@overfullrule=0pt
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This section is a collection of functions to solve regular holonomic
systems in terms of series.
Algorithms are explained in the book [SST].
You can load this package by the command
load("dsolv.rr")$
This package requires Diff
and dmodule
.
To use the functions of the package dsolv
in OpenXM/Risa/Asir,
executing the command load("dsolv.rr")$
is necessary at first.
This package uses ox_sm1
, so the variables you can use
is as same as those you can use in the package sm1
.
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1.1.1 dsolv_dual  
1.1.2 dsolv_starting_term 
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dsolv_dual
:: Grobner dual of f.
List
List
Algorithm:
This is an implementation of Algorithm 2.3.14 of the book [SST].
If we replace variables x, y, ... in the output by log(x), log(y), ...,
then these polynomials in log are solutions of the system of differential
equations f_(x>x*dx, y>y*dy, ...)
.
[435] dsolv_dual([yx^2,y+x^2],[x,y]); [x,1] [436] dsolv_act(y*dysm1.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^2x^3,x^2*y^2],[x,y]); [[[y^2x^3,y^4,x^2*y^2],[y,x]]] [440] dsolv_dual([y^2x^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.
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dsolv_starting_term
:: Find the starting term of the solutions of the regular holonomic system f to the direction w.
List
List
Dsolv_message_starting_term
to 1,
then this function outputs messages during the computation.
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*dx11,x4*dx4+x2*dx2+x1*dx1, x4*dx4+x3*dx3+x2*dx2, dx2*dx5+dx1*dx3,dx5^2dx2*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+x31,x5+3*x4x21,x5+2*x4+x11,3*x5^2+(8*x46)*x58*x4+3, x5^22*x58*x4^2+1,x5^33*x5^2+3*x51], [x51,x4,x3,x2,x1]]]  root is [ 0 0 0 0 1 ]  dual system is [x5^2+(3/4*x41/2*x31/4*x21/2*x1)*x5+1/8*x4^2 +(1/4*x3+1/4*x1)*x4+1/4*x2*x31/8*x2^2+1/4*x1*x2, x42*x3+3*x22*x1,x5x3+x2x1,1] [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]]]
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Jump to:  D 

Index Entry  Section  

 
D  
dsolv_dual  1.1.1 dsolv_dual  
dsolv_starting_term  1.1.2 dsolv_starting_term  

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