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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) = 103 e/ }\" {2 M) n6 ^
x^2+2*x*y+y^2 = 9

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代码：

f(x,y,y1,y2)=& M4 h4 t& Q# h- k
{
2 E& e9 L) a9 R6 j
y1=(x-y)^2-3*(x-y)-10,
! }, n0 M B- L; Z/ T* w
y2=x^2+2*x*y+y^2-9\" }+ t( g) N8 o4 A
};
, c2 Y, [5 \9 O% a
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) = 04 V+ w4 V% V3 N6 A
-(x1^3)+x1*x2-exp(-x2) = 0

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代码：

f(x1,x2,y1,y2)=
, r3 W( N' k: D* T/ k\" {% L- k' n
{0 d: `4 M: Y8 ?
y1=2*x1-x2^2-exp(-x1),
k- F% O# E1 b- q, C& W$ g
y2=-(x1^3)+x1*x2-exp(-x2)
; E) `! p, @8 I6 I
};, A\" t# x$ `' \% ^) K* T
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% p8 s# O& i, P3 y3 S+ e
-b*sin(a+7*t)+n-40.5696=0& `# `- A) ^2 L$ ^; F
-b*sin(a+8*t)+n-41.0443=0
_7 n- L; Q1 t
-b*sin(a+9*t)+n-41.4190=0

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代码：

!using["fcopt"];
( I2 g6 t V! V1 z/ W \. V
f(a,b,n,t,y1,y2,y3,y4)=
. X% E) y* g' I3 E r# Y
{
3 r- V/ K6 h- ~9 q5 [
y1=-b*sin(a+6*t)+n-40.4945,0 f: u6 P7 `/ b' B8 l6 d
y2=-b*sin(a+7*t)+n-40.5696,
! Q/ M( X( g( N% V
y3=-b*sin(a+8*t)+n-41.0443,
( J% d# {2 M! u' {
y4=-b*sin(a+9*t)+n-41.4190
, U# k; R1 i6 O) \# f
};* Z Q1 O8 u7 M8 U- h/ S7 a9 s
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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