オプション指定 |
|
|
case 0 : $B;X?tB?9`<0%j%9%H$N(B2$BHVL\$N@.J,$rM-M}?t78?tB?9`<0$GJV$9(B. |
case 0 : 指数多項式リストの2番目の成分を有理数係数多項式で返す. |
|
|
case 1 : $B;X?tB?9`<0%j%9%H$N(B2$BHVL\$N@.J,$r@0?t78?t2=%j%9%H$GJV$9(B. |
case 1 : 指数多項式リストの2番目の成分を整数係数化リストで返す. |
|
|
default : case 0 |
default : case 0 |
@end table |
@end table |
|
|
@itemize @bullet |
@itemize @bullet |
@item $BJQ?t$O(B2$B<oN`I,MW(B($BBe?tE*6I=j%3%[%b%m%8!<N`$NJQ?t$HA|$NFHN)JQ?t(B). |
@item 変数は2種類必要(代数的局所コホモロジー類の変数と像の独立変数). |
@var{num/den}$B$NITDj85$H(B@var{var}$B$NITDj85$,>WFM$7$J$$$h$&Cm0U(B. |
@var{num/den}の不定元と@var{var}の不定元が衝突しないよう注意. |
@item taji_alc.fbt()$B$O(B, Res(Rat*exp(z*x))$B$J$k7A$NM-M}7A4X?t$NN1?t$r5a$a$k(B. |
@item taji_alc.fbt()は, Res(Rat*exp(z*x))なる形の有理形関数の留数を求める. |
$B$3$NM-M}7A4X?t$NN1?t$O;X?tB?9`<0$H$J$k$?$a(B, $B;X?tB?9`<0%j%9%H$GJV$9(B. |
この有理形関数の留数は指数多項式となるため, 指数多項式リストで返す. |
@item $BFbIt$N%"%k%4%j%:%`$O(Btaji_alc.residue()$B$H$[$\F1$8$G$"$j(B, $B<B:]$K(Btaji_alc.residue()$B$r8F$S=P$7$F7W;;$r9T$C$F$$$k(B. |
@item 内部のアルゴリズムはtaji_alc.residue()とほぼ同じであり, 実際にtaji_alc.residue()を呼び出して計算を行っている. |
@end itemize |
@end itemize |
|
|
@example |
@example |
Line 884 default : case 0 |
|
Line 884 default : case 0 |
|
@end example |
@end example |
|
|
@table @t |
@table @t |
@item $B;2>H(B |
@item 参照 |
@ref{taji_alc.residue, taji_alc.invfbt} |
@ref{taji_alc.residue, taji_alc.invfbt} |
@end table |
@end table |
|
|
@comment --- ChangeLog $B$r=q$/(B. $BF05!(B. $B%=!<%9%3!<%I$N0LCV(B. $BJQ99F|;~(B $B$J$I(B CVS$B%5!<%P$r8+$k$?$a(B |
@comment --- ChangeLog を書く. 動機. ソースコードの位置. 変更日時 など CVSサーバを見るため |
@comment --- openxm $B$N30It$+$i$N4sM?$b=R$Y$k(B. Credit. |
@comment --- openxm の外部からの寄与も述べる. Credit. |
@noindent |
@noindent |
ChangeLog |
ChangeLog |
@itemize @bullet |
@itemize @bullet |
|
|
|
|
@page |
@page |
@comment **************************************************************** |
@comment **************************************************************** |
@node taji_alc.inv,,, 1$BJQ?tBe?tE*6I=j%3%[%b%m%8!<N`MQ$N4X?t(B |
@node taji_alc.inv,,, 1変数代数的局所コホモロジー類用の関数 |
@subsection @code{taji_alc.invfbt} |
@subsection @code{taji_alc.invfbt} |
@findex taji_alc.invfbt |
@findex taji_alc.invfbt |
|
|
@table @t |
@table @t |
@item taji_alc.invfbt(@var{exppoly},@var{var}) |
@item taji_alc.invfbt(@var{exppoly},@var{var}) |
:: $B;X?tB?9`<0$N5U%U!<%j%(!&%\%l%kJQ49$r9T$&(B. |
:: 指数多項式の逆フーリエ・ボレル変換を行う. |
@end table |
@end table |
|
|
@table @var |
@table @var |
@item return |
@item return |
$BM-M}4X?t(B |
有理関数 |
|
|
@item exppoly |
@item exppoly |
$B;X?tB?9`<0%j%9%H(B |
指数多項式リスト |
@item var |
@item var |
$BITDj85(B ($B;X?tB?9`<0$NFHN)JQ?t(B) |
不定元 (指数多項式の独立変数) |
|
|
@item switch |
@item switch |
$B%*%W%7%g%s;XDj(B |
オプション指定 |
|
|
case 0 : $BM-M}4X?t$GJV$9(B. |
case 0 : 有理関数で返す. |
|
|
case 1 : $BM-M}4X?t$r(B[$BJ,;R(B,$BJ,Jl$r(BQ$B>e$G4{LsJ,2r$7$?%j%9%H(B]$B$J$k%j%9%H$GJV$9(B. |
case 1 : 有理関数を[分子,分母をQ上で既約分解したリスト]なるリストで返す. |
|
|
default : case 0 |
default : case 0 |
@end table |
@end table |
|
|
@itemize @bullet |
@itemize @bullet |
@item $BJQ?t$O(B2$B<oN`I,MW(B($BBe?tE*?t$N:G>.B?9`<0$NJQ?t$H;X?tB?9`<0$NFHN)JQ?t(B). |
@item 変数は2種類必要(代数的数の最小多項式の変数と指数多項式の独立変数). |
$B>WFM$7$J$$$h$&Cm0U(B. |
衝突しないよう注意. |
@item taji_alc.invfbt()$B$O(B, exppoly$B$r(B, Res(Rat*exp(z*x))$B$J$k7A$NN1?tI=<($KJQ49$7(B, Rat$BItJ,$rJV$9(B. |
@item taji_alc.invfbt()は, exppolyを, Res(Rat*exp(z*x))なる形の留数表示に変換し, Rat部分を返す. |
@item taji_alc.fbt()$B$N5U1i;;$G$"$k(B. |
@item taji_alc.fbt()の逆演算である. |
@end itemize |
@end itemize |
|
|
@example |
@example |
|
|
@end example |
@end example |
|
|
@table @t |
@table @t |
@item $B;2>H(B |
@item 参照 |
@ref{taji_alc.fbt} |
@ref{taji_alc.fbt} |
@end table |
@end table |
|
|
@comment --- ChangeLog $B$r=q$/(B. $BF05!(B. $B%=!<%9%3!<%I$N0LCV(B. $BJQ99F|;~(B $B$J$I(B CVS$B%5!<%P$r8+$k$?$a(B |
@comment --- ChangeLog を書く. 動機. ソースコードの位置. 変更日時 など CVSサーバを見るため |
@comment --- openxm $B$N30It$+$i$N4sM?$b=R$Y$k(B. Credit. |
@comment --- openxm の外部からの寄与も述べる. Credit. |
@noindent |
@noindent |
ChangeLog |
ChangeLog |
@itemize @bullet |
@itemize @bullet |
|
|
|
|
|
|
|
|
@comment --- $B$*$^$8$J$$(B --- |
@comment --- おまじない --- |
@node Index,,, Top |
@node Index,,, Top |
@unnumbered Index |
@unnumbered Index |
@printindex fn |
@printindex fn |
|
|
@summarycontents |
@summarycontents |
@contents |
@contents |
@bye |
@bye |
@comment --- $B$*$^$8$J$$=*$j(B --- |
@comment --- おまじない終り --- |