Annotation of OpenXM/src/asir-contrib/packages/doc/Diff.texi, Revision 1.7
1.7 ! takayama 1: @c $OpenXM: OpenXM/src/asir-contrib/packages/doc/Diff.texi,v 1.6 2000/02/20 11:30:44 okutani Exp $
! 2: @node Differential equations (library by Okutani),,, Top
! 3: @section Differential equations (library by Okutani)
1.1 takayama 4: $B%U%!%$%k(B @file{gr}, @file{Matrix} $B$,I,MW$G$9(B.
5:
1.7 ! takayama 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:
1.1 takayama 10: @tex
1.6 okutani 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.
1.1 takayama 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: $$ [ [ [f_{\alpha^{(1)}},[\alpha_{1}^{(1)},\ldots,\alpha_{n}^{(1)}]],\ldots ],\ldots,[ [f_{\alpha^{(s)}},[\alpha_{1}^{(s)},\ldots,\alpha_{n}^{(s)}]],\ldots ] ] $$
1.6 okutani 20: $BNc$($PHyJ,:nMQAG(B$x \partial_{x} + y \partial_{y} + 1$$B$N>l9g$O(B
1.1 takayama 21: $$ [ [x,[1,0]],[y,[0,1]],[1,[0,0]] ] $$
1.6 okutani 22: $B$H$J$j$^$9(B. $B$^$?HyJ,:nMQAG$N%j%9%H$G(B$x \partial_{x} + y \partial_{y} + 1, {\partial_{x}}^{2} + {\partial_{y}}^{2}$$B$rI=$9$H(B
1.1 takayama 23: $$ [ [ [x,[1,0]],[y,[0,1]],[1,[0,0]] ],[ [1,[2,0]],[1,[0,2]] ] ] $$
1.6 okutani 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.
1.1 takayama 29: @end tex
30: @menu
1.7 ! takayama 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::
1.1 takayama 59: @end menu
60:
1.7 ! takayama 61: @node odiff_op_appell4,,, Differential equations (library by Okutani)
! 62: @subsection @code{odiff_op_appell4}
! 63: @findex odiff_op_appell4
1.1 takayama 64: @table @t
1.7 ! takayama 65: @item odiff_op_appell4(@var{a},@var{b},@var{c1},@var{c2},@var{V})
1.1 takayama 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: $B%j%9%H(B
75: @end table
76: @itemize @bullet
1.7 ! takayama 77: @item @code{odiff_op_appell4}$B$NNc(B.
1.1 takayama 78: @end itemize
79: @example
1.7 ! takayama 80: [298] odiff_op_appell4(a,b,c1,c2,[x,y]);
1.1 takayama 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:
1.7 ! takayama 87: @node odiff_op_tosm1,,, Differential equations (library by Okutani)
! 88: @subsection @code{odiff_op_tosm1}
! 89: @findex odiff_op_tosm1
1.6 okutani 90: @table @t
1.7 ! takayama 91: @item odiff_op_tosm1(@var{LL},@var{V})
1.6 okutani 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$^$9(B.
1.7 ! takayama 104: @item @code{odiff_op_tosm1}$B$NNc(B
1.6 okutani 105: @end itemize
106: @example
1.7 ! takayama 107: [299] odiff_op_tosm1([[[x,[2,0]],[-1,[0,0]]],
1.6 okutani 108: [[y,[0,2]],[-1,[0,0]]]],[x,y]);
109: [ + ( + (1) x) dx^2 + ( + (-1)), + ( + (1) y) dy^2 + ( + (-1))]
110:
1.7 ! takayama 111: [300] odiff_op_tosm1([[[x,[1,0]],[y,[0,1]],[1,[0,0]]],
1.6 okutani 112: [[1,[2,0]],[1,[0,2]]]],[x,y]);
113: [ + ( + (1) x) dx + ( + (1) y) dy + ( + (1)), + ( + (1)) dx^2 + ( + (1)) dy^2]
114:
1.7 ! takayama 115: [301] odiff_op_tosm1([[[1/2,[1,0]],[1,[0,0]]],
1.6 okutani 116: [[1/3,[0,1]],[1/4,[0,0]]]],[x,y]);
117: [ + ( + (6)) dx + ( + (12)), + ( + (4)) dy + ( + (3))]
118:
1.7 ! takayama 119: [302] odiff_op_tosm1([[[1/2*x,[1,0]],[1,[0,0]]],
1.6 okutani 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:
1.7 ! takayama 124: @node odiff_op_toasir,,, Differential equations (library by Okutani)
! 125: @subsection @code{odiff_op_toasir}
! 126: @findex odiff_op_toasir
1.6 okutani 127: @table @t
1.7 ! takayama 128: @item odiff_op_toasir(@var{LL},@var{V})
1.6 okutani 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: $B%j%9%H(B
138: @end table
139: @itemize @bullet
1.7 ! takayama 140: @item @code{odiff_op_toasir}$B$NNc(B
1.6 okutani 141: @end itemize
142: @example
1.7 ! takayama 143: [303] odiff_op_toasir([[[1/2*x,[1,0]],[1,[0,0]]],
1.6 okutani 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:
1.7 ! takayama 147: [304] odiff_op_toasir([[[x,[1,0]],[y,[0,1]],[1,[0,0]]],
1.6 okutani 148: [[1,[2,0]],[1,[0,2]]]],[x,y]);
149: [x*dx+y*dy+1,dx^2+dy^2]
150: @end example
151:
1.7 ! takayama 152: @node odiff_op_fromasir,,, Differential equations (library by Okutani)
! 153: @subsection @code{odiff_op_fromasir}
! 154: @findex odiff_op_fromasir
1.6 okutani 155: @table @t
1.7 ! takayama 156: @item odiff_op_fromasir(@var{D_list},@var{V})
1.6 okutani 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: $B%j%9%H(B
166: @end table
167: @itemize @bullet
1.7 ! takayama 168: @item @code{odiff_op_fromasir}$B$NNc(B
1.6 okutani 169: @end itemize
170: @example
1.7 ! takayama 171: [305] odiff_op_fromasir([1/2*x*dx+1,1/3*y*dy+1/4],[x,y]);
1.6 okutani 172: [[[1/2*x,[1,0]],[1,[0,0]]],[[1/3*y,[0,1]],[1/4,[0,0]]]]
173:
1.7 ! takayama 174: [306] odiff_op_fromasir([x*dx+y*dy+1,dx^2+dy^2],[x,y]);
1.6 okutani 175: [[[x,[1,0]],[y,[0,1]],[1,[0,0]]],[[1,[2,0]],[1,[0,2]]]]
176: @end example
177:
1.7 ! takayama 178: @node odiff_act,,, Differential equations (library by Okutani)
! 179: @subsection @code{odiff_act}
! 180: @findex odiff_act
1.1 takayama 181: @table @t
1.7 ! takayama 182: @item odiff_act(@var{L},@var{F},@var{V})
1.1 takayama 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
1.6 okutani 189: $B%j%9%H(B or $BB?9`<0(B
1.1 takayama 190: @item F
191: $BM-M}<0(B
192: @item V
193: $B%j%9%H(B
194: @end table
195: @itemize @bullet
1.7 ! takayama 196: @item @code{odiff_act}$B$NNc(B
1.1 takayama 197: @end itemize
198: @example
1.7 ! takayama 199: [302] odiff_act([[1,[2]]],x^3+x^2+x+1,[x]);
1.1 takayama 200: 6*x+2
201:
1.7 ! takayama 202: [303] odiff_act([[1,[1,0]],[1,[0,1]]],x^2+y^2,[x,y]);
1.1 takayama 203: 2*x+2*y
1.6 okutani 204:
1.7 ! takayama 205: [349] odiff_act(x*dx+y*dy, x^2+x*y+y^2, [x,y]);
1.6 okutani 206: 2*x^2+2*y*x+2*y^2
1.1 takayama 207: @end example
208:
1.7 ! takayama 209: @node odiff_act_appell4,,, Differential equations (library by Okutani)
! 210: @subsection @code{odiff_act_appell4}
! 211: @findex odiff_act_appell4
1.1 takayama 212: @table @t
1.7 ! takayama 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.
1.1 takayama 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: $B%j%9%H(B
225: @end table
226: @itemize @bullet
1.7 ! takayama 227: @item @code{odiff_act_appell4}$B$NNc(B
1.1 takayama 228: @end itemize
229: @example
1.7 ! takayama 230: [303] odiff_act_appell4(1,0,1,1,x^2+y^2,[x,y]);
1.1 takayama 231: [-6*x^2+4*x-6*y^2,-6*x^2-6*y^2+4*y]
232:
1.7 ! takayama 233: [304] odiff_act_appell4(0,0,1,1,x^2+y^2,[x,y]);
1.1 takayama 234: [-4*x^2+4*x-4*y^2,-4*x^2-4*y^2+4*y]
235:
1.7 ! takayama 236: [305] odiff_act_appell4(-2,-2,-1,-1,x^2+y^2,[x,y]);
1.1 takayama 237: [0,0]
238: @end example
239:
1.7 ! takayama 240: @node odiff_poly_solve,,, Differential equations (library by Okutani)
! 241: @subsection @code{odiff_poly_solve}
! 242: @findex odiff_poly_solve
1.1 takayama 243: @table @t
1.7 ! takayama 244: @item odiff_poly_solve(@var{LL},@var{N},@var{V})
1.1 takayama 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: $B%j%9%H(B
256: @end table
257: @itemize @bullet
1.7 ! takayama 258: @item @code{odiff_poly_solve}$B$NNc(B.
1.1 takayama 259: @end itemize
260: @example
1.7 ! takayama 261: [297] odiff_poly_solve([[[x,[1,0]],[-1,[0,0]]],[[y,[0,1]],[-1,[0,0]]]],5,[x,y]);
1.1 takayama 262: [_4*y*x,[_4]]
263:
1.7 ! takayama 264: [298] odiff_poly_solve([[[x,[1,0]],[-2,[0,0]]],[[y,[0,1]],[-2,[0,0]]]],5,[x,y]);
1.1 takayama 265: [_33*y^2*x^2,[_33]]
1.6 okutani 266:
1.7 ! takayama 267: [356] odiff_poly_solve([x*dx+y*dy-3,dx+dy],4,[x,y]);
1.6 okutani 268: [-_126*x^3+3*_126*y*x^2-3*_126*y^2*x+_126*y^3,[_126]]
1.2 okutani 269: @end example
270:
1.7 ! takayama 271: @node odiff_poly_solve_hg1,,, Differential equations (library by Okutani)
! 272: @subsection @code{odiff_poly_solve_hg1}
! 273: @findex odiff_poly_solve_hg1
1.5 okutani 274: @table @t
1.7 ! takayama 275: @item odiff_poly_solve_hg1(@var{a},@var{b},@var{c},@var{V})
1.5 okutani 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: $B%j%9%H(B
285: @end table
286: @itemize @bullet
1.7 ! takayama 287: @item @code{odiff_poly_solve_hg1}$B$NNc(B.
1.5 okutani 288: @end itemize
289: @example
1.7 ! takayama 290: [334] odiff_poly_solve_hg1(-3,-6,-5,[x]);
1.5 okutani 291: [_1*x^6-2*_0*x^3+9/2*_0*x^2-18/5*_0*x+_0,[_0,_1]]
292:
1.7 ! takayama 293: [335] odiff_poly_solve_hg1(-3,-6,-7,[x]);
1.5 okutani 294: [-4/7*_2*x^3+15/7*_2*x^2-18/7*_2*x+_2,[_2]]
295: @end example
296:
1.7 ! takayama 297: @node odiff_poly_solve_appell4,,, Differential equations (library by Okutani)
! 298: @subsection @code{odiff_poly_solve_appell4}
! 299: @findex odiff_poly_solve_appell4
1.2 okutani 300: @table @t
1.7 ! takayama 301: @item odiff_poly_solve_appell4(@var{a},@var{b},@var{c1},@var{c2},@var{V})
1.5 okutani 302: :: F_4$B$,$_$?$9@~7?HyJ,J}Dx<07O$NB?9`<02r$r5a$a$k(B.
1.2 okutani 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: $B%j%9%H(B
311: @end table
312: @itemize @bullet
1.7 ! takayama 313: @item @code{odiff_poly_solve_appell4}$B$NNc(B.
1.2 okutani 314: @end itemize
315: @example
1.7 ! takayama 316: [299] odiff_poly_solve_appell4(-3,1,-1,-1,[x,y]);
1.2 okutani 317: [-_26*x^3+(3*_26*y+_26)*x^2+3*_24*y^2*x-_24*y^3+_24*y^2,[_24,_26]]
318:
1.7 ! takayama 319: [300] odiff_poly_solve_appell4(-3,1,1,-1,[x,y]);
1.2 okutani 320: [-3*_45*y^2*x-_45*y^3+_45*y^2,[_45]]
1.3 okutani 321: @end example
322:
1.7 ! takayama 323: @node odiff_rat_solve,,, Differential equations (library by Okutani)
! 324: @subsection @code{odiff_rat_solve}
! 325: @findex odiff_rat_solve
1.3 okutani 326: @table @t
1.7 ! takayama 327: @item odiff_rat_solve(@var{LL},@var{Dn},@var{N},@var{V})
1.3 okutani 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: $B%j%9%H(B
341: @end table
342: @itemize @bullet
1.7 ! takayama 343: @item @code{odiff_rat_solve}$B$NNc(B.
1.3 okutani 344: @end itemize
345: @example
1.7 ! takayama 346: [333] odiff_rat_solve([[[x,[1]],[1,[0]]]],x,1,[x]);
1.4 okutani 347: [(_8)/(x),[_8]]
1.6 okutani 348:
1.7 ! takayama 349: [361] odiff_rat_solve([x*(1-x)*dx^2+(1-3*x)*dx-1],1-x,2,[x]);
1.6 okutani 350: [(_180)/(-x+1),[_180]]
1.4 okutani 351:
1.7 ! takayama 352: [350] D = odiff_op_appell4(0,0,3,0,[x,y])$
! 353: [351] odiff_rat_solve(D,x^2,2,[x,y]);
1.4 okutani 354: [(_118*x^2-_114*y*x+1/2*_114*y^2+_114*y)/(x^2),[_114,_118]]
1.1 takayama 355: @end example
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