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Diff for /OpenXM/src/asir-contrib/packages/doc/Attic/Diff.texi between version 1.5 and 1.7

version 1.5, 2000/01/31 11:01:00

version 1.7, 2001/06/24 01:14:21

Line 1

@c $OpenXM: OpenXM/src/asir-contrib/packages/doc/Diff.texi,v 1.4 2000/01/21 12:54:39 okutani Exp $

@c $OpenXM: OpenXM/src/asir-contrib/packages/doc/Diff.texi,v 1.6 2000/02/20 11:30:44 okutani Exp $

@node Differential equations,,, $B$=$NB>$NH!?t(B

@node Differential equations (library by Okutani),,, Top

@section Differential equations

@section Differential equations (library by Okutani)

$B%U%!%$%k(B @file{gr}, @file{Matrix} $B$,I,MW$G$9(B.

Yukio Okutani $B;a$K$h$k(B Risa/Asir $B8@8l$G=q$+$l$?O"N)@~7AJPHyJ,J}Dx<0MQ(B

$B$N%i%$%V%i%j$G$9(B.

$B$9$Y$F$N4X?tL>$O(B odiff_ $B$G;O$^$j$^$9(B.

@tex

$B$3$N@a$G>R2p$5$l$k4X?t$G$OHyJ,:nMQAG$O<!$N$h$&$J%j%9%H$GI=8=$7$^$9(B.

$B$3$N@a$G>R2p$5$l$k4X?t$G$OHyJ,:nMQAG$O%j%9%H$^$?$OB?9`<0$GI=8=$5$l$^$9(B.

$B%j%9%H$K$h$kI=8=$O<!$N$h$&$K$J$j$^$9(B.

$$ [ [f_{\alpha},[\alpha_{1},\ldots,\alpha_{n}]],\ldots ] $$

$B$3$l$O(B

$$ \sum_{\alpha}f_{\alpha}\partial^{\alpha} $$

Line 12 $$ \sum_{\alpha}f_{\alpha}\partial^{\alpha} $$

Line 17 $$ \sum_{\alpha}f_{\alpha}\partial^{\alpha} $$

$$ (\sum_{\alpha^{(i)}}f_{\alpha^{(i)}}\partial^{\alpha^{(i)}})\bullet u = 0 \quad (i = 1,\ldots,s) $$

$B$J$I$N$h$&$KJ#?t$NHyJ,:nMQAG$rI=8=$9$k$H$-$OHyJ,:nMQAG$N%j%9%H$r;H$$$^$9(B.

$$ [ [ [f_{\alpha^{(1)}},[\alpha_{1}^{(1)},\ldots,\alpha_{n}^{(1)}]],\ldots ],\ldots,[ [f_{\alpha^{(s)}},[\alpha_{1}^{(s)},\ldots,\alpha_{n}^{(s)}]],\ldots ] ] $$

$BNc$($PHyJ,:nMQAG(B$x dx + y dy + 1$$B$N>l9g$O(B

$BNc$($PHyJ,:nMQAG(B$x \partial_{x} + y \partial_{y} + 1$$B$N>l9g$O(B

$$ [ [x,[1,0]],[y,[0,1]],[1,[0,0]] ] $$

$B$H$J$j$^$9(B. $B$^$?HyJ,:nMQAG$N%j%9%H$G(B$x dx + y dy + 1, dx^2 + dy^2$$B$rI=$9$H(B

$B$H$J$j$^$9(B. $B$^$?HyJ,:nMQAG$N%j%9%H$G(B$x \partial_{x} + y \partial_{y} + 1, {\partial_{x}}^{2} + {\partial_{y}}^{2}$$B$rI=$9$H(B

$$ [ [ [x,[1,0]],[y,[0,1]],[1,[0,0]] ],[ [1,[2,0]],[1,[0,2]] ] ] $$

$B$H$J$j$^$9(B. $B$^$?$3$l$i$NI=8=K!$r;H$&$H$-$OJQ?t%j%9%H$r>o$K0U<1$7$F$$$kI,MW$,$"$j$^$9!#(B

$B$H$J$j$^$9(B. $B$^$?$3$NI=8=K!$r;H$&$H$-$OJQ?t%j%9%H$r>o$K0U<1$7$F$$$kI,MW$,$"$j$^$9(B.

$B<!$KB?9`<0$K$h$kI=8=$K$D$$$F=R$Y$^$9(B. $BJQ?t(B$x$$B$KBP$9$kHyJ,$O(B$dx$$B$GI=8=$5$l$^$9(B.

$BNc$($P(B$x \partial_{x} + y \partial_{y} + 1$$B$K$D$$$F$O(B

$$ x*dx+y*dy+1 $$

$B$HI=8=$5$l$^$9(B.

@end tex

@menu

@c * diff_op_hg1::

@c * odiff_op_hg1::

@c * diff_op_appell1::

@c * odiff_op_appell1::

@c * diff_op_appell2::

@c * odiff_op_appell2::

@c * diff_op_appell3::

@c * odiff_op_appell3::

* diff_op_appell4::

* odiff_op_appell4::

@c * diff_op_selberg2::

@c * odiff_op_selberg2::

@c * diff_op_gkz::

@c * odiff_op_gkz::

* diff_act::

* odiff_op_tosm1::

@c * diff_act_hg1::

* odiff_op_toasir::

@c * diff_act_appell1::

* odiff_op_fromasir::

@c * diff_act_appell2::

* odiff_act::

@c * diff_act_appell3::

@c * odiff_act_hg1::

* diff_act_appell4::

@c * odiff_act_appell1::

@c * diff_act_selberg2::

@c * odiff_act_appell2::

@c * diff_act_gkz::

@c * odiff_act_appell3::

* diff_poly_solve::

* odiff_act_appell4::

* diff_poly_solve_hg1::

@c * odiff_act_selberg2::

@c * diff_poly_solve_appell1::

@c * odiff_act_gkz::

@c * diff_poly_solve_appell2::

* odiff_poly_solve::

@c * diff_poly_solve_appell3::

* odiff_poly_solve_hg1::

* diff_poly_solve_appell4::

@c * odiff_poly_solve_appell1::

@c * diff_poly_solve_selberg2::

@c * odiff_poly_solve_appell2::

@c * diff_poly_solve_gkz::

@c * odiff_poly_solve_appell3::

* diff_rat_solve::

* odiff_poly_solve_appell4::

@c * diff_pseries_appell4::

@c * odiff_poly_solve_selberg2::

@c * odiff_poly_solve_gkz::

* odiff_rat_solve::

@c * odiff_pseries_appell4::

@end menu

@node diff_op_appell4,,, Differential equations

@node odiff_op_appell4,,, Differential equations (library by Okutani)

@subsection @code{diff_op_appell4}

@subsection @code{odiff_op_appell4}

@findex diff_op_appell4

@findex odiff_op_appell4

@table @t

@item diff_op_appell4(@var{a},@var{b},@var{c1},@var{c2},@var{V})

@item odiff_op_appell4(@var{a},@var{b},@var{c1},@var{c2},@var{V})

:: appell $B$N(B F_4 $B$rNm2=$9$kHyJ,:nMQAG$r@8@.$7$^$9(B.

@end table

@table @var

Line 62 $$ [ [ [x,[1,0]],[y,[0,1]],[1,[0,0]] ],[ [1,[2,0]],[1,

Line 74 $$ [ [ [x,[1,0]],[y,[0,1]],[1,[0,0]] ],[ [1,[2,0]],[1,

$B%j%9%H(B

@end table

@itemize @bullet

@item @code{diff_op_appell4}$B$NNc(B.

@item @code{odiff_op_appell4}$B$NNc(B.

@end itemize

@example

[298] diff_op_appell4(a,b,c1,c2,[x,y]);

[298] odiff_op_appell4(a,b,c1,c2,[x,y]);

[ [ [-x^2+x,[2,0]], [-2*y*x,[1,1]], [-y^2,[0,2]],

[(-a-b-1)*x+c1,[1,0]], [(-a-b-1)*y,[0,1]], [-b*a,[0,0]] ],

[ [-y^2+y,[0,2]], [-2*y*x,[1,1]], [-x^2,[2,0]],

[(-a-b-1)*y+c2,[0,1]], [(-a-b-1)*x,[1,0]], [-b*a,[0,0]] ] ]

@end example

@node diff_act,,, Differential equations

@node odiff_op_tosm1,,, Differential equations (library by Okutani)

@subsection @code{diff_act}

@subsection @code{odiff_op_tosm1}

@findex diff_act

@findex odiff_op_tosm1

@table @t

@item diff_act(@var{L},@var{F},@var{V})

@item odiff_op_tosm1(@var{LL},@var{V})

:: $B%j%9%H7A<0$NHyJ,:nMQAG%j%9%H$r(B sm1 $B7A<0$KJQ49$7$^$9(B.

@end table

@table @var

@item return

$B%j%9%H(B

@item LL

$B%j%9%H(B

@item V

$B%j%9%H(B

@end table

@itemize @bullet

@item $BHyJ,:nMQAG$N78?t$O@0?tB?9`<0$KJQ49$5$l$^$9(B.

@item @code{odiff_op_tosm1}$B$NNc(B

@end itemize

@example

[299] odiff_op_tosm1([[[x,[2,0]],[-1,[0,0]]],

[[y,[0,2]],[-1,[0,0]]]],[x,y]);

[ + ( + (1) x) dx^2 + ( + (-1)), + ( + (1) y) dy^2 + ( + (-1))]

[300] odiff_op_tosm1([[[x,[1,0]],[y,[0,1]],[1,[0,0]]],

[[1,[2,0]],[1,[0,2]]]],[x,y]);

[ + ( + (1) x) dx + ( + (1) y) dy + ( + (1)), + ( + (1)) dx^2 + ( + (1)) dy^2]

[301] odiff_op_tosm1([[[1/2,[1,0]],[1,[0,0]]],

[[1/3,[0,1]],[1/4,[0,0]]]],[x,y]);

[ + ( + (6)) dx + ( + (12)), + ( + (4)) dy + ( + (3))]

[302] odiff_op_tosm1([[[1/2*x,[1,0]],[1,[0,0]]],

[[1/3*y,[0,1]],[1/4,[0,0]]]],[x,y]);

[ + ( + (6) x) dx + ( + (12)), + ( + (4) y) dy + ( + (3))]

@end example

@node odiff_op_toasir,,, Differential equations (library by Okutani)

@subsection @code{odiff_op_toasir}

@findex odiff_op_toasir

@table @t

@item odiff_op_toasir(@var{LL},@var{V})

:: $B%j%9%H7A<0$NHyJ,:nMQAG%j%9%H(B @var{LL} $B$r(B @code{asir} $B$NB?9`<0$KJQ49$7$^$9(B.

@end table

@table @var

@item return

$B%j%9%H(B

@item LL

$B%j%9%H(B

@item V

$B%j%9%H(B

@end table

@itemize @bullet

@item @code{odiff_op_toasir}$B$NNc(B

@end itemize

@example

[303] odiff_op_toasir([[[1/2*x,[1,0]],[1,[0,0]]],

[[1/3*y,[0,1]],[1/4,[0,0]]]],[x,y]);

[1/2*x*dx+1,1/3*y*dy+1/4]

[304] odiff_op_toasir([[[x,[1,0]],[y,[0,1]],[1,[0,0]]],

[[1,[2,0]],[1,[0,2]]]],[x,y]);

[x*dx+y*dy+1,dx^2+dy^2]

@end example

@node odiff_op_fromasir,,, Differential equations (library by Okutani)

@subsection @code{odiff_op_fromasir}

@findex odiff_op_fromasir

@table @t

@item odiff_op_fromasir(@var{D_list},@var{V})

:: @code{asir} $B$NB?9`<0$+$i%j%9%H7A<0$NHyJ,:nMQAG%j%9%H$KJQ49$7$^$9(B.

@end table

@table @var

@item return

$B%j%9%H(B

@item D_list

$B%j%9%H(B

@item V

$B%j%9%H(B

@end table

@itemize @bullet

@item @code{odiff_op_fromasir}$B$NNc(B

@end itemize

@example

[305] odiff_op_fromasir([1/2*x*dx+1,1/3*y*dy+1/4],[x,y]);

[[[1/2*x,[1,0]],[1,[0,0]]],[[1/3*y,[0,1]],[1/4,[0,0]]]]

[306] odiff_op_fromasir([x*dx+y*dy+1,dx^2+dy^2],[x,y]);

[[[x,[1,0]],[y,[0,1]],[1,[0,0]]],[[1,[2,0]],[1,[0,2]]]]

@end example

@node odiff_act,,, Differential equations (library by Okutani)

@subsection @code{odiff_act}

@findex odiff_act

@table @t

@item odiff_act(@var{L},@var{F},@var{V})

:: $BHyJ,:nMQAG(B @var{L} $B$rM-M}<0(B @var{F} $B$K:nMQ$5$;$k(B. @var{V} $B$OJQ?t%j%9%H(B.

@end table

@table @var

@item return

$BM-M}<0(B

@item L

$B%j%9%H(B

$B%j%9%H(B or $BB?9`<0(B

@item F

$BM-M}<0(B

@item V

$B%j%9%H(B

@end table

@itemize @bullet

@item @code{diff_act}$B$NNc(B

@item @code{odiff_act}$B$NNc(B

@end itemize

@example

[302] diff_act([[1,[2]]],x^3+x^2+x+1,[x]);

[302] odiff_act([[1,[2]]],x^3+x^2+x+1,[x]);

6*x+2

[303] diff_act([[1,[1,0]],[1,[0,1]]],x^2+y^2,[x,y]);

[303] odiff_act([[1,[1,0]],[1,[0,1]]],x^2+y^2,[x,y]);

2*x+2*y

[349] odiff_act(x*dx+y*dy, x^2+x*y+y^2, [x,y]);

2*x^2+2*y*x+2*y^2

@end example

@node diff_act_appell4,,, Differential equations

@node odiff_act_appell4,,, Differential equations (library by Okutani)

@subsection @code{diff_act_appell4}

@subsection @code{odiff_act_appell4}

@findex diff_act_appell4

@findex odiff_act_appell4

@table @t

@item diff_act_appell4(@var{a},@var{b},@var{c1},@var{c2},@var{F},@var{V})

@item odiff_act_appell4(@var{a},@var{b},@var{c1},@var{c2},@var{F},@var{V})

:: $BHyJ,:nMQAG(B @code{diff_op_appell4} $B$rM-M}<0(B @var{F} $B$K:nMQ$5$;$k(B.

:: $BHyJ,:nMQAG(B @code{odiff_op_appell4} $B$rM-M}<0(B @var{F} $B$K:nMQ$5$;$k(B.

@end table

@table @var

@item return

Line 118 $$ [ [ [x,[1,0]],[y,[0,1]],[1,[0,0]] ],[ [1,[2,0]],[1,

Line 224 $$ [ [ [x,[1,0]],[y,[0,1]],[1,[0,0]] ],[ [1,[2,0]],[1,

$B%j%9%H(B

@end table

@itemize @bullet

@item @code{diff_act_appell4}$B$NNc(B

@item @code{odiff_act_appell4}$B$NNc(B

@end itemize

@example

[303] diff_act_appell4(1,0,1,1,x^2+y^2,[x,y]);

[303] odiff_act_appell4(1,0,1,1,x^2+y^2,[x,y]);

[-6*x^2+4*x-6*y^2,-6*x^2-6*y^2+4*y]

[304] diff_act_appell4(0,0,1,1,x^2+y^2,[x,y]);

[304] odiff_act_appell4(0,0,1,1,x^2+y^2,[x,y]);

[-4*x^2+4*x-4*y^2,-4*x^2-4*y^2+4*y]

[305] diff_act_appell4(-2,-2,-1,-1,x^2+y^2,[x,y]);

[305] odiff_act_appell4(-2,-2,-1,-1,x^2+y^2,[x,y]);

[0,0]

@end example

@node diff_poly_solve,,, Differential equations

@node odiff_poly_solve,,, Differential equations (library by Okutani)

@subsection @code{diff_poly_solve}

@subsection @code{odiff_poly_solve}

@findex diff_poly_solve

@findex odiff_poly_solve

@table @t

@item diff_poly_solve(@var{LL},@var{N},@var{V})

@item odiff_poly_solve(@var{LL},@var{N},@var{V})

:: $BM?$($i$l$?@~7?HyJ,J}Dx<07O$N(B @var{N} $B<!0J2<$NB?9`<02r$r5a$a$k(B.

@end table

@table @var

Line 149 $$ [ [ [x,[1,0]],[y,[0,1]],[1,[0,0]] ],[ [1,[2,0]],[1,

Line 255 $$ [ [ [x,[1,0]],[y,[0,1]],[1,[0,0]] ],[ [1,[2,0]],[1,

$B%j%9%H(B

@end table

@itemize @bullet

@item @code{diff_poly_solve}$B$NNc(B.

@item @code{odiff_poly_solve}$B$NNc(B.

@end itemize

@example

[297] diff_poly_solve([[[x,[1,0]],[-1,[0,0]]],[[y,[0,1]],[-1,[0,0]]]],5,[x,y]);

[297] odiff_poly_solve([[[x,[1,0]],[-1,[0,0]]],[[y,[0,1]],[-1,[0,0]]]],5,[x,y]);

[_4*y*x,[_4]]

[298] diff_poly_solve([[[x,[1,0]],[-2,[0,0]]],[[y,[0,1]],[-2,[0,0]]]],5,[x,y]);

[298] odiff_poly_solve([[[x,[1,0]],[-2,[0,0]]],[[y,[0,1]],[-2,[0,0]]]],5,[x,y]);

[_33*y^2*x^2,[_33]]

[356] odiff_poly_solve([x*dx+y*dy-3,dx+dy],4,[x,y]);

[-_126*x^3+3*_126*y*x^2-3*_126*y^2*x+_126*y^3,[_126]]

@end example

@node diff_poly_solve_hg1,,, Differential equations

@node odiff_poly_solve_hg1,,, Differential equations (library by Okutani)

@subsection @code{diff_poly_solve_hg1}

@subsection @code{odiff_poly_solve_hg1}

@findex diff_poly_solve_hg1

@findex odiff_poly_solve_hg1

@table @t

@item diff_poly_solve_hg1(@var{a},@var{b},@var{c},@var{V})

@item odiff_poly_solve_hg1(@var{a},@var{b},@var{c},@var{V})

:: $B%,%&%9$ND64v2?HyJ,J}Dx<0$NB?9`<02r$r5a$a$k(B.

@end table

@table @var

Line 175 $$ [ [ [x,[1,0]],[y,[0,1]],[1,[0,0]] ],[ [1,[2,0]],[1,

Line 284 $$ [ [ [x,[1,0]],[y,[0,1]],[1,[0,0]] ],[ [1,[2,0]],[1,

$B%j%9%H(B

@end table

@itemize @bullet

@item @code{diff_poly_solve_hg1}$B$NNc(B.

@item @code{odiff_poly_solve_hg1}$B$NNc(B.

@end itemize

@example

[334] diff_poly_solve_hg1(-3,-6,-5,[x]);

[334] odiff_poly_solve_hg1(-3,-6,-5,[x]);

[_1*x^6-2*_0*x^3+9/2*_0*x^2-18/5*_0*x+_0,[_0,_1]]

[335] diff_poly_solve_hg1(-3,-6,-7,[x]);

[335] odiff_poly_solve_hg1(-3,-6,-7,[x]);

[-4/7*_2*x^3+15/7*_2*x^2-18/7*_2*x+_2,[_2]]

@end example

@node diff_poly_solve_appell4,,, Differential equations

@node odiff_poly_solve_appell4,,, Differential equations (library by Okutani)

@subsection @code{diff_poly_solve_appell4}

@subsection @code{odiff_poly_solve_appell4}

@findex diff_poly_solve_appell4

@findex odiff_poly_solve_appell4

@table @t

@item diff_poly_solve_appell4(@var{a},@var{b},@var{c1},@var{c2},@var{V})

@item odiff_poly_solve_appell4(@var{a},@var{b},@var{c1},@var{c2},@var{V})

:: F_4$B$,$_$?$9@~7?HyJ,J}Dx<07O$NB?9`<02r$r5a$a$k(B.

@end table

@table @var

Line 201 $$ [ [ [x,[1,0]],[y,[0,1]],[1,[0,0]] ],[ [1,[2,0]],[1,

Line 310 $$ [ [ [x,[1,0]],[y,[0,1]],[1,[0,0]] ],[ [1,[2,0]],[1,

$B%j%9%H(B

@end table

@itemize @bullet

@item @code{diff_poly_solve_appell4}$B$NNc(B.

@item @code{odiff_poly_solve_appell4}$B$NNc(B.

@end itemize

@example

[299] diff_poly_solve_appell4(-3,1,-1,-1,[x,y]);

[299] odiff_poly_solve_appell4(-3,1,-1,-1,[x,y]);

[-_26*x^3+(3*_26*y+_26)*x^2+3*_24*y^2*x-_24*y^3+_24*y^2,[_24,_26]]

[300] diff_poly_solve_appell4(-3,1,1,-1,[x,y]);

[300] odiff_poly_solve_appell4(-3,1,1,-1,[x,y]);

[-3*_45*y^2*x-_45*y^3+_45*y^2,[_45]]

@end example

@node diff_rat_solve,,, Differential equations

@node odiff_rat_solve,,, Differential equations (library by Okutani)

@subsection @code{diff_rat_solve}

@subsection @code{odiff_rat_solve}

@findex diff_rat_solve

@findex odiff_rat_solve

@table @t

@item diff_rat_solve(@var{LL},@var{Dn},@var{N},@var{V})

@item odiff_rat_solve(@var{LL},@var{Dn},@var{N},@var{V})

:: $BM?$($i$l$?@~7?HyJ,J}Dx<07O$NJ,Jl$,(B @var{Dn}, $BJ,;R$,(B @var{N} $B<!0J2<$NB?9`<0$G$"$k$h$&$J2r$r5a$a$k(B.

@end table

@table @var

Line 231 $$ [ [ [x,[1,0]],[y,[0,1]],[1,[0,0]] ],[ [1,[2,0]],[1,

Line 340 $$ [ [ [x,[1,0]],[y,[0,1]],[1,[0,0]] ],[ [1,[2,0]],[1,

$B%j%9%H(B

@end table

@itemize @bullet

@item @code{diff_rat_solve}$B$NNc(B.

@item @code{odiff_rat_solve}$B$NNc(B.

@end itemize

@example

[333] diff_rat_solve([[[x,[1]],[1,[0]]]],x,1,[x]);

[333] odiff_rat_solve([[[x,[1]],[1,[0]]]],x,1,[x]);

[(_8)/(x),[_8]]

[350] D = diff_op_appell4(0,0,3,0,[x,y])$

[361] odiff_rat_solve([x*(1-x)*dx^2+(1-3*x)*dx-1],1-x,2,[x]);

[351] diff_rat_solve(D,x^2,2,[x,y]);

[(_180)/(-x+1),[_180]]

[350] D = odiff_op_appell4(0,0,3,0,[x,y])$

[351] odiff_rat_solve(D,x^2,2,[x,y]);

[(_118*x^2-_114*y*x+1/2*_114*y^2+_114*y)/(x^2),[_114,_118]]

@end example

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