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Annotation of OpenXM/src/asir-contrib/packages/doc/dsolv/dsolv.oxw, Revision 1.2

1.2     ! takayama    1: /* $OpenXM: OpenXM/src/asir-contrib/packages/doc/dsolv/dsolv.oxw,v 1.1 2005/04/11 11:13:32 takayama Exp $ */
1.1       takayama    2: /* dsolv.oxweave */
                      3: /*&C
                      4: @node DSOLV Functions,,, Top
                      5: */
                      6:
                      7: /*&en
                      8:
                      9: @chapter DSOLV Functions
                     10:
                     11: This section is a collection of functions to solve regular holonomic
                     12: systems in terms of series.
                     13: Algorithms are explained in the book [SST].
                     14: You can load this package by the command
                     15: @code{load("dsolv.rr")$}
                     16: This package requires @code{Diff} and @code{dmodule}.
                     17:
                     18: To use the functions of the package @code{dsolv} in OpenXM/Risa/Asir,
                     19: executing the command @code{load("dsolv.rr")$}
                     20: is necessary at first.
                     21:
                     22:
                     23: This package uses @code{ox_sm1}, so the variables you can use
                     24: is as same as those you can use in the package @code{sm1}.
                     25:
                     26: @section Functions
                     27:
                     28: */
                     29:
                     30: /*&ja
                     31:
                     32: @chapter DSOLV $BH!?t(B
                     33:
                     34: $B$3$N@a$O@5B'%[%m%N%_%C%/7O$r5i?t$G2r$/$?$a$N(B
                     35: $BH!?t$r$"$D$a$F$"$k(B.
                     36: $B%"%k%4%j%:%`$K$D$$$F$O(B [SST] $B$K@bL@$,$"$k(B.
                     37: $B$3$N%Q%C%1!<%8$O<!$N%3%^%s%I(B @code{load("dsolv.rr");}
                     38: $B$G%m!<%I$G$-$k(B.
                     39: $B$3$N%Q%C%1!<%8$O(B @code{Diff} $B$*$h$S(B @code{Dmodule} $B$r;HMQ$9$k(B.
                     40:
                     41: OpenXM/Risa/Asir $B$G$NMxMQ$K$"$?$C$F$O(B,
                     42: @example
                     43: load("dsolv.rr");$
                     44: @end example
                     45: $B$,;O$a$KI,MW(B.
                     46:
                     47: $B$3$N%Q%C%1!<%8$O(B @code{ox_sm1} $B$rMxMQ$7$F$$$k(B.
                     48: $B$7$?$,$C$F;HMQ$G$-$kJQ?t$O(B @code{sm1} $B%Q%C%1!<%8$HF1MM$NJQ?t$7$+$D$+$($J$$(B.
                     49:
                     50: @section $BH!?t0lMw(B
                     51:
                     52: */
                     53:
                     54: /*&C
                     55: @menu
                     56: * dsolv_dual::
                     57: * dsolv_starting_term::
                     58: @end menu
                     59: */
                     60:
                     61:
                     62: /*&en
                     63: @node dsolv_dual,,, DSOLV Functions
                     64: @subsection @code{dsolv_dual}
                     65: @findex dsolv_dual
                     66: @table @t
                     67: @item dsolv_dual(@var{f},@var{v})
                     68: ::  Grobner dual of @var{f}.
                     69: @end table
                     70:
                     71: @table @var
                     72: @item return
                     73: List
                     74: @item f, v
                     75: List
                     76: @end table
                     77:
                     78: @itemize @bullet
                     79: @item  It returns the Grobner dual of @var{f} in the ring of polynomials
                     80: with variables @var{v}.
                     81: @item The ideal generated by @var{f} must be primary to the maximal ideal
                     82: generated by @var{v}.
                     83: If it is not primary to the maximal ideal, then this function falls into
                     84: an infinite loop.
                     85: @end itemize
                     86:
                     87:
                     88: @noindent
                     89: Algorithm:
                     90: This is an implementation of Algorithm 2.3.14 of the book [SST].
                     91: If we replace variables x, y, ... in the output by log(x), log(y), ...,
                     92: then these polynomials in log are solutions of the system of differential
                     93: equations @var{f}@code{_(x->x*dx, y->y*dy, ...)}.
                     94:
                     95: */
                     96:
                     97: /*&ja
                     98: @node dsolv_dual,,, DSOLV Functions
                     99: @subsection @code{dsolv_dual}
                    100: @findex dsolv_dual
                    101: @table @t
                    102: @item dsolv_dual(@var{f},@var{v})
                    103: ::  @var{f} $B$N%0%l%V%JAPBP(B
                    104: @end table
                    105:
                    106: @table @var
                    107: @item $BLa$jCM(B
                    108: $B%j%9%H(B
                    109: @item f, v
                    110: $B%j%9%H(B
                    111: @end table
                    112:
                    113: @itemize @bullet
                    114: @item  $BJQ?t(B @var{v} $B>e$NB?9`<04D$K$*$$$F(B,
                    115: @var{f}  $B$N%0%l%V%JAPBP$r5a$a$k(B.
                    116: @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
                    117: $BBP$7$F(B, primary $B$G$J$$$H$$$1$J$$(B.
                    118: primary $B$G$J$$>l9g(B, $B$3$NH!?t$OL58B%k!<%W$K$*$A$$$k(B.
                    119: @end itemize
                    120:
                    121: @noindent
                    122: Algorithm:
                    123: $B$3$NH!?t$OK\(B [SST] $B$N(B Algorithm 2.3.14  $B$N<BAu$G$"$k(B.
                    124: $B=PNOCf$NJQ?t(B x, y, ... $B$r$=$l$>$l(B log(x), log(y), ..., $B$G$*$-$+$($k$H(B,
                    125: $B$3$l$i$N(B log $BB?9`<0$O(B,
                    126: @var{f}@code{_(x->x*dx, y->y*dy, ...)}
                    127: $B$G@8@.$5$l$kHyJ,J}Dx<07O(B
                    128: $B$N2r$H$J$C$F$$$k(B.
                    129: */
                    130:
                    131: /*&C
                    132:
                    133: @example
                    134:
                    135:
                    136: [435] dsolv_dual([y-x^2,y+x^2],[x,y]);
                    137: [x,1]
1.2     ! takayama  138: [436] dsolv_act(y*dy-sm1.mul(x*dx,x*dx,[x,y]),log(x),[x,y]);
1.1       takayama  139: 0
1.2     ! takayama  140: [437] dsolv_act(y*dy+sm1.mul(x*dx,x*dx,[x,y]),log(x),[x,y]);
1.1       takayama  141: 0
                    142:
                    143: [439] primadec([y^2-x^3,x^2*y^2],[x,y]);
                    144: [[[y^2-x^3,y^4,x^2*y^2],[y,x]]]
                    145: [440] dsolv_dual([y^2-x^3,x^2*y^2],[x,y]);
                    146: [x*y^3+1/4*x^4*y, x^2*y, x*y^2+1/12*x^4, y^3+x^3*y,
                    147:  x^2, x*y, y^2+1/3*x^3, x, y, 1]
                    148:
                    149: [441] dsolv_test_dual();
                    150:   Output is  omitted.
                    151:
                    152: @end example
                    153:
                    154: */
                    155:
                    156:
                    157: /*&en
                    158:
                    159: @node dsolv_starting_term,,, DSOLV Functions
                    160: @subsection @code{dsolv_starting_term}
                    161: @findex dsolv_starting_term
                    162: @table @t
                    163: @item dsolv_starting_term(@var{f},@var{v},@var{w})
                    164: ::  Find the starting term of the solutions of
                    165: the regular holonomic system @var{f}
                    166: to the direction @var{w}.
                    167: @end table
                    168:
                    169: @table @var
                    170: @item return
                    171: List
                    172: @item f, v, w
                    173: List
                    174: @end table
                    175:
                    176: @itemize @bullet
                    177: @item Find the starting term of the solutions of
                    178: the regular holonomic system @var{f}
                    179: to the direction @var{w}.
                    180: @item The return value is of the form
                    181: [[@var{e1}, @var{e2}, ...],
                    182:  [@var{s1}, @var{s2}, ...]]
                    183: where @var{e1} is an exponent vector and @var{s1} is the corresponding
                    184: solution set, and so on.
                    185: @item If you set @code{Dsolv_message_starting_term} to 1,
                    186: then this function outputs messages during the computation.
                    187: @end itemize
                    188:
                    189: */
                    190:
                    191: /*&ja
                    192:
                    193: @node dsolv_starting_term,,, DSOLV Functions
                    194: @subsection @code{dsolv_starting_term}
                    195: @findex dsolv_starting_term
                    196: @table @t
                    197: @item dsolv_starting_term(@var{f},@var{v},@var{w})
                    198: :: $B@5B'%[%m%N%_%C%/7O(B @var{f} $B$NJ}8~(B @var{w} $B$G$N5i?t2r$N(B
                    199: Staring terms $B$r7W;;$9$k(B. $B$3$3$G(B, @var{v} $B$OJQ?t$N=89g(B.
                    200: @end table
                    201:
                    202: @table @var
                    203: @item $BLa$jCM(B
                    204: $B%j%9%H(B
                    205: @item f, v, w
                    206: $B%j%9%H(B
                    207: @end table
                    208:
                    209: @itemize @bullet
                    210: @item $B@5B'%[%m%N%_%C%/7O(B @var{f} $B$NJ}8~(B @var{w} $B$G$N5i?t2r$N(B
                    211: Staring terms $B$r7W;;$9$k(B. $B$3$3$G(B, @var{v} $B$OJQ?t$N=89g(B.
                    212: @item $BLa$jCM$O<!$N7A$r$7$F$$$k(B:
                    213: [[@var{e1}, @var{e2}, ...],
                    214:  [@var{s1}, @var{s2}, ...]]
                    215: $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
                    216: $BBP1~$9$k2r$N=89g(B,   $B0J2<F1MM(B.
                    217: @item $BJQ?t(B @code{Dsolv_message_starting_term} $B$r(B 1 $B$K$7$F$*$/$H(B,
                    218: $B$3$NH!?t$O7W;;$NESCf$K$$$m$$$m$H%a%C%;!<%8$r=PNO$9$k(B.
                    219: @end itemize
                    220:
                    221:
                    222: */
                    223:
                    224: /*&C
                    225:
                    226: @noindent
                    227: Algorithm: Saito, Sturmfels, Takayama, Grobner Deformations of Hypergeometric
                    228: Differential Equations ([SST]), Chapter 2.
                    229:
                    230:
                    231: @example
1.2     ! takayama  232: [1076]   F = sm1.gkz( [ [[1,1,1,1,1],[1,1,0,-1,0],[0,1,1,-1,0]], [1,0,0]]);
1.1       takayama  233: [[x5*dx5+x4*dx4+x3*dx3+x2*dx2+x1*dx1-1,-x4*dx4+x2*dx2+x1*dx1,
                    234:   -x4*dx4+x3*dx3+x2*dx2,
                    235:   -dx2*dx5+dx1*dx3,dx5^2-dx2*dx4],[x1,x2,x3,x4,x5]]
                    236: [1077]  A= dsolv_starting_term(F[0],F[1],[1,1,1,1,0])$
                    237: Computing the initial ideal.
                    238: Done.
                    239: Computing a primary ideal decomposition.
                    240: Primary ideal decomposition of the initial Frobenius ideal
                    241: to the direction [1,1,1,1,0] is
                    242: [[[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,
                    243:    x5^2-2*x5-8*x4^2+1,x5^3-3*x5^2+3*x5-1],
                    244:  [x5-1,x4,x3,x2,x1]]]
                    245:
                    246: ----------- root is [ 0 0 0 0 1 ]
                    247: ----------- dual system is
                    248: [x5^2+(-3/4*x4-1/2*x3-1/4*x2-1/2*x1)*x5+1/8*x4^2
                    249:  +(1/4*x3+1/4*x1)*x4+1/4*x2*x3-1/8*x2^2+1/4*x1*x2,
                    250:  x4-2*x3+3*x2-2*x1,x5-x3+x2-x1,1]
                    251:
                    252: [1078] A[0];
                    253: [[ 0 0 0 0 1 ]]
                    254: [1079] map(fctr,A[1][0]);
                    255: [[[1/8,1],[x5,1],[log(x2)+log(x4)-2*log(x5),1],
                    256:           [2*log(x1)-log(x2)+2*log(x3)+log(x4)-4*log(x5),1]],
                    257:  [[1,1],[x5,1],[-2*log(x1)+3*log(x2)-2*log(x3)+log(x4),1]],
                    258:  [[1,1],[x5,1],[-log(x1)+log(x2)-log(x3)+log(x5),1]],
                    259:  [[1,1],[x5,1]]]
                    260:
                    261: @end example
                    262:
                    263: */
                    264:

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