Annotation of OpenXM/src/asir-contrib/packages/doc/dsolv.oxweave, Revision 1.1
1.1 ! takayama 1: /* $OpenXM$ */
! 2: /* dsolv.oxweave */
! 3: /*&eg-texi
! 4: @node DSOLV Functions,,, Top
! 5:
! 6: @chapter DSOLV Functions
! 7:
! 8: This section is a collection of functions to solve regular holonomic
! 9: systems in terms of series.
! 10: Algorithms are explained in the book [SST].
! 11: You can load this package by the command
! 12: @code{load("dsolv");}
! 13: This package requires @code{Diff} and @code{dmodule}.
! 14:
! 15: This package uses @code{ox_sm1}, so the variables you can use
! 16: is as same as those you can use in the package @code{sm1}.
! 17:
! 18: @section Functions
! 19:
! 20: */
! 21:
! 22: /*&jp-texi
! 23: @node DSOLV $BH!?t(B,,, Top
! 24:
! 25: @chapter DSOLV $BH!?t(B
! 26:
! 27: $B$3$N@a$O@5B'%[%m%N%_%C%/7O$r5i?t$G2r$/$?$a$N(B
! 28: $BH!?t$r$"$D$a$F$"$k(B.
! 29: $B%"%k%4%j%:%`$K$D$$$F$O(B [SST] $B$K@bL@$,$"$k(B.
! 30: $B$3$N%Q%C%1!<%8$O<!$N%3%^%s%I(B @code{load("dsolv");}
! 31: $B$G%m!<%I$G$-$k(B.
! 32: $B$3$N%Q%C%1!<%8$O(B @code{Diff} $B$*$h$S(B @code{dmodule} $B$r;HMQ$9$k(B.
! 33:
! 34: $B$3$N%Q%C%1!<%8$O(B @code{ox_sm1} $B$rMxMQ$7$F$$$k(B.
! 35: $B$7$?$,$C$F;HMQ$G$-$kJQ?t$O(B @code{sm1} $B%Q%C%1!<%8$HF1MM$NJQ?t$7$+$D$+$($J$$(B.
! 36:
! 37: @section Functions
! 38:
! 39: */
! 40:
! 41: /*&eg-texi
! 42: @menu
! 43: * dsolv_dual::
! 44: @end menu
! 45: @node dsolv_dual,,, DSOLV Functions
! 46: @subsection @code{dsolv_dual}
! 47: @findex dsolv_dual
! 48: @table @t
! 49: @item dsolv_dual(@var{f},@var{v})
! 50: :: Grobner dual of @var{f}.
! 51: @end table
! 52:
! 53: @table @var
! 54: @item return
! 55: List
! 56: @item f, v
! 57: List
! 58: @end table
! 59:
! 60: @itemize @bullet
! 61: @item It returns the Grobner dual of @var{f} in the ring of polynomials
! 62: with variables @var{v}.
! 63: @item The ideal generated by @var{f} must be primary to the maximal ideal
! 64: generated by @var{v}.
! 65: If it is not primary to the maximal ideal, then this function falls into
! 66: an infinite loop.
! 67: @item This is an implementation of Algorithm 2.3.14 of the book [SST].
! 68: If we replace variables x, y, ... in the output by log(x), log(y), ...,
! 69: then these polynomials in log are solutions of the system of differential
! 70: equations @code{map(subst,@var{f},x,x*dx, y,y*dy, ...)}.
! 71: @end itemize
! 72: */
! 73:
! 74: /*&jp-texi
! 75: @menu
! 76: * dsolv_dual::
! 77: @end menu
! 78: @node dsolv_dual,,, DSOLV $BH!?t(B
! 79: @subsection @code{dsolv_dual}
! 80: @findex dsolv_dual
! 81: @table @t
! 82: @item dsolv_dual(@var{f},@var{v})
! 83: :: @var{f} $B$N%0%l%V%JAPBP(B
! 84: @end table
! 85:
! 86: @table @var
! 87: @item $BLa$jCM(B
! 88: $B%j%9%H(B
! 89: @item f, v
! 90: $B%j%9%H(B
! 91: @end table
! 92:
! 93: @itemize @bullet
! 94: @item $BJQ?t(B @var{v} $B>e$NB?9`<04D$K$*$$$F(B,
! 95: @var{f} $B$N%0%l%V%JAPBP$r5a$a$k(B.
! 96: @item @var{f} $B$G@8@.$5$l$k%$%G%"%k$O(B, @var{v} $B$G@8@.$5$l$k6KBg%$%G%"%k$K(B
! 97: $BBP$7$F(B, primary $B$G$J$$$H$$$1$J$$(B.
! 98: primary $B$G$J$$>l9g(B, $B$3$NH!?t$OL58B%k!<%W$K$*$A$$$k(B.
! 99: @item $B$3$NH!?t$OK\(B [SST] $B$N(B Algorithm 2.3.14 $B$N<BAu$G$"$k(B.
! 100: $B=PNOCf$NJQ?t(B x, y, ... $B$r$=$l$>$l(B log(x), log(y), ..., $B$G$*$-$+$($k$H(B,
! 101: $B$3$l$i$N(B log $BB?9`<0$O(B,
! 102: @code{map(subst,@var{f},x,x*dx, y,y*dy, ...)} $B$G@8@.$5$l$kHyJ,J}Dx<07O(B
! 103: $B$N2r$H$J$C$F$$$k(B.
! 104: @end itemize
! 105: */
! 106:
! 107: /*&C-texi
! 108:
! 109: @example
! 110:
! 111:
! 112: [435] dsolv_dual([y-x^2,y+x^2],[x,y]);
! 113: [x,1]
! 114: [436] dsolv_act(y*dy-sm1_mul(x*dx,x*dx,[x,y]),log(x),[x,y]);
! 115: 0
! 116: [437] dsolv_act(y*dy+sm1_mul(x*dx,x*dx,[x,y]),log(x),[x,y]);
! 117: 0
! 118:
! 119: [439] primadec([y^2-x^3,x^2*y^2],[x,y]);
! 120: [[[y^2-x^3,y^4,x^2*y^2],[y,x]]]
! 121: [440] dsolv_dual([y^2-x^3,x^2*y^2],[x,y]);
! 122: [x*y^3+1/4*x^4*y, x^2*y, x*y^2+1/12*x^4, y^3+x^3*y,
! 123: x^2, x*y, y^2+1/3*x^3, x, y, 1]
! 124:
! 125: [441] dsolv_test_dual();
! 126: Output is omitted.
! 127:
! 128: @end example
! 129:
! 130: */
! 131:
! 132:
! 133: /*&eg-texi
! 134:
! 135: @menu
! 136: * dsolv_starting_term::
! 137: @end menu
! 138: @node dsolv_starting_term,,, DSOLV Functions
! 139: @subsection @code{dsolv_starting_term}
! 140: @findex dsolv_starting_term
! 141: @table @t
! 142: @item dsolv_starting_term(@var{f},@var{v},@var{w})
! 143: :: Find the starting term of the solutions of
! 144: the regular holonomic system @var{f}
! 145: to the direction @var{w}.
! 146: @end table
! 147:
! 148: @table @var
! 149: @item return
! 150: List
! 151: @item f, v, w
! 152: List
! 153: @end table
! 154:
! 155: @itemize @bullet
! 156: @item Find the starting term of the solutions of
! 157: the regular holonomic system @var{f}
! 158: to the direction @var{w}.
! 159: @item The return value is of the form
! 160: [[@var{e1}, @var{e2}, ...],
! 161: [@var{s1}, @var{s2}, ...]]
! 162: where @var{e1} is an exponent vector and @var{s1} is the corresponding
! 163: solution set, and so on.
! 164: @item If you set @code{Dsolv_message_starting_term} to 1,
! 165: then this function outputs messages during the computation.
! 166: @end itemize
! 167:
! 168: */
! 169:
! 170: /*&jp-texi
! 171:
! 172: @menu
! 173: * dsolv_starting_term::
! 174: @end menu
! 175: @node dsolv_starting_term,,, DSOLV $BH!?t(B
! 176: @subsection @code{dsolv_starting_term}
! 177: @findex dsolv_starting_term
! 178: @table @t
! 179: @item dsolv_starting_term(@var{f},@var{v},@var{w})
! 180: :: $B@5B'%[%m%N%_%C%/7O(B @var{f} $B$NJ}8~(B @var{w} $B$G$N5i?t2r$N(B
! 181: Staring terms $B$r7W;;$9$k(B. $B$3$3$G(B, @var{v} $B$OJQ?t$N=89g(B.
! 182: @end table
! 183:
! 184: @table @var
! 185: @item $BLa$jCM(B
! 186: $B%j%9%H(B
! 187: @item f, v, w
! 188: $B%j%9%H(B
! 189: @end table
! 190:
! 191: @itemize @bullet
! 192: @item $B@5B'%[%m%N%_%C%/7O(B @var{f} $B$NJ}8~(B @var{w} $B$G$N5i?t2r$N(B
! 193: Staring terms $B$r7W;;$9$k(B. $B$3$3$G(B, @var{v} $B$OJQ?t$N=89g(B.
! 194: @item $BLa$jCM$O<!$N7A$r$7$F$$$k(B:
! 195: [[@var{e1}, @var{e2}, ...],
! 196: [@var{s1}, @var{s2}, ...]]
! 197: $B$3$3$G(B @var{e1} $B$O(B exponent $B%Y%/%H%k$G$"$j(B @var{s1} $B$O$3$N%Y%/%H%k$K(B
! 198: $BBP1~$9$k2r$N=89g(B, $B0J2<F1MM(B.
! 199: @item $BJQ?t(B @code{Dsolv_message_starting_term} $B$r(B 1 $B$K$7$F$*$/$H(B,
! 200: $B$3$NH!?t$O7W;;$NESCf$K$$$m$$$m$H%a%C%;!<%8$r=PNO$9$k(B.
! 201: @end itemize
! 202:
! 203: */
! 204:
! 205: /*&C-texi
! 206:
! 207: @example
! 208: [1076] F = sm1_gkz( [ [[1,1,1,1,1],[1,1,0,-1,0],[0,1,1,-1,0]], [1,0,0]]);
! 209: [[x5*dx5+x4*dx4+x3*dx3+x2*dx2+x1*dx1-1,-x4*dx4+x2*dx2+x1*dx1,
! 210: -x4*dx4+x3*dx3+x2*dx2,
! 211: -dx2*dx5+dx1*dx3,dx5^2-dx2*dx4],[x1,x2,x3,x4,x5]]
! 212: [1077] A= dsolv_starting_term(F[0],F[1],[1,1,1,1,0])$
! 213: Computing the initial ideal.
! 214: Done.
! 215: Computing a primary ideal decomposition.
! 216: Primary ideal decomposition of the initial Frobenius ideal
! 217: to the direction [1,1,1,1,0] is
! 218: [[[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,
! 219: x5^2-2*x5-8*x4^2+1,x5^3-3*x5^2+3*x5-1],
! 220: [x5-1,x4,x3,x2,x1]]]
! 221:
! 222: ----------- root is [ 0 0 0 0 1 ]
! 223: ----------- dual system is
! 224: [x5^2+(-3/4*x4-1/2*x3-1/4*x2-1/2*x1)*x5+1/8*x4^2
! 225: +(1/4*x3+1/4*x1)*x4+1/4*x2*x3-1/8*x2^2+1/4*x1*x2,
! 226: x4-2*x3+3*x2-2*x1,x5-x3+x2-x1,1]
! 227:
! 228: [1078] A[0];
! 229: [[ 0 0 0 0 1 ]]
! 230: [1079] map(fctr,A[1][0]);
! 231: [[[1/8,1],[x5,1],[log(x2)+log(x4)-2*log(x5),1],
! 232: [2*log(x1)-log(x2)+2*log(x3)+log(x4)-4*log(x5),1]],
! 233: [[1,1],[x5,1],[-2*log(x1)+3*log(x2)-2*log(x3)+log(x4),1]],
! 234: [[1,1],[x5,1],[-log(x1)+log(x2)-log(x3)+log(x5),1]],
! 235: [[1,1],[x5,1]]]
! 236:
! 237: @end example
! 238:
! 239: */
! 240:
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