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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) = 10
5 m/ m5 r0 q4 ]# m! P( z4 u6 S0 J
x^2+2*x*y+y^2 = 9

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代码：

f(x,y,y1,y2)=) C) @4 Y5 q$ C: [+ Z# |
{5 p# |2 [+ {3 h3 f3 q6 l4 Z
y1=(x-y)^2-3*(x-y)-10,3 t/ h, \& Z( Q. E& X
y2=x^2+2*x*y+y^2-9
* b% b1 {/ Y3 ]8 D8 w( L# `
};% H8 I+ Z- c. s# l
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 u0 s- [) t, J3 x
-(x1^3)+x1*x2-exp(-x2) = 0

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代码：

f(x1,x2,y1,y2)=
3 j$ g# \. _* t2 E$ r& V4 t# L
{; G, _, I' @5 {( e
y1=2*x1-x2^2-exp(-x1),8 }- X1 z: I- H r, L7 j
y2=-(x1^3)+x1*x2-exp(-x2)2 C) Q6 z/ A2 Z4 v6 } M0 ^
};
2 U$ Z2 r O5 ~% K R( f3 ~7 g
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
2 y3 g7 S y( I7 f( k+ G: Z
-b*sin(a+7*t)+n-40.5696=0
* D& r( f( K) ~4 W
-b*sin(a+8*t)+n-41.0443=0
. |- G- f7 Y9 n' k
-b*sin(a+9*t)+n-41.4190=0

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代码：

!using["fcopt"];
& B' w& e e+ |1 ?( p2 n/ L\\" J
f(a,b,n,t,y1,y2,y3,y4)=
' _, P. C+ I$ x( w3 [+ d8 Z' J
{
, ]3 d* v7 }) x* x# X( y
y1=-b*sin(a+6*t)+n-40.4945,7 K; q, U: d, p& n
y2=-b*sin(a+7*t)+n-40.5696,$ A0 V* k- ?6 s% G1 r( q
y3=-b*sin(a+8*t)+n-41.0443,' D- N2 w6 y: c6 E0 O* G
y4=-b*sin(a+9*t)+n-41.4190
; b, W+ F C5 i7 m# u' f
};
6 ?4 a4 n! ^! D4 F/ T
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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