数学建模社区-数学中国
标题:
求方程的全部解
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作者:
forcal
时间:
2011-1-9 11:24
标题:
求方程的全部解
Forcal优化库FcOpt中新增函数fcopt::isolve，试图求解方程的全部解。正在测试修改，请大家多提意见。
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参考：
http://www.forcal.net/sysm/forcal9/fchtm/fcopt.htm
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例子1：求方程的全部解：f(x)=x^6-5*x^5+3*x^4+x^3-7*x^2+7*x-20;
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f(x)=x^6-5*x^5+3*x^4+x^3-7*x^2+7*x-20;
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fcopt::isolve[HFor("f")];
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-1.402463030422577 3.552713678800501e-015
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4.333755446919994 -2.984279490192421e-013
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2.
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说明：前面2行，每行中前一个数是解，后一个数是误差。最后的2说明有两个解。
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作者:
wgxzy_1015
时间:
2011-1-9 11:34
谢谢................
作者:
forcal
时间:
2011-1-9 11:35
例子2：求方程的全部解（当y为1时，求x的全部解）：f(x,y)=(x^2+y^2)^3-36*(x^2-y^2)^2;
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!using("fcopt");
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f(x,y)=(x^2+y^2)^3-36*(x^2-y^2)^2;
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isolve[HFor("f"),optpara,1];
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-5.530393535267971 1. -3.637978807091713e-012
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-1.32936186218296 1. -7.105427357601002e-015
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-0.8047014256478491 1. -1.776356839400251e-015
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0.8047014256478491 1. -1.776356839400251e-015
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1.32936186218296 1. -7.105427357601002e-015
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5.530393535267971 1. -3.637978807091713e-012
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作者:
forcal
时间:
2011-1-9 11:37
例子3：求方程的全部解（x取1.1，求y的全部解）：f(x,y)=(x^2+y^2)^3-36*(x^2-y^2)^2;
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!using("fcopt");
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f(x,y)=(x^2+y^2)^3-36*(x^2-y^2)^2;
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isolve[HFor("f"), optthis,1, optpara,1.1];
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1.1 -5.413569621185505 0.
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1.1 -1.52029233271966 0.
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1.1 -0.8671650244626143 8.881784197001252e-016
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1.1 0.8671650244626143 8.881784197001252e-016
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1.1 1.52029233271966 0.
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1.1 5.413569621185504 0.
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6.
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作者:
forcal
时间:
2011-1-9 12:08
例子4：为隐函数绘图准备数据（x取1～10，y取1～10，z取1～10）：f(x,y,z)=(x+y+z)*(x*y+x*z+y*z)-10*x*y*z;
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!using("fcopt");
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f(x,y,z)=(x+y+z)*(x*y+x*z+y*z)-10*x*y*z;
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init(:x,y,i)={
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i=0,
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x=1,while{x<=10,
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y=1,while{y<=10,
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i=i+isolve[HFor("f"), optrange,1,10, optthis,2, optpara,x,y],
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y=y+0.5
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},
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x=x+0.5
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},
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i
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};
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1. 1. 2. 0.
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1. 1.5 2.226208734813002 0.
' @$ ?* s7 c+ D5 B3 z# Z l
1. 2. 1. 0.
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1. 2. 1.999999999999999 0.
. [" v+ c5 U4 S5 z# K/ V0 z
1.5 1. 2.226208734813 0.
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1.5 1.5 3. 0.
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1.5 2. 3.30716411545796 0.
% f+ ?1 b8 c" v* u
1.5 2.5 1.135767284302283 0.
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1.5 2.5 3.301732715697718 0.
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1.5 3. 1.499999999999999 0.
7 h8 T/ h: D: w7 o1 ]) R) V1 J6 r% `2 S
1.5 3. 3.000000000000004 0.
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2. 1. 1. 0.
2 |% N+ n ~, O7 l5 n) L' G
2. 1. 1.999999999999996 0.
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2. 1.5 3.30716411545796 0.
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2. 2. 1. 0.
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2. 2. 4.000000000000001 0.
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2. 2.5 1.149218940641788 0.
3 N1 U6 n0 n$ a
2. 2.5 4.350781059358212 0.
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2. 3. 1.347582530373998 0.
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2. 3. 4.452417469626003 0.
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2. 3.5 1.611932882193128 0.
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2. 3.5 4.342612572352325 0.
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2. 4. 1.999999999999999 0.
$ M, i1 G w0 H' M; F$ R/ w
2. 4. 3.999999999999999 0.
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2.5 1.5 1.135767284302283 0.
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2.5 1.5 3.301732715697717 0.
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2.5 2. 1.149218940641788 0.
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2.5 2. 4.350781059358212 0.
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2.5 2.5 1.25 0.
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2.5 2.5 4.999999999999998 0.
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2.5 3. 1.39451558049041 0.
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2.5 3. 5.378211692236864 5.684341886080802e-014
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2.5 3.5 1.577202478216426 0.
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2.5 3.5 5.547797521783575 0.
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2.5 4. 1.804522914534426 0.
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2.5 4. 5.541630931619419 0.
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2.5 4.5 2.09541126457111 0.
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2.5 4.5 5.368874449714605 0.
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2.5 5. 2.5 0.
- u) [$ | L% C% K! Y7 O! [2 u
2.5 5. 4.999999999999998 0.
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2.5 5.5 3.292157134434847 0.
& N% p( k! z5 k, ^- z- p9 p- U
2.5 5.5 4.176592865565161 0.
/ j ~2 k1 A& o, ?+ `. O
3. 1.5 1.499999999999999 0.
! z# i& t2 ?5 G E( O
3. 1.5 3.000000000000004 0.
7 H8 d; E; y, `! C
3. 2. 1.347582530373997 0.
7 J# [+ m# I5 ?
3. 2. 4.452417469626003 0.
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3. 2.5 1.39451558049041 0.
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3. 2.5 5.378211692236861 -5.684341886080802e-014
+ ^% N& M6 Y W p
3. 3. 1.5 0.
- E1 y8 M! z* ]
3. 3. 6.000000000000001 0.
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3. 3.5 1.641371232518948 0.
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3. 3.5 6.397090305942587 0.
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3. 4. 1.814243197655509 -2.842170943040401e-014
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3. 4. 6.614328230915922 0.
* m8 {+ @. c. }
3. 4.5 2.021373795560996 0.
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3. 4.5 6.678626204439006 0.
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3. 5. 2.271534568604568 0.
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3. 5. 6.603465431395431 0.
" p: w; q+ J* d, |; w+ H; B
3. 5.5 2.583225948048685 0.
2 P1 J8 W+ Z: R# {" v
3. 5.5 6.387362287245431 0.
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3. 6. 3. 0.
* @; d( `. D1 m
3. 6. 6.000000000000004 0.
7 s9 Y m( D* X, A. I8 ?
3. 6.5 3.692013773129524 0.
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3. 6.5 5.281670437396789 0.
3 j5 ]0 M1 w+ a- Q
3.5 2. 1.611932882193128 0.
8 j" e/ ?; r2 u4 W. E+ z6 B
3.5 2. 4.342612572352325 0.
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3.5 2.5 1.577202478216427 0.
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3.5 2.5 5.547797521783575 0.
3 u- a# m3 p: Y: X; U
3.5 3. 1.641371232518947 0.
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3.5 3. 6.397090305942591 0.
8 g" g: l: ~" n7 i9 V) V
3.5 3.5 1.75 0.
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3.5 3.5 7. 0.
}9 C. g4 _1 C2 \1 u/ V
3.5 4. 1.889112461544294 0.
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3.5 4. 7.410887538455708 0.
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3.5 4.5 2.055177184113076 0.
0 @" b) q, c! P' \. \! Q7 b
3.5 4.5 7.663572815886926 0.
6 q% [# I7 {0 w8 G2 Q2 l, ^: @( `
3.5 5. 2.249335186315033 0.
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3.5 5. 7.78007657839085 0.
3 I. b" a- U% p2 W$ |' V, B J
3.5 5.5 2.476297487447864 0.
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3.5 5.5 7.773702512552139 0.
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3.5 6. 2.745300211529267 0.
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3.5 6. 7.649436630576002 0.
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3.5 6.5 3.073844239487252 0.
' _, E( O2 |2 { B0 k
3.5 6.5 7.401155760512754 0.
4 V8 h6 C6 k% n I" w- c" q
3.5 7. 3.5 0.
$ ~2 A( Q' x4 E- m) s, w% G0 \
3.5 7. 7. 0.
2 F: J3 Q6 C/ T6 W2 o* J8 |+ c
3.5 7.5 4.146248634662254 0.
4 m1 o) D9 }! ~; q) |; E
3.5 7.5 6.331024092610477 0.
& b: L9 G; J8 j6 P; V9 M
4. 2. 2.000000000000001 0.
2 G9 x: q( b5 G0 h0 E0 t/ P
4. 2. 3.999999999999997 0.
3 n# O' {7 C2 ~& m( v. t
4. 2.5 1.804522914534426 0.
- j# U6 L1 O4 P: z1 |
4. 2.5 5.541630931619421 0.
" R: L; }- e2 l4 S& F
4. 3. 1.814243197655509 0.
3 r4 c+ c! F& L) Y9 p
4. 3. 6.614328230915922 0.
; C3 T) }. _- a: g6 f9 w" k/ y
4. 3.5 1.889112461544296 0.
0 [! H2 E7 h' {) n q/ [, s$ o
4. 3.5 7.410887538455708 0.
9 T2 A7 e! W" l7 W: T7 h: {# B$ ]
4. 4. 2. 0.
) @- C, U2 b( H8 T+ |
4. 4. 8. 0.
( O, ^& K; p M/ o" D
4. 4.5 2.137407852977751 0.
y- p* n/ x0 y& j, j
4. 4.5 8.421415676434013 0.
/ u# } }9 o8 f+ I
4. 5. 2.298437881283576 0.
5 I: a* k6 P& v
4. 5. 8.701562118716424 0.
+ p# I3 D1 _3 X5 ] w
4. 5.5 2.483445700017514 0.
7 q& X S: d. ]2 }. c- K4 L) G! P
4. 5.5 8.858659563140384 0.
9 h- k! U0 v9 K) s( ?, e. E
4. 6. 2.695165060747996 0.
, t! W/ I' L0 a" w) t7 S8 C
4. 6. 8.904834939252009 0.
/ U2 C* t, m8 w" }/ \, H- H
4. 6.5 2.938916502606828 0.
5 Z; U+ X$ t: T9 D1 l4 C
4. 6.5 8.846797783107462 0.
0 t; ^: q9 z; @5 h( t6 A% H y
4. 7. 3.223865764386256 0.
9 ^* W: F8 p8 q9 k
4. 7. 8.685225144704656 0.
1 d* D, R* x& U
4. 7.5 3.566387492405245 0.
4 V, J1 c( F9 ^* D
4. 7.5 8.411873377159974 0.
* [; d- |( u2 o |5 I- \# n
4. 8. 4.000000000000002 0.
+ r* ^! B9 R) o4 b
4. 8. 7.999999999999995 0.
" {: q+ o' I0 J5 e
4. 8.5 4.618832614156824 0.
+ r* ]- l# V. h" V3 }, Z
4. 8.5 7.361167385843178 0.
6 C5 P# C1 ]) u
4.5 2.5 2.095411264571111 0.
& N9 e; j K) o' |
4.5 2.5 5.368874449714603 0.
% L3 \# Q$ d6 k( U( S6 _
4.5 3. 2.021373795560996 0.
& ] p; J! K$ Y/ M! _) v' Z) D
4.5 3. 6.678626204439004 0.
$ _/ H" q/ N" X/ y; W8 F
4.5 3.5 2.055177184113076 0.
: u, H! w4 e$ [0 r: M' r! M# C
4.5 3.5 7.663572815886925 0.
: ]6 ^) @" n6 X0 |+ _( x. |. ^6 t
4.5 4. 2.137407852977751 0.
- ^' \ e, k: l0 k% q/ A* Q* N6 N
4.5 4. 8.421415676434013 0.
4 k7 b3 h/ M& ]& u5 v# @: P7 {
4.5 4.5 2.25 0.
$ I, i1 e x$ t9 E$ L* f
4.5 4.5 9. 0.
, W' j X6 N1 Y2 R) \1 [
4.5 5. 2.386074153449669 0.
$ H) h. J! f1 l& V4 |/ b% {
4.5 5. 9.429715320234545 0.
8 b" V4 W* e& T/ @; {
4.5 5.5 2.543211996792689 0.
- A* E3 Y# o! j
4.5 5.5 9.731788003207315 0.
; Y$ J, i" M' F* V
4.5 6. 2.721364796483262 0.
8 ~. V; _4 c9 J
4.5 6. 9.92149234637388 0.
5 F y, a" n4 z6 `- L, y9 A
4.5 6.5 2.922185008548723 0.
! u- ]( g% W0 C" ]. f% j
4.5 7. 3.149000780229209 0.
% Y& U ]+ Q7 W
4.5 7.5 3.407301852906852 0.
3 `, s0 ~; B1 L. T; O0 N
4.5 7.5 9.905198147093154 0.
5 x% W- z/ B5 p& l) [; K( y0 w
4.5 8. 3.705986018674346 0.
' K) Y8 _ v1 b# `! @' n- p
4.5 8. 9.714013981325651 0.
% w5 g; A% E* `
4.5 8.5 4.060312832145664 0.
5 N. V8 k' h* y9 X2 P
4.5 8.5 9.420456398623564 0.
7 K. w, W& Y! K. c
4.5 9. 4.5 0.
) u( j/ @( X) }/ W2 Z7 h
4.5 9. 9.000000000000005 0.
+ `" J, P9 ` s
4.5 9.5 5.100326474258905 0.
@0 o! Z0 u& y0 L
4.5 9.5 8.381816382883958 0.
4 P3 Q" f$ a; P2 b/ T/ ^3 _# R
4.5 10. 6.402193416458818 0.
) h* L' S7 J( ~$ H9 s0 O! C
4.5 10. 7.028841066299776 4.547473508864641e-013
4 A6 z# k, [& A3 c
5. 2.5 2.499999999999997 0.
9 h: y/ [: V: O1 |" \% r h
5. 2.5 4.999999999999998 0.
( `" E/ B7 Q8 N3 G" C! E. ? z
5. 3. 2.271534568604568 0.
& X a, e7 v/ ~4 ~
5. 3. 6.603465431395431 0.
/ e5 n" V4 T/ O6 R L1 o" }
5. 3.5 2.249335186315031 0.
5 \* R) L. K$ V1 ]! R" \
5. 3.5 7.78007657839085 0.
& |9 S+ u4 W# d2 }' R
5. 4. 2.298437881283576 0.
: S& A! a$ W: M% L
5. 4. 8.701562118716424 0.
, G7 N( M$ z A& G D
5. 4.5 2.386074153449668 0.
9 I& A3 [: X' V z$ w# a8 k0 V
5. 4.5 9.429715320234545 0.
% f( |9 Z7 q% \1 Q5 S* Y; g
5. 5. 2.5 0.
- ~5 Y- z& Y' d2 |9 X* `
5. 5. 9.999999999999993 0.
: E0 E* p* U K5 X3 B7 o9 f' n
5. 5.5 2.635001373375082 0.
4 m9 S7 f2 L- f c
5. 6. 2.789031160980821 0.
9 n- J4 w& K4 f: _* F6 t$ ?
5. 6.5 2.961827475384206 0.
, V1 _; v2 J' l5 B! e
5. 7. 3.154404956432853 0.
! p( U4 T2 z% D* z$ V" K- q, z% M
5. 7.5 3.368956325934996 0.
0 I) H9 e3 [% j
5. 8. 3.609045829068848 0.
( y1 D; N8 L% I9 G# W7 ^/ S
5. 8.5 3.880150804346176 0.
# g; m1 z2 J& V3 g
5. 9. 4.190822529142221 0.
' Q: e" f2 E& m1 V2 ]
5. 9.5 4.555265557190405 0.
% r5 _) \0 d# q/ x6 w1 v
5. 10. 4.999999999999997 0.
% G" T3 y! s# ]5 l( Y' w" O8 e$ L
5. 10. 10. 0.
7 y7 b# ^. Z1 j2 r
5.5 2.5 3.292157134434847 0.
. Y5 n- f3 v! Z
5.5 2.5 4.176592865565167 0.
# e7 z7 M, L. d! k* @0 B( q
5.5 3. 2.583225948048685 0.
. }1 N' L: G1 T, Q( V' J- X
5.5 3. 6.387362287245431 0.
7 i* R; F7 t( z
5.5 3.5 2.476297487447864 0.
: w k) b9 `& Z4 _7 Z
5.5 3.5 7.773702512552133 0.
9 F; ~5 D, P$ s) \/ e/ g( W1 y
5.5 4. 2.483445700017513 0.
: s3 B, ?9 _ ^ a
5.5 4. 8.858659563140384 0.
4 ]3 e3 W# q; w
5.5 4.5 2.543211996792689 0.
! @' i( T1 `9 C$ a0 J$ w- y
5.5 4.5 9.731788003207313 0.
: W$ b! M$ t. w; f& N7 C( j
5.5 5. 2.635001373375081 0.
- R! E) a1 ]3 I/ _, k. C
5.5 5.5 2.749999999999999 0.
! v4 v8 f3 b9 _' V
5.5 6. 2.884119329429114 0.
2 X: G# |6 E0 i9 T
5.5 6.5 3.035604845963815 0.
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6. 3.5 7.649436630575996 0.
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6. 4. 2.695165060747994 0.
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6. 4.5 2.721364796483262 0.
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6. 4.5 9.92149234637388 0.
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6. 7.5 3.447656821925362 2.273736754432321e-013
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6. 8. 3.628486395311015 0.
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6.5 3.5 7.401155760512746 0.
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6.5 4. 2.938916502606827 0.
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6.5 6.5 3.25 0.
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6.5 9.5 4.268736592110795 0.
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7.5 4.5 9.905198147093145 0.
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7.5 5.5 3.390173520212255 0.
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7.5 6.5 3.530313399868477 0.
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7.5 7. 3.632214429249618 0.
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7.5 7.5 3.750000000000003 0.
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7.5 9. 4.183546741471229 0.
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7.5 9.5 4.353157178625922 0.
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7.5 10. 4.535607994138771 0.
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8. 10. 4.596875762567152 0.
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8.5 4. 4.618832614156823 0.
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8.5 4. 7.361167385843172 0.
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8.5 4.5 4.060312832145666 0.
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8.5 4.5 9.420456398623564 0.
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8.5 5.5 3.822111065390603 0.
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8.5 6. 3.826272163735231 0.
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8.5 8. 4.13133382165941 0.
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9. 4.5 4.499999999999997 0.
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9. 7.5 4.183546741471229 0.
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9.5 9.5 4.75 0.
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10. 4.5 7.028841066299777 4.547473508864641e-013
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10. 9.5 4.880354401607461 0.
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10. 10. 5. 0.
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330.
+ F, ?4 h* O* @. V. w3 l
) }* v8 g* F# Y9 P/ d
作者:
forcal
时间:
2011-1-11 06:30
isolve函数正在测试阶段，请大家赐题测试，以帮助改进。方程中可包含任意内容，包括积分、微分、微分方程等等。谢谢！
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