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求方程组全部解

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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
3 y, s( h8 S+ C, H+ t
x^2+2*x*y+y^2 = 9

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代码：

f(x,y,y1,y2)=/ a! x) k, F% b7 ^& ~2 r. B
{) e6 W+ o' G2 x- A d- g, ]
y1=(x-y)^2-3*(x-y)-10,- o* w* t! _3 ]4 T7 C
y2=x^2+2*x*y+y^2-9
; d1 x2 q$ s/ z d c/ H8 t9 k/ H
};' B( |$ c- C+ B w2 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) = 0
# g/ ^+ h( O7 A$ G; O# I2 d
-(x1^3)+x1*x2-exp(-x2) = 0

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代码：

f(x1,x2,y1,y2)=* k. o; ]. N& l* j
{6 l6 ~; o\" E7 H
y1=2*x1-x2^2-exp(-x1),& ^: _. Q# N! O4 c L' z3 N/ t, o* Z6 o
y2=-(x1^3)+x1*x2-exp(-x2)7 M: P; Q) e& w' M) s' s6 y! F
};
8 X. ^\" E7 g- R- f
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=03 P) A: E( A7 }- ^: `) I4 d2 G2 p
-b*sin(a+7*t)+n-40.5696=0
, X4 {0 ^ {; c\" n
-b*sin(a+8*t)+n-41.0443=0
) J2 c8 s% }' s4 T3 w2 t# Y
-b*sin(a+9*t)+n-41.4190=0

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代码：

!using["fcopt"];
) {% z' N. d0 A1 H
f(a,b,n,t,y1,y2,y3,y4)=1 X) a6 |5 e. H o9 b( u- i3 W
{
\\" `& e+ J6 b I) c8 \; ]5 `
y1=-b*sin(a+6*t)+n-40.4945,
& X C( F: V& s) e3 R
y2=-b*sin(a+7*t)+n-40.5696,
8 g' J: k x* r
y3=-b*sin(a+8*t)+n-41.0443,
2 i' I& }- x$ l: k
y4=-b*sin(a+9*t)+n-41.41900 N: G( A# M7 {. A0 A
};\\" U# M# @ }: H- 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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