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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
% P9 C% t3 E8 X6 f) t; @( L f
x^2+2*x*y+y^2 = 9

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代码：

f(x,y,y1,y2)=& t2 r7 z( O7 u& R% l
{
8 Z2 `\" e3 v B& b' ]8 E
y1=(x-y)^2-3*(x-y)-10,6 ?# Y; ~6 L0 q3 J
y2=x^2+2*x*y+y^2-9) b' w4 I. V, V2 w\" V) n W# \4 I
};' Y2 g1 K7 b\" s' }# t
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
9 H6 W) j7 H& [
-(x1^3)+x1*x2-exp(-x2) = 0

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代码：

f(x1,x2,y1,y2)=
, F) R. g! z+ \9 |$ e
{; {4 E' b+ x7 L$ O. `6 ~% h3 w
y1=2*x1-x2^2-exp(-x1),; Q4 O0 ]5 |7 T3 u3 B/ ~; ]3 E# j* O4 J! n
y2=-(x1^3)+x1*x2-exp(-x2)
' h, e8 x0 T: _& A5 Z
};
. c7 l5 c% I d' w# b) j0 w
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* x! j7 y\" ^# v, J: X5 C' C' _4 C
-b*sin(a+7*t)+n-40.5696=0
# L: x: [: d) k. U5 x+ o) a( h* B
-b*sin(a+8*t)+n-41.0443=0
4 t2 |4 A\" A2 L5 Y2 T
-b*sin(a+9*t)+n-41.4190=0

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代码：

!using["fcopt"];1 Q$ r9 q5 N8 m) V7 a7 x$ g
f(a,b,n,t,y1,y2,y3,y4)=
0 k( P% P- |7 Q: ?6 ]
{
, h+ S1 U$ z6 J0 z$ w
y1=-b*sin(a+6*t)+n-40.4945,
3 X* z9 e, p! L( h) K1 u& @- O
y2=-b*sin(a+7*t)+n-40.5696,% }* L, D* j& w% {
y3=-b*sin(a+8*t)+n-41.0443,
4 J$ c* L0 @3 G2 J/ }
y4=-b*sin(a+9*t)+n-41.4190+ _' d0 _7 u! \1 O# G9 C# ~
};
# g4 O3 P7 n$ z- x: W( m
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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