查看: 10428|回复: 4

求方程组全部解

[复制链接]

字体大小: 正常放大

forcal

45 主题	3 听众	282 积分

升级 91%

TA的每日心情

	难过 2012-8-27 18:22

签到天数: 1 天

[LV.1]初来乍到

电梯直达

1^#

发表于 2011-1-15 17:04 |只看该作者 |倒序浏览

|招呼Ta 关注Ta

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) = 109 R! D' i, c) G/ o$ t
x^2+2*x*y+y^2 = 9

复制代码

代码：

f(x,y,y1,y2)=6 f) V( {( V& K' A5 @* i3 T& _& ^
{
: f1 m+ s( D, H/ T6 a
y1=(x-y)^2-3*(x-y)-10,% R' B% C D( ~6 g
y2=x^2+2*x*y+y^2-9\" L( P, f2 w8 Z4 h\" ]& r
};
' S3 p+ C/ o2 b' o5 E* d2 I
fcopt::solve[HFor("f")];

复制代码

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
+ T5 ?) L6 W3 t4 K
-(x1^3)+x1*x2-exp(-x2) = 0

复制代码

代码：

f(x1,x2,y1,y2)=
4 c0 F6 F: W, [, r: C5 ~. x
{% s6 o2 O5 ~* S ?( d: w
y1=2*x1-x2^2-exp(-x1),, E- T4 |# z4 o
y2=-(x1^3)+x1*x2-exp(-x2)9 T: O) R8 D( S; F, z0 d1 b h
}; [6 P ~1 ~3 s ~
fcopt::solve[HFor("f")];

复制代码

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$ j\" a1 t) i8 d4 ]1 y
-b*sin(a+7*t)+n-40.5696=0% p\" H9 i) c2 y& _+ u
-b*sin(a+8*t)+n-41.0443=0
* a& g1 ~' `7 r1 {, K/ j( G
-b*sin(a+9*t)+n-41.4190=0

复制代码

代码：

!using["fcopt"];
, p\\" \, z: W! J* W' R% U! ^
f(a,b,n,t,y1,y2,y3,y4)=
: b' @9 d+ Z0 B' V
{
( h\\" Z3 o6 h, N/ }0 p. K1 \
y1=-b*sin(a+6*t)+n-40.4945,
: j. t. P3 u9 i5 K1 s
y2=-b*sin(a+7*t)+n-40.5696,0 y9 q: j0 w\\" v# [
y3=-b*sin(a+8*t)+n-41.0443,
( A( I- |8 s& P9 M6 ^) X' H- ]\\" I7 Y
y4=-b*sin(a+9*t)+n-41.4190
; G8 E% Q; ~% Y9 q$ ?
};
; v6 T% O( I0 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.

zan