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求方程组全部解

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发表于 2011-1-15 17:04 |只看该作者 |倒序浏览

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Forcal优化库FcOpt中新增函数fcopt::solve，试图求解方程的全部解。正在测试修改，请大家多提意见。
参考：http://www.forcal.net/sysm/forcal9/fchtm/fcopt.htm

例子1：解方程组：

(x-y)^2-3*(x-y) = 10: k6 z8 p' [/ Z' Z) ~
x^2+2*x*y+y^2 = 9

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代码：

f(x,y,y1,y2)=
2 {6 j7 p0 x- S
{$ O: i! h4 O. h/ l! K4 b
y1=(x-y)^2-3*(x-y)-10,, x' S8 n1 P! ^% ~- b( C4 p
y2=x^2+2*x*y+y^2-9
$ |/ ~' s1 ], c- J; C9 K8 ?
};- g0 t- r+ d V ~* f: D- c8 z2 c/ ]
fcopt::solve[HFor("f")];

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结果：
0.5                      2.5                      0.
-2.5                   -0.5                   0.
1.000000000225044       -4.00000000022569       2.231017652693784e-009
4.000000000395746       -1.00000000039106       3.894538219597456e-009
4.

例子2：解方程组：

2*x1-x2^2-exp(-x1) = 0+ U\" z8 T# h- X/ k& Q/ L
-(x1^3)+x1*x2-exp(-x2) = 0

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代码：

f(x1,x2,y1,y2)=
z1 u& K/ }\" X\" N
{. k+ _3 j3 y: B# l0 ?- V\" k9 H
y1=2*x1-x2^2-exp(-x1),\" d& ] U1 Q3 X' X, k$ Y6 C
y2=-(x1^3)+x1*x2-exp(-x2)
5 `8 W# g& i0 \; E. S5 E% W
}; A2 Q5 c( ]6 _9 Z* v4 q8 o
fcopt::solve[HFor("f")];

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结果：
0.7914550065632104 1.062885264188035 0.
0.9977869653328695 1.275491849454102 3.925231146709438e-017
2.

例子3：解方程组：t取-7～7

-b*sin(a+6*t)+n-40.4945=0
# _. Z' l/ R/ c. H$ }; {% c& p! u
-b*sin(a+7*t)+n-40.5696=0& {2 w( Z# ]) a* a
-b*sin(a+8*t)+n-41.0443=0
# }6 |4 q1 e6 t) p+ d\" r/ O
-b*sin(a+9*t)+n-41.4190=0

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代码：

!using["fcopt"];
2 A h% \7 G Y7 a
f(a,b,n,t,y1,y2,y3,y4)=4 E$ c2 T$ f& {\\" a
{
# }+ X; w# E' f- P+ X
y1=-b*sin(a+6*t)+n-40.4945,1 r1 N$ y9 h2 [2 |
y2=-b*sin(a+7*t)+n-40.5696,# J$ Y+ k7 j- d. p! e# n
y3=-b*sin(a+8*t)+n-41.0443,. b, q. j- M; t$ P F
y4=-b*sin(a+9*t)+n-41.4190% a p: I5 n! t# L5 V& i; |
};( w9 E\\" V. u/ A- |( Q4 T; O( I
solve[HFor("f"), optrange,-1e50,1e50,-1e50,1e50,-1e50,1e50,-7,7];

一种可能的结果（该方程组有无穷解）：
-2.140093203561007       -0.4915300827061839    40.94928398718974       1.077226214994063       3.552713678800501e-015
-11.56487116433041       0.491530082706186       40.94928398718974       1.077226214994066       5.024295867788081e-015
-8.423278510740103       -0.4915300827061995    40.94928398718977       1.077226214993991       8.702335715267317e-015
2555.116326818533       -0.4915300827062283    40.94928398718988       1.07722621499373       4.819135301037582e-014
1.001499450023601       0.4915300827059401       40.94928398718962       -5.205959092184797       1.64387405750109e-013
-17.84805647151125       0.4915300827056817       40.9492839871897       1.077226214994272       3.642354617502926e-013
3146.874339449554       -0.4915300825865869    40.94928398712157       -1.077226215397079       1.198690006101687e-010
4.14309210834897       -0.4915300817987574    40.94928398665894       -5.205959092793353       8.618584276014861e-010
5628.732535974947       -0.491530080064976       40.9492839770687       -1.077226245248003       7.394104227928194e-009
1934.219575147075       -0.4915300766540718    40.94928398081019       -1.077226212465366       8.617217026839414e-009
10.

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