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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) = 104 r2 T; t- z- F7 }4 d
x^2+2*x*y+y^2 = 9

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代码：

f(x,y,y1,y2)=
& E7 C6 o5 ~* i& E0 W& r G0 `
{
& i/ s3 O6 ]$ |* N5 Q
y1=(x-y)^2-3*(x-y)-10,8 y) a4 s3 D' |/ `
y2=x^2+2*x*y+y^2-9; ?& X8 p% a& Q- U1 p. U- F
};- |9 d& i8 b# c3 b- v1 C, 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) r4 I8 O) x- p y
-(x1^3)+x1*x2-exp(-x2) = 0

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代码：

f(x1,x2,y1,y2)=6 e. C* g2 m+ Y& k9 f* Q0 M5 F/ L7 `
{
4 G( @- G; i3 r\" R8 w# ^! ~
y1=2*x1-x2^2-exp(-x1),
5 ]( e; f7 W/ n U* d7 q, u0 {0 E' T
y2=-(x1^3)+x1*x2-exp(-x2)0 x/ |) A5 j4 X8 a* A4 u
};
2 f0 @( i4 a0 Z# v& Y
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 X- i! }6 i0 k6 g; L# ^
-b*sin(a+7*t)+n-40.5696=0
' f2 F( f1 Y4 i0 F: t. S: U) c
-b*sin(a+8*t)+n-41.0443=02 K4 V8 ^\" S& l% r: S9 n/ v
-b*sin(a+9*t)+n-41.4190=0

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代码：

!using["fcopt"];
& ?9 A) W; e; D o3 V& |
f(a,b,n,t,y1,y2,y3,y4)=9 A$ V8 p. I% H
{
: o1 z3 {% g$ k; u3 [+ `7 e
y1=-b*sin(a+6*t)+n-40.4945,
. c0 j* {0 r; S) u
y2=-b*sin(a+7*t)+n-40.5696,
' Y3 s1 C8 T9 b4 M
y3=-b*sin(a+8*t)+n-41.0443,
2 u6 s7 W+ M, ], r% X1 [
y4=-b*sin(a+9*t)+n-41.41904 B* r+ ~+ D9 q ?7 S# W/ h& }
};
( E- k$ R' G5 w& X! w* }
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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