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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 |正序浏览

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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% {/ c1 G\" _+ _5 ?4 G
x^2+2*x*y+y^2 = 9

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代码：

f(x,y,y1,y2)=
6 ]. Y0 ~1 a. z; e$ Q4 s+ P
{, b! c b& W1 K- i' C
y1=(x-y)^2-3*(x-y)-10,
( A. L* E7 L* i% h) a0 p; \
y2=x^2+2*x*y+y^2-9$ o( O( \4 `$ ^6 n* ?5 b8 p; D
};\" B, V3 O) L' V9 m: D3 |\" `
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: }# v2 J2 Q4 e& y2 K) D
-(x1^3)+x1*x2-exp(-x2) = 0

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代码：

f(x1,x2,y1,y2)=- P1 M1 |/ e6 a7 N' n
{- x6 U1 n1 s. q$ {9 ~1 _9 M
y1=2*x1-x2^2-exp(-x1),
# J; q6 n5 @- p0 `' N
y2=-(x1^3)+x1*x2-exp(-x2)8 H9 ^* A: R+ f4 a
};
6 ?1 J5 Z; l$ c) d! R
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
& F: t2 z0 W- ]% r+ A
-b*sin(a+7*t)+n-40.5696=0; K8 P3 y2 S9 E7 K2 o
-b*sin(a+8*t)+n-41.0443=0
/ j9 t1 i\" f. f1 X9 x1 T: r
-b*sin(a+9*t)+n-41.4190=0

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代码：

!using["fcopt"];
- p |& T/ x- D4 Q0 Y5 }# b
f(a,b,n,t,y1,y2,y3,y4)=1 f- M& @\\" W0 `! C
{8 ], u( H0 k) M8 ]
y1=-b*sin(a+6*t)+n-40.4945,
5 O$ T( N8 s6 L- r$ T! p
y2=-b*sin(a+7*t)+n-40.5696,; t% K; ^6 O5 A9 W\\" o* Q3 ]
y3=-b*sin(a+8*t)+n-41.0443,
) q9 O. X: `* ?9 A8 p
y4=-b*sin(a+9*t)+n-41.4190( z) H\\" ?& z& R2 r- Q& z I( }
};\\" a x) }3 Q3 F+ _ |\\" K
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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