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求方程组全部解

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发表于 2011-1-15 17:04 |显示全部楼层 |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
; Z& ]+ K5 F0 \
x^2+2*x*y+y^2 = 9

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代码：

f(x,y,y1,y2)=
, a+ F: F1 W) _8 Y' A) E8 T
{
- v2 Q( u/ e2 H2 |
y1=(x-y)^2-3*(x-y)-10, V6 v. n9 W0 F2 f1 L
y2=x^2+2*x*y+y^2-9: T; U( T; z# V3 M$ }
};0 A8 k9 e1 m# p0 ?
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- W! X$ F( R$ p1 R
-(x1^3)+x1*x2-exp(-x2) = 0

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代码：

f(x1,x2,y1,y2)=
4 ?4 W$ D' H: D! ]
{! ^6 `6 z% u8 M. ^3 [0 F\" E! C
y1=2*x1-x2^2-exp(-x1),# l$ `5 V. ]. F( P3 I
y2=-(x1^3)+x1*x2-exp(-x2)
( b; Q3 g$ c# R& J# n( h
};
( d. o& ?( A# f8 F- D- u- C
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
- [\" h$ a0 z* M) Q( a( G
-b*sin(a+7*t)+n-40.5696=05 d% r) a3 ?3 D7 S: |; ~2 C
-b*sin(a+8*t)+n-41.0443=0
. Y R9 F5 h/ d6 ^9 l) @
-b*sin(a+9*t)+n-41.4190=0

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代码：

!using["fcopt"];
: r4 M- D( q1 Q3 ~$ J0 J9 `
f(a,b,n,t,y1,y2,y3,y4)=% D7 }; U1 G. L; M$ {( V
{6 D/ ~8 n, A' l3 k% V# q
y1=-b*sin(a+6*t)+n-40.4945,
% d2 b s+ h( N/ y5 O) W; ?- u
y2=-b*sin(a+7*t)+n-40.5696,0 [\\" V: G/ M: F' ^
y3=-b*sin(a+8*t)+n-41.0443,
( p8 ?+ E; g: p- x
y4=-b*sin(a+9*t)+n-41.4190$ Z& Q2 ]- s; A/ A6 S
};
2 ~8 A4 t: x. M* 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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