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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 |正序浏览

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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
9 K6 d9 v/ ]$ Z9 }, L
x^2+2*x*y+y^2 = 9

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代码：

f(x,y,y1,y2)=. Q. K% K0 s6 U6 T* e4 P$ k5 r
{, Q& w7 D, Y3 h: p3 k* Z
y1=(x-y)^2-3*(x-y)-10,
% h Y* \% u' C- [ p, P
y2=x^2+2*x*y+y^2-9( h/ c3 B4 \ @6 g
};5 j. g9 Q1 y/ P5 O) F$ w
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: @$ h) I\" n7 e$ h }3 q6 W
-(x1^3)+x1*x2-exp(-x2) = 0

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代码：

f(x1,x2,y1,y2)=* n, B2 T! T5 P& r U$ T( |, l
{% a, }6 Y9 b- X. V5 f
y1=2*x1-x2^2-exp(-x1),9 G: `7 P, r8 b3 l6 s( i [1 n9 C
y2=-(x1^3)+x1*x2-exp(-x2)- F$ D5 R- S4 d: u
};
9 c( H m) R1 s$ G3 N5 ]2 P0 s
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
) |2 e8 h9 x4 \1 k2 s
-b*sin(a+7*t)+n-40.5696=0
4 k O# E3 F6 H8 h# }
-b*sin(a+8*t)+n-41.0443=0& B$ f' X1 ~5 ]5 _0 q o! a
-b*sin(a+9*t)+n-41.4190=0

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代码：

!using["fcopt"];, O) ?' j\\" G\\" k& ?9 ~/ C! t
f(a,b,n,t,y1,y2,y3,y4)=
' Y; o; s* ^7 j9 B% X2 V6 T c
{
7 Q+ ], D- i3 b- B0 X
y1=-b*sin(a+6*t)+n-40.4945, ^0 I6 F6 t, D; R; Z\\" w2 \
y2=-b*sin(a+7*t)+n-40.5696,
; z; i( B; X; @# S9 {7 m
y3=-b*sin(a+8*t)+n-41.0443,
# J( q. f, N$ Q# u5 W
y4=-b*sin(a+9*t)+n-41.4190
/ Y/ G& N2 Z7 d* h9 L
};
6 x& { d; S4 Y7 N/ I) [: 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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