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1.2     ! takayama    1: @c $OpenXM: OpenXM/src/asir-contrib/packages/doc/Diff-ja.texi,v 1.1 2002/07/29 02:56:26 takayama Exp $
1.1       takayama    2: @node Differential equations (library by Okutani),,, Top
                      3: @chapter Differential equations (library by Okutani)
1.2     ! takayama    4: $B%U%!%$%k(B @file{gr}, @file{Matrix}, @file{Diff} $B$,I,MW$G$9(B.
1.1       takayama    5:
                      6: Yukio Okutani $B;a$K$h$k(B Risa/Asir $B8@8l$G=q$+$l$?O"N)@~7AJPHyJ,J}Dx<0MQ(B
                      7: $B$N%i%$%V%i%j$G$9(B.
                      8: $B$9$Y$F$N4X?tL>$O(B odiff_ $B$G;O$^$j$^$9(B.
                      9:
                     10: @tex
                     11: $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.
                     12: $B%j%9%H$K$h$kI=8=$O<!$N$h$&$K$J$j$^$9(B.
                     13: $$ [ [f_{\alpha},[\alpha_{1},\ldots,\alpha_{n}]],\ldots ] $$
                     14: $B$3$l$O(B
                     15: $$ \sum_{\alpha}f_{\alpha}\partial^{\alpha} $$
                     16: $B$H$$$&0UL#$G$9(B. $B@~7?JPHyJ,J}Dx<07O(B
                     17: $$ (\sum_{\alpha^{(i)}}f_{\alpha^{(i)}}\partial^{\alpha^{(i)}})\bullet u = 0 \quad (i = 1,\ldots,s) $$
                     18: $B$J$I$N$h$&$KJ#?t$NHyJ,:nMQAG$rI=8=$9$k$H$-$OHyJ,:nMQAG$N%j%9%H$r;H$$$^$9(B.
                     19: $$ [ [ [f_{\alpha^{(1)}},[\alpha_{1}^{(1)},\ldots,\alpha_{n}^{(1)}]],\ldots ],\ldots,[ [f_{\alpha^{(s)}},[\alpha_{1}^{(s)},\ldots,\alpha_{n}^{(s)}]],\ldots ] ] $$
                     20: $BNc$($PHyJ,:nMQAG(B$x \partial_{x} + y \partial_{y} + 1$$B$N>l9g$O(B
                     21: $$ [ [x,[1,0]],[y,[0,1]],[1,[0,0]] ] $$
                     22: $B$H$J$j$^$9(B. $B$^$?HyJ,:nMQAG$N%j%9%H$G(B$x \partial_{x} + y \partial_{y} + 1, {\partial_{x}}^{2} + {\partial_{y}}^{2}$$B$rI=$9$H(B
                     23: $$ [ [ [x,[1,0]],[y,[0,1]],[1,[0,0]] ],[ [1,[2,0]],[1,[0,2]] ] ] $$
                     24: $B$H$J$j$^$9(B. $B$^$?$3$NI=8=K!$r;H$&$H$-$OJQ?t%j%9%H$r>o$K0U<1$7$F$$$kI,MW$,$"$j$^$9(B.
                     25: $B<!$KB?9`<0$K$h$kI=8=$K$D$$$F=R$Y$^$9(B. $BJQ?t(B$x$$B$KBP$9$kHyJ,$O(B$dx$$B$GI=8=$5$l$^$9(B.
                     26: $BNc$($P(B$x \partial_{x} + y \partial_{y} + 1$$B$K$D$$$F$O(B
                     27: $$ x*dx+y*dy+1 $$
                     28: $B$HI=8=$5$l$^$9(B.
                     29: @end tex
                     30: @menu
                     31: @c * odiff_op_hg1::
                     32: @c * odiff_op_appell1::
                     33: @c * odiff_op_appell2::
                     34: @c * odiff_op_appell3::
                     35: * odiff_op_appell4::
                     36: @c * odiff_op_selberg2::
                     37: @c * odiff_op_gkz::
                     38: * odiff_op_tosm1::
                     39: * odiff_op_toasir::
                     40: * odiff_op_fromasir::
                     41: * odiff_act::
                     42: @c * odiff_act_hg1::
                     43: @c * odiff_act_appell1::
                     44: @c * odiff_act_appell2::
                     45: @c * odiff_act_appell3::
                     46: * odiff_act_appell4::
                     47: @c * odiff_act_selberg2::
                     48: @c * odiff_act_gkz::
                     49: * odiff_poly_solve::
                     50: * odiff_poly_solve_hg1::
                     51: @c * odiff_poly_solve_appell1::
                     52: @c * odiff_poly_solve_appell2::
                     53: @c * odiff_poly_solve_appell3::
                     54: * odiff_poly_solve_appell4::
                     55: @c * odiff_poly_solve_selberg2::
                     56: @c * odiff_poly_solve_gkz::
                     57: * odiff_rat_solve::
                     58: @c * odiff_pseries_appell4::
                     59: @end menu
                     60:
                     61: @node odiff_op_appell4,,, Differential equations (library by Okutani)
                     62: @subsection @code{odiff_op_appell4}
                     63: @findex odiff_op_appell4
                     64: @table @t
                     65: @item odiff_op_appell4(@var{a},@var{b},@var{c1},@var{c2},@var{V})
                     66: ::  appell $B$N(B F_4 $B$rNm2=$9$kHyJ,:nMQAG$r@8@.$7$^$9(B.
                     67: @end table
                     68: @table @var
                     69: @item return
                     70: $B%j%9%H(B
                     71: @item a, b, c1, c2
                     72: $BM-M}<0(B
                     73: @item V
                     74: $B%j%9%H(B
                     75: @end table
                     76: @itemize @bullet
                     77: @item  @code{odiff_op_appell4}$B$NNc(B.
                     78: @end itemize
                     79: @example
                     80: [298] odiff_op_appell4(a,b,c1,c2,[x,y]);
                     81: [ [ [-x^2+x,[2,0]], [-2*y*x,[1,1]], [-y^2,[0,2]],
                     82:     [(-a-b-1)*x+c1,[1,0]], [(-a-b-1)*y,[0,1]], [-b*a,[0,0]] ],
                     83:   [ [-y^2+y,[0,2]], [-2*y*x,[1,1]], [-x^2,[2,0]],
                     84:     [(-a-b-1)*y+c2,[0,1]], [(-a-b-1)*x,[1,0]], [-b*a,[0,0]] ] ]
                     85: @end example
                     86:
                     87: @node odiff_op_tosm1,,, Differential equations (library by Okutani)
                     88: @subsection @code{odiff_op_tosm1}
                     89: @findex odiff_op_tosm1
                     90: @table @t
                     91: @item odiff_op_tosm1(@var{LL},@var{V})
                     92: ::  $B%j%9%H7A<0$NHyJ,:nMQAG%j%9%H$r(B sm1 $B7A<0$KJQ49$7$^$9(B.
                     93: @end table
                     94: @table @var
                     95: @item return
                     96: $B%j%9%H(B
                     97: @item LL
                     98: $B%j%9%H(B
                     99: @item V
                    100: $B%j%9%H(B
                    101: @end table
                    102: @itemize @bullet
                    103: @item  $BHyJ,:nMQAG$N78?t$O@0?tB?9`<0$KJQ49$5$l$^$9(B.
                    104: @item  @code{odiff_op_tosm1}$B$NNc(B
                    105: @end itemize
                    106: @example
                    107: [299] odiff_op_tosm1([[[x,[2,0]],[-1,[0,0]]],
                    108:                           [[y,[0,2]],[-1,[0,0]]]],[x,y]);
                    109: [ + ( + (1) x) dx^2 + ( + (-1)), + ( + (1) y) dy^2 + ( + (-1))]
                    110:
                    111: [300] odiff_op_tosm1([[[x,[1,0]],[y,[0,1]],[1,[0,0]]],
                    112:                           [[1,[2,0]],[1,[0,2]]]],[x,y]);
                    113: [ + ( + (1) x) dx + ( + (1) y) dy + ( + (1)), + ( + (1)) dx^2 + ( + (1)) dy^2]
                    114:
                    115: [301] odiff_op_tosm1([[[1/2,[1,0]],[1,[0,0]]],
                    116:                           [[1/3,[0,1]],[1/4,[0,0]]]],[x,y]);
                    117: [ + ( + (6)) dx + ( + (12)), + ( + (4)) dy + ( + (3))]
                    118:
                    119: [302] odiff_op_tosm1([[[1/2*x,[1,0]],[1,[0,0]]],
                    120:                           [[1/3*y,[0,1]],[1/4,[0,0]]]],[x,y]);
                    121: [ + ( + (6) x) dx + ( + (12)), + ( + (4) y) dy + ( + (3))]
                    122: @end example
                    123:
                    124: @node odiff_op_toasir,,, Differential equations (library by Okutani)
                    125: @subsection @code{odiff_op_toasir}
                    126: @findex odiff_op_toasir
                    127: @table @t
                    128: @item odiff_op_toasir(@var{LL},@var{V})
                    129: ::  $B%j%9%H7A<0$NHyJ,:nMQAG%j%9%H(B @var{LL} $B$r(B @code{asir} $B$NB?9`<0$KJQ49$7$^$9(B.
                    130: @end table
                    131: @table @var
                    132: @item return
                    133: $B%j%9%H(B
                    134: @item LL
                    135: $B%j%9%H(B
                    136: @item V
                    137: $B%j%9%H(B
                    138: @end table
                    139: @itemize @bullet
                    140: @item  @code{odiff_op_toasir}$B$NNc(B
                    141: @end itemize
                    142: @example
                    143: [303] odiff_op_toasir([[[1/2*x,[1,0]],[1,[0,0]]],
                    144:                            [[1/3*y,[0,1]],[1/4,[0,0]]]],[x,y]);
                    145: [1/2*x*dx+1,1/3*y*dy+1/4]
                    146:
                    147: [304] odiff_op_toasir([[[x,[1,0]],[y,[0,1]],[1,[0,0]]],
                    148:                            [[1,[2,0]],[1,[0,2]]]],[x,y]);
                    149: [x*dx+y*dy+1,dx^2+dy^2]
                    150: @end example
                    151:
                    152: @node odiff_op_fromasir,,, Differential equations (library by Okutani)
                    153: @subsection @code{odiff_op_fromasir}
                    154: @findex odiff_op_fromasir
                    155: @table @t
                    156: @item odiff_op_fromasir(@var{D_list},@var{V})
                    157: ::  @code{asir} $B$NB?9`<0$+$i%j%9%H7A<0$NHyJ,:nMQAG%j%9%H$KJQ49$7$^$9(B.
                    158: @end table
                    159: @table @var
                    160: @item return
                    161: $B%j%9%H(B
                    162: @item D_list
                    163: $B%j%9%H(B
                    164: @item V
                    165: $B%j%9%H(B
                    166: @end table
                    167: @itemize @bullet
                    168: @item  @code{odiff_op_fromasir}$B$NNc(B
                    169: @end itemize
                    170: @example
                    171: [305] odiff_op_fromasir([1/2*x*dx+1,1/3*y*dy+1/4],[x,y]);
                    172: [[[1/2*x,[1,0]],[1,[0,0]]],[[1/3*y,[0,1]],[1/4,[0,0]]]]
                    173:
                    174: [306] odiff_op_fromasir([x*dx+y*dy+1,dx^2+dy^2],[x,y]);
                    175: [[[x,[1,0]],[y,[0,1]],[1,[0,0]]],[[1,[2,0]],[1,[0,2]]]]
                    176: @end example
                    177:
                    178: @node odiff_act,,, Differential equations (library by Okutani)
                    179: @subsection @code{odiff_act}
                    180: @findex odiff_act
                    181: @table @t
                    182: @item odiff_act(@var{L},@var{F},@var{V})
                    183: ::  $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.
                    184: @end table
                    185: @table @var
                    186: @item return
                    187: $BM-M}<0(B
                    188: @item L
                    189: $B%j%9%H(B or $BB?9`<0(B
                    190: @item F
                    191: $BM-M}<0(B
                    192: @item V
                    193: $B%j%9%H(B
                    194: @end table
                    195: @itemize @bullet
                    196: @item  @code{odiff_act}$B$NNc(B
                    197: @end itemize
                    198: @example
                    199: [302] odiff_act([[1,[2]]],x^3+x^2+x+1,[x]);
                    200: 6*x+2
                    201:
                    202: [303] odiff_act([[1,[1,0]],[1,[0,1]]],x^2+y^2,[x,y]);
                    203: 2*x+2*y
                    204:
                    205: [349] odiff_act(x*dx+y*dy, x^2+x*y+y^2, [x,y]);
                    206: 2*x^2+2*y*x+2*y^2
                    207: @end example
                    208:
                    209: @node odiff_act_appell4,,, Differential equations (library by Okutani)
                    210: @subsection @code{odiff_act_appell4}
                    211: @findex odiff_act_appell4
                    212: @table @t
                    213: @item odiff_act_appell4(@var{a},@var{b},@var{c1},@var{c2},@var{F},@var{V})
                    214: ::  $BHyJ,:nMQAG(B @code{odiff_op_appell4} $B$rM-M}<0(B @var{F} $B$K:nMQ$5$;$k(B.
                    215: @end table
                    216: @table @var
                    217: @item return
                    218: $B%j%9%H(B
                    219: @item a, b, c1, c2
                    220: $BM-M}<0(B
                    221: @item F
                    222: $BM-M}<0(B
                    223: @item V
                    224: $B%j%9%H(B
                    225: @end table
                    226: @itemize @bullet
                    227: @item  @code{odiff_act_appell4}$B$NNc(B
                    228: @end itemize
                    229: @example
                    230: [303] odiff_act_appell4(1,0,1,1,x^2+y^2,[x,y]);
                    231: [-6*x^2+4*x-6*y^2,-6*x^2-6*y^2+4*y]
                    232:
                    233: [304] odiff_act_appell4(0,0,1,1,x^2+y^2,[x,y]);
                    234: [-4*x^2+4*x-4*y^2,-4*x^2-4*y^2+4*y]
                    235:
                    236: [305] odiff_act_appell4(-2,-2,-1,-1,x^2+y^2,[x,y]);
                    237: [0,0]
                    238: @end example
                    239:
                    240: @node odiff_poly_solve,,, Differential equations (library by Okutani)
                    241: @subsection @code{odiff_poly_solve}
                    242: @findex odiff_poly_solve
                    243: @table @t
                    244: @item odiff_poly_solve(@var{LL},@var{N},@var{V})
                    245: ::  $BM?$($i$l$?@~7?HyJ,J}Dx<07O$N(B @var{N} $B<!0J2<$NB?9`<02r$r5a$a$k(B.
                    246: @end table
                    247: @table @var
                    248: @item return
                    249: $B%j%9%H(B
                    250: @item LL
                    251: $B%j%9%H(B
                    252: @item N
                    253: $B@0?t(B
                    254: @item V
                    255: $B%j%9%H(B
                    256: @end table
                    257: @itemize @bullet
                    258: @item  @code{odiff_poly_solve}$B$NNc(B.
                    259: @end itemize
                    260: @example
                    261: [297] odiff_poly_solve([[[x,[1,0]],[-1,[0,0]]],[[y,[0,1]],[-1,[0,0]]]],5,[x,y]);
                    262: [_4*y*x,[_4]]
                    263:
                    264: [298] odiff_poly_solve([[[x,[1,0]],[-2,[0,0]]],[[y,[0,1]],[-2,[0,0]]]],5,[x,y]);
                    265: [_33*y^2*x^2,[_33]]
                    266:
                    267: [356] odiff_poly_solve([x*dx+y*dy-3,dx+dy],4,[x,y]);
                    268: [-_126*x^3+3*_126*y*x^2-3*_126*y^2*x+_126*y^3,[_126]]
                    269: @end example
                    270:
                    271: @node odiff_poly_solve_hg1,,, Differential equations (library by Okutani)
                    272: @subsection @code{odiff_poly_solve_hg1}
                    273: @findex odiff_poly_solve_hg1
                    274: @table @t
                    275: @item odiff_poly_solve_hg1(@var{a},@var{b},@var{c},@var{V})
                    276: ::  $B%,%&%9$ND64v2?HyJ,J}Dx<0$NB?9`<02r$r5a$a$k(B.
                    277: @end table
                    278: @table @var
                    279: @item return
                    280: $B%j%9%H(B
                    281: @item a, b, c
                    282: $BM-M}<0(B
                    283: @item V
                    284: $B%j%9%H(B
                    285: @end table
                    286: @itemize @bullet
                    287: @item  @code{odiff_poly_solve_hg1}$B$NNc(B.
                    288: @end itemize
                    289: @example
                    290: [334] odiff_poly_solve_hg1(-3,-6,-5,[x]);
                    291: [_1*x^6-2*_0*x^3+9/2*_0*x^2-18/5*_0*x+_0,[_0,_1]]
                    292:
                    293: [335] odiff_poly_solve_hg1(-3,-6,-7,[x]);
                    294: [-4/7*_2*x^3+15/7*_2*x^2-18/7*_2*x+_2,[_2]]
                    295: @end example
                    296:
                    297: @node odiff_poly_solve_appell4,,, Differential equations (library by Okutani)
                    298: @subsection @code{odiff_poly_solve_appell4}
                    299: @findex odiff_poly_solve_appell4
                    300: @table @t
                    301: @item odiff_poly_solve_appell4(@var{a},@var{b},@var{c1},@var{c2},@var{V})
                    302: ::  F_4$B$,$_$?$9@~7?HyJ,J}Dx<07O$NB?9`<02r$r5a$a$k(B.
                    303: @end table
                    304: @table @var
                    305: @item return
                    306: $B%j%9%H(B
                    307: @item a, b, c1, c2
                    308: $BM-M}<0(B
                    309: @item V
                    310: $B%j%9%H(B
                    311: @end table
                    312: @itemize @bullet
                    313: @item  @code{odiff_poly_solve_appell4}$B$NNc(B.
                    314: @end itemize
                    315: @example
                    316: [299] odiff_poly_solve_appell4(-3,1,-1,-1,[x,y]);
                    317: [-_26*x^3+(3*_26*y+_26)*x^2+3*_24*y^2*x-_24*y^3+_24*y^2,[_24,_26]]
                    318:
                    319: [300] odiff_poly_solve_appell4(-3,1,1,-1,[x,y]);
                    320: [-3*_45*y^2*x-_45*y^3+_45*y^2,[_45]]
                    321: @end example
                    322:
                    323: @node odiff_rat_solve,,, Differential equations (library by Okutani)
                    324: @subsection @code{odiff_rat_solve}
                    325: @findex odiff_rat_solve
                    326: @table @t
                    327: @item odiff_rat_solve(@var{LL},@var{Dn},@var{N},@var{V})
                    328: ::  $BM?$($i$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.
                    329: @end table
                    330: @table @var
                    331: @item return
                    332: $B%j%9%H(B
                    333: @item LL
                    334: $B%j%9%H(B
                    335: @item Dn
                    336: $BM-M}<0(B
                    337: @item N
                    338: $B@0?t(B
                    339: @item V
                    340: $B%j%9%H(B
                    341: @end table
                    342: @itemize @bullet
                    343: @item  @code{odiff_rat_solve}$B$NNc(B.
                    344: @end itemize
                    345: @example
                    346: [333] odiff_rat_solve([[[x,[1]],[1,[0]]]],x,1,[x]);
                    347: [(_8)/(x),[_8]]
                    348:
                    349: [361] odiff_rat_solve([x*(1-x)*dx^2+(1-3*x)*dx-1],1-x,2,[x]);
                    350: [(_180)/(-x+1),[_180]]
                    351:
                    352: [350] D = odiff_op_appell4(0,0,3,0,[x,y])$
                    353: [351] odiff_rat_solve(D,x^2,2,[x,y]);
                    354: [(_118*x^2-_114*y*x+1/2*_114*y^2+_114*y)/(x^2),[_114,_118]]
                    355: @end example