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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) = 10
& P& ^0 Y+ R* d9 H8 D! `8 A; |% Q\" m
x^2+2*x*y+y^2 = 9

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代码：

f(x,y,y1,y2)=' T$ F6 Z2 }\" R- Y
{
; H) }9 [1 Z7 E9 Q
y1=(x-y)^2-3*(x-y)-10,
& a. A. }2 ~+ F) O3 T/ V5 \
y2=x^2+2*x*y+y^2-95 G3 m+ j, y* }7 m
};
% ^. [& H/ ~0 S4 n& f
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% U! Y& J. ~, r- |) b8 h
-(x1^3)+x1*x2-exp(-x2) = 0

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代码：

f(x1,x2,y1,y2)=! a; B$ X5 m) |( i8 I# s
{2 p/ b5 I9 P3 r7 d! ^
y1=2*x1-x2^2-exp(-x1),8 [6 W8 ?+ ~$ \ m# C x) O7 F4 v
y2=-(x1^3)+x1*x2-exp(-x2)1 {/ ^( J8 j5 T) i: f5 v
};5 c4 ~- s% }. B, j6 A1 a7 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+ V5 y\" F+ K! p* Z/ A; y
-b*sin(a+7*t)+n-40.5696=0
' N1 _$ b A8 o0 w/ w* S
-b*sin(a+8*t)+n-41.0443=0
( F5 a' z1 b) I8 \7 k6 N& H
-b*sin(a+9*t)+n-41.4190=0

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代码：

!using["fcopt"];9 T3 f3 u0 l1 ?0 v$ ] ^
f(a,b,n,t,y1,y2,y3,y4)=
: h8 \+ {, h$ R; R! v
{
\\" V2 S. M: t* S1 I5 b
y1=-b*sin(a+6*t)+n-40.4945,. F9 t- c8 _$ m
y2=-b*sin(a+7*t)+n-40.5696,8 U M3 Z. x- l9 A
y3=-b*sin(a+8*t)+n-41.0443,* h8 S* @' v$ }) J. ]5 ]
y4=-b*sin(a+9*t)+n-41.4190\\" s4 U- U9 p s O& x3 J9 J; e
};3 o1 z5 k* _% N9 y3 F- q
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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