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求方程组全部解

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forcal

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	难过 2012-8-27 18:22

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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) = 101 ?& {\" {. I9 w* i& b
x^2+2*x*y+y^2 = 9

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代码：

f(x,y,y1,y2)=
! V) ]/ A. n( m$ E5 D2 V# T
{
; @3 D2 H+ R1 v
y1=(x-y)^2-3*(x-y)-10,
; t+ f: D6 |8 |, ^# y% O8 D3 C8 N
y2=x^2+2*x*y+y^2-9
, o; @' W2 C Y; x7 d: q1 x
};
. w% G, q% J7 w$ ]1 }
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
2 `2 Q l\" K0 }% z/ F' J9 ]
-(x1^3)+x1*x2-exp(-x2) = 0

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代码：

f(x1,x2,y1,y2)=6 F% a\" t9 j+ `& Q9 ?. M
{
/ [7 Z. _' B# q
y1=2*x1-x2^2-exp(-x1),\" O: V3 Q) f4 P4 J; v
y2=-(x1^3)+x1*x2-exp(-x2)! `$ t* @* i; P
};
0 I7 i' [$ A& @
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( [0 z, N# z8 ]- a6 u9 \
-b*sin(a+7*t)+n-40.5696=0
# t/ F2 t1 ~- C: c; m% m1 a
-b*sin(a+8*t)+n-41.0443=0
( r. H6 @/ D* `' d/ N\" G0 A# M
-b*sin(a+9*t)+n-41.4190=0

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代码：

!using["fcopt"];
\\" B$ G2 i2 A, m5 }
f(a,b,n,t,y1,y2,y3,y4)=
~+ {) }/ u2 y3 w2 z
{ a4 g% J$ s; j( a
y1=-b*sin(a+6*t)+n-40.4945,\\" n) _8 Z0 ?/ X% A2 e
y2=-b*sin(a+7*t)+n-40.5696,: s% Y- V) ^/ Y0 A* _2 F' s2 U
y3=-b*sin(a+8*t)+n-41.0443,
! E8 K8 a. m% d. N4 O. Z
y4=-b*sin(a+9*t)+n-41.4190* G# Q* s- d0 {* @
};
' g$ }1 I\\" w. x3 n/ j
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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