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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
% `\" c( a- n& }1 R7 D
x^2+2*x*y+y^2 = 9

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代码：

f(x,y,y1,y2)=
' s- T7 ~2 Q4 p! q* l- K
{
- L4 r4 |# m4 } c4 c( w5 a
y1=(x-y)^2-3*(x-y)-10,8 F) x* U. D+ l S% P, T8 E7 a0 }# d
y2=x^2+2*x*y+y^2-94 v) N. I9 `5 ]1 |* L
};
+ a5 ?\" a+ o3 c2 J& G( o
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/ P. f- f# j3 B8 t\" Y& d$ F5 t# |
-(x1^3)+x1*x2-exp(-x2) = 0

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代码：

f(x1,x2,y1,y2)=
$ Q t7 ^3 v: }; k0 G! k j
{3 P1 a* D! q' k0 N, E P
y1=2*x1-x2^2-exp(-x1),' B- L7 ~- ?; d9 b% R
y2=-(x1^3)+x1*x2-exp(-x2)
3 c3 V6 w+ ?6 Q
};
% ^6 G; s- x) N' m
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
N# x. f' E5 v$ j& @, N) G
-b*sin(a+7*t)+n-40.5696=0
' f- S! v; @( l# B+ W
-b*sin(a+8*t)+n-41.0443=00 x% q# a6 p& C\" h; p
-b*sin(a+9*t)+n-41.4190=0

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代码：

!using["fcopt"];) m* n/ ?- j\\" T# W
f(a,b,n,t,y1,y2,y3,y4)=
' @& ]5 I( m* w/ b% Z: `6 j
{& C+ j\\" }2 w7 G2 M5 n
y1=-b*sin(a+6*t)+n-40.4945,% V' B' ~ l2 `1 p7 m: j
y2=-b*sin(a+7*t)+n-40.5696,
' i+ A2 o+ g8 ^, R( B/ y9 F, r
y3=-b*sin(a+8*t)+n-41.0443,
& m$ ]' O1 K# c: t: z
y4=-b*sin(a+9*t)+n-41.4190+ ]$ K' N; v, R4 b L
};
! A* Z\\" U' t5 A% V
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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