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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
\" ~) H9 `: w2 }
x^2+2*x*y+y^2 = 9

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代码：

f(x,y,y1,y2)=- }* Q$ v\" \) m T4 l$ ]
{
# D B/ S! h! q* I; t
y1=(x-y)^2-3*(x-y)-10,
2 K9 g i( w$ i+ d7 Y
y2=x^2+2*x*y+y^2-9$ Z& l( x/ A* ~ [5 I( ^7 ~0 ~9 ^
};
9 R4 H+ V8 @1 M3 _8 P0 S7 |
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 t, k* C) K( V
-(x1^3)+x1*x2-exp(-x2) = 0

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代码：

f(x1,x2,y1,y2)=
6 M5 S9 K6 R# l\" @\" B5 f
{. ?5 L6 V' W3 E
y1=2*x1-x2^2-exp(-x1),
, T+ s/ o8 X2 y9 V0 U. {6 U) b
y2=-(x1^3)+x1*x2-exp(-x2)
0 p# Z7 D/ L) e2 J( }6 F8 ]! p- c
};% r5 `$ O8 v+ M0 u3 G' V
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=09 t& B( }! ~6 i7 F$ S
-b*sin(a+7*t)+n-40.5696=0) c3 W- v: T; j6 D. L4 r$ v
-b*sin(a+8*t)+n-41.0443=0
9 w$ R: k0 H' _( E$ y0 D% U+ V: z8 k
-b*sin(a+9*t)+n-41.4190=0

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代码：

!using["fcopt"];: M9 ]# q\\" l3 `' }. h
f(a,b,n,t,y1,y2,y3,y4)=% K6 j. c) C0 S8 R3 o
{
' a, {( j. g7 n- d* V( w; s
y1=-b*sin(a+6*t)+n-40.4945,
y2=-b*sin(a+7*t)+n-40.5696,
+ H, t, h3 a+ X4 ?' o% A( P8 u
y3=-b*sin(a+8*t)+n-41.0443,+ |7 E/ G: Z3 ?: Q
y4=-b*sin(a+9*t)+n-41.4190
. K+ `' t( q\\" e- h8 h
};5 E9 U+ R7 ]% ~# _/ Z
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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