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求方程组全部解

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forcal

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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) = 107 U- S% o) g! Q6 L0 q\" r
x^2+2*x*y+y^2 = 9

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代码：

f(x,y,y1,y2)=2 r2 Y8 Z, l# |. t j- [6 d
{
6 T1 A( l/ ~\" |+ Z0 F
y1=(x-y)^2-3*(x-y)-10,, W \1 t1 S9 a0 e
y2=x^2+2*x*y+y^2-9
, y& R t0 C1 M8 d. Y
};
0 n* m6 {* u6 K* q1 F/ g( {& E
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
- j# f* P3 R! z$ t% Q- U' R9 c# a
-(x1^3)+x1*x2-exp(-x2) = 0

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代码：

f(x1,x2,y1,y2)=5 j& q* i0 x0 V8 T
{3 S! `. i6 P9 `7 n\" f% n
y1=2*x1-x2^2-exp(-x1),\" o* J: d) N. E4 x# W
y2=-(x1^3)+x1*x2-exp(-x2)
' O7 x& p4 `) [% O, t; q( O# v
};
! ~8 l- P/ x( Z
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=06 I0 S0 g$ w( Y( e1 A
-b*sin(a+7*t)+n-40.5696=0
& ]. [+ Y6 {( P! ^- h# K
-b*sin(a+8*t)+n-41.0443=0( P* p+ e8 |) M6 {1 P7 G
-b*sin(a+9*t)+n-41.4190=0

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代码：

!using["fcopt"];/ `9 r' A9 ]7 Y/ C) y/ {
f(a,b,n,t,y1,y2,y3,y4)=
8 [$ l* {\\" ?3 c# _( j\\" L4 m9 e
{
3 V y6 A7 o( Q% p' |
y1=-b*sin(a+6*t)+n-40.4945,( M d+ F8 S/ m3 d3 s- y
y2=-b*sin(a+7*t)+n-40.5696,
3 `, k6 H. x8 k9 I5 @/ E6 o! I9 S
y3=-b*sin(a+8*t)+n-41.0443,; @) p6 c/ d+ q% \( R
y4=-b*sin(a+9*t)+n-41.4190
Q3 r2 l, a! ]# |* F- P* A8 x6 E
};0 T7 p% {$ l4 f/ E: D& U) `
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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