/* $OpenXM: OpenXM/src/asir-contrib/packages/doc/dsolv.oxweave,v 1.2 2000/02/07 04:46:25 takayama Exp $ */
/* dsolv.oxweave */
/*&eg-texi
@node DSOLV Functions,,, Top

@chapter DSOLV Functions

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
@code{load("dsolv");}
This package requires @code{Diff} and @code{dmodule}.

This package uses @code{ox_sm1}, so the variables you can use
is as same as those you can use in the package @code{sm1}.

@section Functions

*/

/*&jp-texi
@node DSOLV 函数,,, Top

@chapter 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 @code{load("dsolv");}
でロードできる.
このパッケージは @code{Diff} および @code{dmodule} を使用する.

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

@section 函数一覧

*/

/*&eg-texi
@menu
* dsolv_dual::
@end menu
@node dsolv_dual,,, DSOLV Functions
@subsection @code{dsolv_dual}
@findex dsolv_dual
@table @t
@item dsolv_dual(@var{f},@var{v})
::  Grobner dual of @var{f}.
@end table

@table @var
@item return
List
@item f, v
List
@end table

@itemize @bullet
@item  It returns the Grobner dual of @var{f} in the ring of polynomials
with variables @var{v}.
@item The ideal generated by @var{f} must be primary to the maximal ideal
generated by @var{v}.
If it is not primary to the maximal ideal, then this function falls into
an infinite loop.
@item 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 @var{f}@code{_(x->x*dx, y->y*dy, ...)}.
@end itemize
*/

/*&jp-texi
@menu
* dsolv_dual::
@end menu
@node dsolv_dual,,, DSOLV 函数
@subsection @code{dsolv_dual}
@findex dsolv_dual
@table @t
@item dsolv_dual(@var{f},@var{v})
::  @var{f} のグレブナ双対
@end table

@table @var
@item 戻り値
リスト
@item f, v
リスト
@end table

@itemize @bullet
@item  変数 @var{v} 上の多項式環において,
@var{f}  のグレブナ双対を求める.
@item @var{f} で生成されるイデアルは, @var{v} で生成される極大イデアルに
対して, primary でないといけない.
primary でない場合, この函数は無限ループにおちいる.
@item 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, 
@var{f}@code{_(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.
@end itemize
*/

/*&C-texi

@example


[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.

@end example

*/


/*&eg-texi

@menu
* dsolv_starting_term::
@end menu
@node dsolv_starting_term,,, DSOLV Functions
@subsection @code{dsolv_starting_term}
@findex dsolv_starting_term
@table @t
@item dsolv_starting_term(@var{f},@var{v},@var{w})
::  Find the starting term of the solutions of 
the regular holonomic system @var{f}
to the direction @var{w}.
@end table

@table @var
@item return
List
@item f, v, w
List
@end table

@itemize @bullet
@item Find the starting term of the solutions of 
the regular holonomic system @var{f}
to the direction @var{w}.
@item The return value is of the form 
[[@var{e1}, @var{e2}, ...],
 [@var{s1}, @var{s2}, ...]]
where @var{e1} is an exponent vector and @var{s1} is the corresponding
solution set, and so on.
@item If you set @code{Dsolv_message_starting_term} to 1,
then this function outputs messages during the computation.
@end itemize

*/

/*&jp-texi

@menu
* dsolv_starting_term::
@end menu
@node dsolv_starting_term,,, DSOLV 函数
@subsection @code{dsolv_starting_term}
@findex dsolv_starting_term
@table @t
@item dsolv_starting_term(@var{f},@var{v},@var{w})
:: 正則ホロノミック系 @var{f} の方向 @var{w} での級数解の
Staring terms を計算する. ここで, @var{v} は変数の集合.
@end table

@table @var
@item 戻り値
リスト
@item f, v, w
リスト
@end table

@itemize @bullet
@item 正則ホロノミック系 @var{f} の方向 @var{w} での級数解の
Staring terms を計算する. ここで, @var{v} は変数の集合.
@item 戻り値は次の形をしている:
[[@var{e1}, @var{e2}, ...],
 [@var{s1}, @var{s2}, ...]]
ここで @var{e1} は exponent ベクトルであり @var{s1} はこのベクトルに
対応する解の集合,   以下同様.
@item 変数 @code{Dsolv_message_starting_term} を 1 にしておくと,
この函数は計算の途中にいろいろとメッセージを出力する.
@end itemize

*/

/*&C-texi

@example
[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]]]

@end example

*/