Dsolv マニュアル

Edition : auto generated by oxgentexi on October 19, 2017

OpenXM.org

@overfullrule=0pt

[ << ] [ < ] [ Up ] [ > ] [ >> ]         [Top] [Contents] [Index] [ ? ]

1 DSOLV 函数

この節は正則ホロノミック系を級数で解くための 函数をあつめてある. アルゴリズムについては [SST] に説明がある. このパッケージは次のコマンド load("dsolv.rr"); でロードできる. このパッケージは Diff および Dmodule を使用する.

OpenXM/Risa/Asir での利用にあたっては,

load("dsolv.rr");$

が始めに必要.

このパッケージは ox_sm1 を利用している. したがって使用できる変数は sm1 パッケージと同様の変数しかつかえない.


[ << ] [ < ] [ Up ] [ > ] [ >> ]         [Top] [Contents] [Index] [ ? ]

1.1 函数一覧


[ << ] [ < ] [ Up ] [ > ] [ >> ]         [Top] [Contents] [Index] [ ? ]

1.1.1 dsolv_dual

dsolv_dual(f,v)

:: f のグレブナ双対

戻り値

リスト

f, v

リスト

Algorithm: この函数は本 [SST] の Algorithm 2.3.14 の実装である. 出力中の変数 x, y, ... をそれぞれ log(x), log(y), ..., でおきかえると, これらの log 多項式は, f_(x->x*dx, y->y*dy, ...) で生成される微分方程式系 の解となっている.


[435] dsolv_dual([y-x^2,y+x^2],[x,y]);
[x,1]
[436] dsolv_act(y*dy-sm1.mul(x*dx,x*dx,[x,y]),log(x),[x,y]);
0
[437] dsolv_act(y*dy+sm1.mul(x*dx,x*dx,[x,y]),log(x),[x,y]);
0

[439] primadec([y^2-x^3,x^2*y^2],[x,y]);
[[[y^2-x^3,y^4,x^2*y^2],[y,x]]]
[440] dsolv_dual([y^2-x^3,x^2*y^2],[x,y]);
[x*y^3+1/4*x^4*y, x^2*y, x*y^2+1/12*x^4, y^3+x^3*y,
 x^2, x*y, y^2+1/3*x^3, x, y, 1]

[441] dsolv_test_dual();
  Output is  omitted.


[ << ] [ < ] [ Up ] [ > ] [ >> ]         [Top] [Contents] [Index] [ ? ]

1.1.2 dsolv_starting_term

dsolv_starting_term(f,v,w)

:: 正則ホロノミック系 f の方向 w での級数解の Staring terms を計算する. ここで, v は変数の集合.

戻り値

リスト

f, v, w

リスト

Algorithm: Saito, Sturmfels, Takayama, Grobner Deformations of Hypergeometric Differential Equations ([SST]), Chapter 2.

[1076]   F = sm1.gkz( [ [[1,1,1,1,1],[1,1,0,-1,0],[0,1,1,-1,0]], [1,0,0]]);
[[x5*dx5+x4*dx4+x3*dx3+x2*dx2+x1*dx1-1,-x4*dx4+x2*dx2+x1*dx1,
  -x4*dx4+x3*dx3+x2*dx2,
  -dx2*dx5+dx1*dx3,dx5^2-dx2*dx4],[x1,x2,x3,x4,x5]]
[1077]  A= dsolv_starting_term(F[0],F[1],[1,1,1,1,0])$
Computing the initial ideal.
Done.
Computing a primary ideal decomposition.
Primary ideal decomposition of the initial Frobenius ideal 
to the direction [1,1,1,1,0] is 
[[[x5+2*x4+x3-1,x5+3*x4-x2-1,x5+2*x4+x1-1,3*x5^2+(8*x4-6)*x5-8*x4+3,
   x5^2-2*x5-8*x4^2+1,x5^3-3*x5^2+3*x5-1],
 [x5-1,x4,x3,x2,x1]]]
 
----------- root is [ 0 0 0 0 1 ]
----------- dual system is 
[x5^2+(-3/4*x4-1/2*x3-1/4*x2-1/2*x1)*x5+1/8*x4^2
 +(1/4*x3+1/4*x1)*x4+1/4*x2*x3-1/8*x2^2+1/4*x1*x2,
 x4-2*x3+3*x2-2*x1,x5-x3+x2-x1,1]
  
[1078] A[0];
[[ 0 0 0 0 1 ]]
[1079] map(fctr,A[1][0]);
[[[1/8,1],[x5,1],[log(x2)+log(x4)-2*log(x5),1],
          [2*log(x1)-log(x2)+2*log(x3)+log(x4)-4*log(x5),1]],
 [[1,1],[x5,1],[-2*log(x1)+3*log(x2)-2*log(x3)+log(x4),1]],
 [[1,1],[x5,1],[-log(x1)+log(x2)-log(x3)+log(x5),1]],
 [[1,1],[x5,1]]]


[ << ] [ < ] [ Up ] [ > ] [ >> ]         [Top] [Contents] [Index] [ ? ]

Index

Jump to:   D  
Index Entry  Section

D
dsolv_dual 1.1.1 dsolv_dual
dsolv_starting_term 1.1.2 dsolv_starting_term

Jump to:   D  

[Top] [Contents] [Index] [ ? ]

Table of Contents


[Top] [Contents] [Index] [ ? ]

Short Table of Contents


[Top] [Contents] [Index] [ ? ]

About This Document

This document was generated on October 19, 2017 using texi2html 5.0.

The buttons in the navigation panels have the following meaning:

Button Name Go to From 1.2.3 go to
[ << ] FastBack Beginning of this chapter or previous chapter 1
[ < ] Back Previous section in reading order 1.2.2
[ Up ] Up Up section 1.2
[ > ] Forward Next section in reading order 1.2.4
[ >> ] FastForward Next chapter 2
[Top] Top Cover (top) of document  
[Contents] Contents Table of contents  
[Index] Index Index  
[ ? ] About About (help)  

where the Example assumes that the current position is at Subsubsection One-Two-Three of a document of the following structure:


This document was generated on October 19, 2017 using texi2html 5.0.