Annotation of OpenXM/src/asir-contrib/packages/doc/ok_diff/ok_diff.oxw, Revision 1.1
1.1 ! takayama 1: @c $OpenXM$
! 2: /*&C
! 3: @node Differential equations (library by Okutani),,, Top
! 4: @chapter Differential equations (library by Okutani)
! 5:
! 6: */
! 7:
! 8: /*&en
! 9:
! 10: The following functions will be imported.
! 11: odiff_act,
! 12: odiff_act_appell4,
! 13: odiff_op_appell4,
! 14: odiff_op_fromasir,
! 15: odiff_op_toasir,
! 16: odiff_op_tosm1,
! 17: odiff_poly_solve,
! 18: odiff_poly_solve_appell4,
! 19: odiff_poly_solve_hg1,
! 20: odiff_rat_solve.
! 21:
! 22: English manual has not been written.
! 23:
! 24: */
! 25:
! 26: /*&ja
! 27: $B%U%!%$%k(B @file{gr}, @file{ok_matrix.rr}, @file{ok_diff.rr} $B$,I,MW$G$9(B.
! 28:
! 29: Yukio Okutani $B;a$K$h$k(B Risa/Asir $B8@8l$G=q$+$l$?O"N)@~7AJPHyJ,J}Dx<0MQ(B
! 30: $B$N%i%$%V%i%j$G$9(B.
! 31: $B$9$Y$F$N4X?tL>$O(B odiff_ $B$G;O$^$j$^$9(B.
! 32:
! 33: @tex
! 34: $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.
! 35: $B%j%9%H$K$h$kI=8=$O<!$N$h$&$K$J$j$^$9(B.
! 36: $$ [ [f_{\alpha},[\alpha_{1},\ldots,\alpha_{n}]],\ldots ] $$
! 37: $B$3$l$O(B
! 38: $$ \sum_{\alpha}f_{\alpha}\partial^{\alpha} $$
! 39: $B$H$$$&0UL#$G$9(B. $B@~7?JPHyJ,J}Dx<07O(B
! 40: $$ (\sum_{\alpha^{(i)}}f_{\alpha^{(i)}}\partial^{\alpha^{(i)}})\bullet u = 0 \quad (i = 1,\ldots,s) $$
! 41: $B$J$I$N$h$&$KJ#?t$NHyJ,:nMQAG$rI=8=$9$k$H$-$OHyJ,:nMQAG$N%j%9%H$r;H$$$^$9(B.
! 42: $$ [ [ [f_{\alpha^{(1)}},[\alpha_{1}^{(1)},\ldots,\alpha_{n}^{(1)}]],\ldots ],\ldots,[ [f_{\alpha^{(s)}},[\alpha_{1}^{(s)},\ldots,\alpha_{n}^{(s)}]],\ldots ] ] $$
! 43: $BNc$($PHyJ,:nMQAG(B$x \partial_{x} + y \partial_{y} + 1$$B$N>l9g$O(B
! 44: $$ [ [x,[1,0]],[y,[0,1]],[1,[0,0]] ] $$
! 45: $B$H$J$j$^$9(B. $B$^$?HyJ,:nMQAG$N%j%9%H$G(B$x \partial_{x} + y \partial_{y} + 1, {\partial_{x}}^{2} + {\partial_{y}}^{2}$$B$rI=$9$H(B
! 46: $$ [ [ [x,[1,0]],[y,[0,1]],[1,[0,0]] ],[ [1,[2,0]],[1,[0,2]] ] ] $$
! 47: $B$H$J$j$^$9(B. $B$^$?$3$NI=8=K!$r;H$&$H$-$OJQ?t%j%9%H$r>o$K0U<1$7$F$$$kI,MW$,$"$j$^$9(B.
! 48: $B<!$KB?9`<0$K$h$kI=8=$K$D$$$F=R$Y$^$9(B. $BJQ?t(B$x$$B$KBP$9$kHyJ,$O(B$dx$$B$GI=8=$5$l$^$9(B.
! 49: $BNc$($P(B$x \partial_{x} + y \partial_{y} + 1$$B$K$D$$$F$O(B
! 50: $$ x*dx+y*dy+1 $$
! 51: $B$HI=8=$5$l$^$9(B.
! 52: @end tex
! 53: @menu
! 54: @c * odiff_op_hg1::
! 55: @c * odiff_op_appell1::
! 56: @c * odiff_op_appell2::
! 57: @c * odiff_op_appell3::
! 58: * odiff_op_appell4::
! 59: @c * odiff_op_selberg2::
! 60: @c * odiff_op_gkz::
! 61: * odiff_op_tosm1::
! 62: * odiff_op_toasir::
! 63: * odiff_op_fromasir::
! 64: * odiff_act::
! 65: @c * odiff_act_hg1::
! 66: @c * odiff_act_appell1::
! 67: @c * odiff_act_appell2::
! 68: @c * odiff_act_appell3::
! 69: * odiff_act_appell4::
! 70: @c * odiff_act_selberg2::
! 71: @c * odiff_act_gkz::
! 72: * odiff_poly_solve::
! 73: * odiff_poly_solve_hg1::
! 74: @c * odiff_poly_solve_appell1::
! 75: @c * odiff_poly_solve_appell2::
! 76: @c * odiff_poly_solve_appell3::
! 77: * odiff_poly_solve_appell4::
! 78: @c * odiff_poly_solve_selberg2::
! 79: @c * odiff_poly_solve_gkz::
! 80: * odiff_rat_solve::
! 81: @c * odiff_pseries_appell4::
! 82: @end menu
! 83:
! 84: @node odiff_op_appell4,,, Differential equations (library by Okutani)
! 85: @subsection @code{odiff_op_appell4}
! 86: @findex odiff_op_appell4
! 87: @table @t
! 88: @item odiff_op_appell4(@var{a},@var{b},@var{c1},@var{c2},@var{V})
! 89: :: appell $B$N(B F_4 $B$rNm2=$9$kHyJ,:nMQAG$r@8@.$7$^$9(B.
! 90: @end table
! 91: @table @var
! 92: @item return
! 93: $B%j%9%H(B
! 94: @item a, b, c1, c2
! 95: $BM-M}<0(B
! 96: @item V
! 97: $B%j%9%H(B
! 98: @end table
! 99: @itemize @bullet
! 100: @item @code{odiff_op_appell4}$B$NNc(B.
! 101: @end itemize
! 102: @example
! 103: [298] odiff_op_appell4(a,b,c1,c2,[x,y]);
! 104: [ [ [-x^2+x,[2,0]], [-2*y*x,[1,1]], [-y^2,[0,2]],
! 105: [(-a-b-1)*x+c1,[1,0]], [(-a-b-1)*y,[0,1]], [-b*a,[0,0]] ],
! 106: [ [-y^2+y,[0,2]], [-2*y*x,[1,1]], [-x^2,[2,0]],
! 107: [(-a-b-1)*y+c2,[0,1]], [(-a-b-1)*x,[1,0]], [-b*a,[0,0]] ] ]
! 108: @end example
! 109:
! 110: @node odiff_op_tosm1,,, Differential equations (library by Okutani)
! 111: @subsection @code{odiff_op_tosm1}
! 112: @findex odiff_op_tosm1
! 113: @table @t
! 114: @item odiff_op_tosm1(@var{LL},@var{V})
! 115: :: $B%j%9%H7A<0$NHyJ,:nMQAG%j%9%H$r(B sm1 $B7A<0$KJQ49$7$^$9(B.
! 116: @end table
! 117: @table @var
! 118: @item return
! 119: $B%j%9%H(B
! 120: @item LL
! 121: $B%j%9%H(B
! 122: @item V
! 123: $B%j%9%H(B
! 124: @end table
! 125: @itemize @bullet
! 126: @item $BHyJ,:nMQAG$N78?t$O@0?tB?9`<0$KJQ49$5$l$^$9(B.
! 127: @item @code{odiff_op_tosm1}$B$NNc(B
! 128: @end itemize
! 129: @example
! 130: [299] odiff_op_tosm1([[[x,[2,0]],[-1,[0,0]]],
! 131: [[y,[0,2]],[-1,[0,0]]]],[x,y]);
! 132: [ + ( + (1) x) dx^2 + ( + (-1)), + ( + (1) y) dy^2 + ( + (-1))]
! 133:
! 134: [300] odiff_op_tosm1([[[x,[1,0]],[y,[0,1]],[1,[0,0]]],
! 135: [[1,[2,0]],[1,[0,2]]]],[x,y]);
! 136: [ + ( + (1) x) dx + ( + (1) y) dy + ( + (1)), + ( + (1)) dx^2 + ( + (1)) dy^2]
! 137:
! 138: [301] odiff_op_tosm1([[[1/2,[1,0]],[1,[0,0]]],
! 139: [[1/3,[0,1]],[1/4,[0,0]]]],[x,y]);
! 140: [ + ( + (6)) dx + ( + (12)), + ( + (4)) dy + ( + (3))]
! 141:
! 142: [302] odiff_op_tosm1([[[1/2*x,[1,0]],[1,[0,0]]],
! 143: [[1/3*y,[0,1]],[1/4,[0,0]]]],[x,y]);
! 144: [ + ( + (6) x) dx + ( + (12)), + ( + (4) y) dy + ( + (3))]
! 145: @end example
! 146:
! 147: @node odiff_op_toasir,,, Differential equations (library by Okutani)
! 148: @subsection @code{odiff_op_toasir}
! 149: @findex odiff_op_toasir
! 150: @table @t
! 151: @item odiff_op_toasir(@var{LL},@var{V})
! 152: :: $B%j%9%H7A<0$NHyJ,:nMQAG%j%9%H(B @var{LL} $B$r(B @code{asir} $B$NB?9`<0$KJQ49$7$^$9(B.
! 153: @end table
! 154: @table @var
! 155: @item return
! 156: $B%j%9%H(B
! 157: @item LL
! 158: $B%j%9%H(B
! 159: @item V
! 160: $B%j%9%H(B
! 161: @end table
! 162: @itemize @bullet
! 163: @item @code{odiff_op_toasir}$B$NNc(B
! 164: @end itemize
! 165: @example
! 166: [303] odiff_op_toasir([[[1/2*x,[1,0]],[1,[0,0]]],
! 167: [[1/3*y,[0,1]],[1/4,[0,0]]]],[x,y]);
! 168: [1/2*x*dx+1,1/3*y*dy+1/4]
! 169:
! 170: [304] odiff_op_toasir([[[x,[1,0]],[y,[0,1]],[1,[0,0]]],
! 171: [[1,[2,0]],[1,[0,2]]]],[x,y]);
! 172: [x*dx+y*dy+1,dx^2+dy^2]
! 173: @end example
! 174:
! 175: @node odiff_op_fromasir,,, Differential equations (library by Okutani)
! 176: @subsection @code{odiff_op_fromasir}
! 177: @findex odiff_op_fromasir
! 178: @table @t
! 179: @item odiff_op_fromasir(@var{D_list},@var{V})
! 180: :: @code{asir} $B$NB?9`<0$+$i%j%9%H7A<0$NHyJ,:nMQAG%j%9%H$KJQ49$7$^$9(B.
! 181: @end table
! 182: @table @var
! 183: @item return
! 184: $B%j%9%H(B
! 185: @item D_list
! 186: $B%j%9%H(B
! 187: @item V
! 188: $B%j%9%H(B
! 189: @end table
! 190: @itemize @bullet
! 191: @item @code{odiff_op_fromasir}$B$NNc(B
! 192: @end itemize
! 193: @example
! 194: [305] odiff_op_fromasir([1/2*x*dx+1,1/3*y*dy+1/4],[x,y]);
! 195: [[[1/2*x,[1,0]],[1,[0,0]]],[[1/3*y,[0,1]],[1/4,[0,0]]]]
! 196:
! 197: [306] odiff_op_fromasir([x*dx+y*dy+1,dx^2+dy^2],[x,y]);
! 198: [[[x,[1,0]],[y,[0,1]],[1,[0,0]]],[[1,[2,0]],[1,[0,2]]]]
! 199: @end example
! 200:
! 201: @node odiff_act,,, Differential equations (library by Okutani)
! 202: @subsection @code{odiff_act}
! 203: @findex odiff_act
! 204: @table @t
! 205: @item odiff_act(@var{L},@var{F},@var{V})
! 206: :: $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.
! 207: @end table
! 208: @table @var
! 209: @item return
! 210: $BM-M}<0(B
! 211: @item L
! 212: $B%j%9%H(B or $BB?9`<0(B
! 213: @item F
! 214: $BM-M}<0(B
! 215: @item V
! 216: $B%j%9%H(B
! 217: @end table
! 218: @itemize @bullet
! 219: @item @code{odiff_act}$B$NNc(B
! 220: @end itemize
! 221: @example
! 222: [302] odiff_act([[1,[2]]],x^3+x^2+x+1,[x]);
! 223: 6*x+2
! 224:
! 225: [303] odiff_act([[1,[1,0]],[1,[0,1]]],x^2+y^2,[x,y]);
! 226: 2*x+2*y
! 227:
! 228: [349] odiff_act(x*dx+y*dy, x^2+x*y+y^2, [x,y]);
! 229: 2*x^2+2*y*x+2*y^2
! 230: @end example
! 231:
! 232: @node odiff_act_appell4,,, Differential equations (library by Okutani)
! 233: @subsection @code{odiff_act_appell4}
! 234: @findex odiff_act_appell4
! 235: @table @t
! 236: @item odiff_act_appell4(@var{a},@var{b},@var{c1},@var{c2},@var{F},@var{V})
! 237: :: $BHyJ,:nMQAG(B @code{odiff_op_appell4} $B$rM-M}<0(B @var{F} $B$K:nMQ$5$;$k(B.
! 238: @end table
! 239: @table @var
! 240: @item return
! 241: $B%j%9%H(B
! 242: @item a, b, c1, c2
! 243: $BM-M}<0(B
! 244: @item F
! 245: $BM-M}<0(B
! 246: @item V
! 247: $B%j%9%H(B
! 248: @end table
! 249: @itemize @bullet
! 250: @item @code{odiff_act_appell4}$B$NNc(B
! 251: @end itemize
! 252: @example
! 253: [303] odiff_act_appell4(1,0,1,1,x^2+y^2,[x,y]);
! 254: [-6*x^2+4*x-6*y^2,-6*x^2-6*y^2+4*y]
! 255:
! 256: [304] odiff_act_appell4(0,0,1,1,x^2+y^2,[x,y]);
! 257: [-4*x^2+4*x-4*y^2,-4*x^2-4*y^2+4*y]
! 258:
! 259: [305] odiff_act_appell4(-2,-2,-1,-1,x^2+y^2,[x,y]);
! 260: [0,0]
! 261: @end example
! 262:
! 263: @node odiff_poly_solve,,, Differential equations (library by Okutani)
! 264: @subsection @code{odiff_poly_solve}
! 265: @findex odiff_poly_solve
! 266: @table @t
! 267: @item odiff_poly_solve(@var{LL},@var{N},@var{V})
! 268: :: $BM?$($i$l$?@~7?HyJ,J}Dx<07O$N(B @var{N} $B<!0J2<$NB?9`<02r$r5a$a$k(B.
! 269: @end table
! 270: @table @var
! 271: @item return
! 272: $B%j%9%H(B
! 273: @item LL
! 274: $B%j%9%H(B
! 275: @item N
! 276: $B@0?t(B
! 277: @item V
! 278: $B%j%9%H(B
! 279: @end table
! 280: @itemize @bullet
! 281: @item @code{odiff_poly_solve}$B$NNc(B.
! 282: @end itemize
! 283: @example
! 284: [297] odiff_poly_solve([[[x,[1,0]],[-1,[0,0]]],[[y,[0,1]],[-1,[0,0]]]],5,[x,y]);
! 285: [_4*y*x,[_4]]
! 286:
! 287: [298] odiff_poly_solve([[[x,[1,0]],[-2,[0,0]]],[[y,[0,1]],[-2,[0,0]]]],5,[x,y]);
! 288: [_33*y^2*x^2,[_33]]
! 289:
! 290: [356] odiff_poly_solve([x*dx+y*dy-3,dx+dy],4,[x,y]);
! 291: [-_126*x^3+3*_126*y*x^2-3*_126*y^2*x+_126*y^3,[_126]]
! 292: @end example
! 293:
! 294: @node odiff_poly_solve_hg1,,, Differential equations (library by Okutani)
! 295: @subsection @code{odiff_poly_solve_hg1}
! 296: @findex odiff_poly_solve_hg1
! 297: @table @t
! 298: @item odiff_poly_solve_hg1(@var{a},@var{b},@var{c},@var{V})
! 299: :: $B%,%&%9$ND64v2?HyJ,J}Dx<0$NB?9`<02r$r5a$a$k(B.
! 300: @end table
! 301: @table @var
! 302: @item return
! 303: $B%j%9%H(B
! 304: @item a, b, c
! 305: $BM-M}<0(B
! 306: @item V
! 307: $B%j%9%H(B
! 308: @end table
! 309: @itemize @bullet
! 310: @item @code{odiff_poly_solve_hg1}$B$NNc(B.
! 311: @end itemize
! 312: @example
! 313: [334] odiff_poly_solve_hg1(-3,-6,-5,[x]);
! 314: [_1*x^6-2*_0*x^3+9/2*_0*x^2-18/5*_0*x+_0,[_0,_1]]
! 315:
! 316: [335] odiff_poly_solve_hg1(-3,-6,-7,[x]);
! 317: [-4/7*_2*x^3+15/7*_2*x^2-18/7*_2*x+_2,[_2]]
! 318: @end example
! 319:
! 320: @node odiff_poly_solve_appell4,,, Differential equations (library by Okutani)
! 321: @subsection @code{odiff_poly_solve_appell4}
! 322: @findex odiff_poly_solve_appell4
! 323: @table @t
! 324: @item odiff_poly_solve_appell4(@var{a},@var{b},@var{c1},@var{c2},@var{V})
! 325: :: F_4$B$,$_$?$9@~7?HyJ,J}Dx<07O$NB?9`<02r$r5a$a$k(B.
! 326: @end table
! 327: @table @var
! 328: @item return
! 329: $B%j%9%H(B
! 330: @item a, b, c1, c2
! 331: $BM-M}<0(B
! 332: @item V
! 333: $B%j%9%H(B
! 334: @end table
! 335: @itemize @bullet
! 336: @item @code{odiff_poly_solve_appell4}$B$NNc(B.
! 337: @end itemize
! 338: @example
! 339: [299] odiff_poly_solve_appell4(-3,1,-1,-1,[x,y]);
! 340: [-_26*x^3+(3*_26*y+_26)*x^2+3*_24*y^2*x-_24*y^3+_24*y^2,[_24,_26]]
! 341:
! 342: [300] odiff_poly_solve_appell4(-3,1,1,-1,[x,y]);
! 343: [-3*_45*y^2*x-_45*y^3+_45*y^2,[_45]]
! 344: @end example
! 345:
! 346: @node odiff_rat_solve,,, Differential equations (library by Okutani)
! 347: @subsection @code{odiff_rat_solve}
! 348: @findex odiff_rat_solve
! 349: @table @t
! 350: @item odiff_rat_solve(@var{LL},@var{Dn},@var{N},@var{V})
! 351: :: $BM?$($i$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.
! 352: @end table
! 353: @table @var
! 354: @item return
! 355: $B%j%9%H(B
! 356: @item LL
! 357: $B%j%9%H(B
! 358: @item Dn
! 359: $BM-M}<0(B
! 360: @item N
! 361: $B@0?t(B
! 362: @item V
! 363: $B%j%9%H(B
! 364: @end table
! 365: @itemize @bullet
! 366: @item @code{odiff_rat_solve}$B$NNc(B.
! 367: @end itemize
! 368: @example
! 369: [333] odiff_rat_solve([[[x,[1]],[1,[0]]]],x,1,[x]);
! 370: [(_8)/(x),[_8]]
! 371:
! 372: [361] odiff_rat_solve([x*(1-x)*dx^2+(1-3*x)*dx-1],1-x,2,[x]);
! 373: [(_180)/(-x+1),[_180]]
! 374:
! 375: [350] D = odiff_op_appell4(0,0,3,0,[x,y])$
! 376: [351] odiff_rat_solve(D,x^2,2,[x,y]);
! 377: [(_118*x^2-_114*y*x+1/2*_114*y^2+_114*y)/(x^2),[_114,_118]]
! 378: @end example
! 379: */
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