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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复制代码
-1.402463030422577 3.552713678800501e-015
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4.333755446919994 -2.984279490192421e-013
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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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复制代码
-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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6.
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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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复制代码
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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复制代码
1. 1. 2. 0.
8 z3 m2 }5 n; I$ Y9 N, }
1. 1.5 2.226208734813002 0.
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1. 2. 1. 0.
/ ~0 g* u4 C+ n
1. 2. 1.999999999999999 0.
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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.
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1.5 2.5 1.135767284302283 0.
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1.5 2.5 3.301732715697718 0.
7 y' k, L4 D- W6 w
1.5 3. 1.499999999999999 0.
; G$ \ h3 |. @+ s, n& | _5 |7 z
1.5 3. 3.000000000000004 0.
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2. 1. 1. 0.
; S. l3 s- P+ C6 H1 @! e; K: F
2. 1. 1.999999999999996 0.
' u4 q3 @( K' C) c
2. 1.5 3.30716411545796 0.
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2. 2. 1. 0.
( U8 r+ ^/ D3 T0 q" w2 s, u. {
2. 2. 4.000000000000001 0.
1 ]9 C3 a2 a3 B" ~% |* X
2. 2.5 1.149218940641788 0.
0 C. N4 n5 i5 j/ L) ^5 }9 D5 P6 @$ w
2. 2.5 4.350781059358212 0.
/ [* ~; w0 _' @
2. 3. 1.347582530373998 0.
% U8 ?+ r- H6 h' d: ^- h
2. 3. 4.452417469626003 0.
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2. 3.5 1.611932882193128 0.
w- T4 V8 K' r% q+ v
2. 3.5 4.342612572352325 0.
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2. 4. 1.999999999999999 0.
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2. 4. 3.999999999999999 0.
- e2 i! I" }$ I/ X
2.5 1.5 1.135767284302283 0.
+ _+ s, x, C9 M$ ]0 |# U0 n
2.5 1.5 3.301732715697717 0.
* m9 |* H2 G3 ]) b: w- Y4 B4 b; S' f
2.5 2. 1.149218940641788 0.
; b* G1 h- t% Y* S* m3 z
2.5 2. 4.350781059358212 0.
4 G7 A0 V5 G+ A! F9 Q
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.
+ l% I/ u; |3 D+ f R
2.5 4.5 2.09541126457111 0.
: }! H3 R K& [4 c+ ~7 s
2.5 4.5 5.368874449714605 0.
: Q2 H6 d( q+ ?1 E, i& k& o
2.5 5. 2.5 0.
( ]; C9 t3 I, b9 |$ ?
2.5 5. 4.999999999999998 0.
2 n: |" ~+ ]7 [" {7 j, I6 ?
2.5 5.5 3.292157134434847 0.
0 F9 Z; O2 z4 x& I% t7 R/ X5 H
2.5 5.5 4.176592865565161 0.
7 K' E" k9 Z3 C; p) ]
3. 1.5 1.499999999999999 0.
: l* x5 X3 z) r" }
3. 1.5 3.000000000000004 0.
( E1 }% W* V) {
3. 2. 1.347582530373997 0.
1 S" T$ T" q1 ? k1 U2 s
3. 2. 4.452417469626003 0.
' z; z* V1 N; b9 q0 x
3. 2.5 1.39451558049041 0.
! L+ U) k+ s' q
3. 2.5 5.378211692236861 -5.684341886080802e-014
) h+ L! J4 B2 N0 O! A, s/ S: K
3. 3. 1.5 0.
0 \0 [, c# R4 o
3. 3. 6.000000000000001 0.
1 _6 V# X2 P& @4 M
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.
# x Q% s+ C b, {
3. 4.5 2.021373795560996 0.
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3. 4.5 6.678626204439006 0.
1 y# a! z, j4 A
3. 5. 2.271534568604568 0.
: q- e W$ S7 b/ c9 M
3. 5. 6.603465431395431 0.
2 F4 u: Z$ W: e3 {' f0 Y% [/ b
3. 5.5 2.583225948048685 0.
6 |6 E% o, q9 \% ^1 H& ]
3. 5.5 6.387362287245431 0.
! P+ c/ K1 a3 b8 R- ^8 k8 h f
3. 6. 3. 0.
* k P x0 f! A! ^6 ^5 B/ J8 i
3. 6. 6.000000000000004 0.
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3. 6.5 3.692013773129524 0.
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3. 6.5 5.281670437396789 0.
% [1 b v, L; \% z L% P6 C$ ^5 [& x
3.5 2. 1.611932882193128 0.
7 \* Z2 a( c) n3 Q' z
3.5 2. 4.342612572352325 0.
! M$ J) ~! ]! ]. l% Y: H6 Q! [' X
3.5 2.5 1.577202478216427 0.
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3.5 2.5 5.547797521783575 0.
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3.5 3. 1.641371232518947 0.
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3.5 3. 6.397090305942591 0.
& ?% o. D. k$ p& ^
3.5 3.5 1.75 0.
# I1 q, ^& o) d9 z1 [
3.5 3.5 7. 0.
6 B+ J9 ^/ a, W$ Z" K
3.5 4. 1.889112461544294 0.
K. o t+ d/ E" Z/ u, E
3.5 4. 7.410887538455708 0.
2 ]% i# J! j R2 Z$ h
3.5 4.5 2.055177184113076 0.
m8 Q+ \/ F, G. Y6 B
3.5 4.5 7.663572815886926 0.
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3.5 5. 2.249335186315033 0.
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3.5 5. 7.78007657839085 0.
) d- K7 s" D9 S/ x
3.5 5.5 2.476297487447864 0.
: ~" A6 ^+ [3 [1 b5 I
3.5 5.5 7.773702512552139 0.
6 w, u4 j: R! A! H$ r6 u" g
3.5 6. 2.745300211529267 0.
$ m7 X- ~2 W, ]9 h
3.5 6. 7.649436630576002 0.
4 e& Q* G7 E$ Q+ l; R
3.5 6.5 3.073844239487252 0.
; h0 \, m( B D" m9 H
3.5 6.5 7.401155760512754 0.
( P$ v2 G2 R, B I; e! J# N
3.5 7. 3.5 0.
# T6 x" q, A# Q) d x" Z5 v# p% Q
3.5 7. 7. 0.
) I7 i4 I! p! R7 k- ~% b: \7 q* S( E
3.5 7.5 4.146248634662254 0.
, T' r) V9 ` c& `( _
3.5 7.5 6.331024092610477 0.
8 \3 [# W' V) b
4. 2. 2.000000000000001 0.
% ^9 q, g, r6 v+ F1 P' Y; Y: H
4. 2. 3.999999999999997 0.
1 J, B% F/ Y3 v7 K" h! [
4. 2.5 1.804522914534426 0.
' H l8 P" } F5 N4 \
4. 2.5 5.541630931619421 0.
7 f! _4 U% P) g6 y% u
4. 3. 1.814243197655509 0.
) f( V8 b4 @2 _- R8 S
4. 3. 6.614328230915922 0.
4 j" ~2 l5 x! P! z* W9 ^6 k
4. 3.5 1.889112461544296 0.
u' _, i& s3 o9 g: g) ?. y
4. 3.5 7.410887538455708 0.
+ ~: o% Z# B# i& i
4. 4. 2. 0.
6 }$ C2 e% j h5 t" k! {, W _' m, m
4. 4. 8. 0.
6 G4 v3 H9 I) k
4. 4.5 2.137407852977751 0.
+ L# f" Y' [- I2 I0 _$ y" `7 v
4. 4.5 8.421415676434013 0.
( p" `) T* o0 m4 ^% ?6 o
4. 5. 2.298437881283576 0.
4 l1 _; a6 \4 j9 c6 r
4. 5. 8.701562118716424 0.
; }! E9 W# z2 c
4. 5.5 2.483445700017514 0.
4 t3 Z- k' |2 \9 m% s+ j
4. 5.5 8.858659563140384 0.
( C7 q$ v* x" S( _
4. 6. 2.695165060747996 0.
" c* ?+ y: ]1 o1 U2 E: K" F
4. 6. 8.904834939252009 0.
( L( ^+ s1 L2 H7 A
4. 6.5 2.938916502606828 0.
$ w( Z: L, {; @; H5 {7 Z
4. 6.5 8.846797783107462 0.
- e8 _& w5 Q7 |
4. 7. 3.223865764386256 0.
4 S- a+ G H' t
4. 7. 8.685225144704656 0.
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4. 7.5 3.566387492405245 0.
& K. L( `- V: n2 m$ O/ d9 d: b( x
4. 7.5 8.411873377159974 0.
5 K1 A9 c3 o. X8 G
4. 8. 4.000000000000002 0.
" Q) T7 F2 ]8 O; V$ \0 Z
4. 8. 7.999999999999995 0.
7 v% g5 Z% F5 U2 ^6 t7 \
4. 8.5 4.618832614156824 0.
% |( u- C- c' k" B& E
4. 8.5 7.361167385843178 0.
- _' g0 u) I; l7 r" J" w' u: K
4.5 2.5 2.095411264571111 0.
+ ]. J8 }) ?( E$ g; J; d
4.5 2.5 5.368874449714603 0.
) z/ d$ B, u) B: }
4.5 3. 2.021373795560996 0.
( u) v5 F+ O* N2 U! O; v* O
4.5 3. 6.678626204439004 0.
: N+ i- v6 B: n6 ^
4.5 3.5 2.055177184113076 0.
u, K" e8 {" F7 @: ^ D5 t6 I
4.5 3.5 7.663572815886925 0.
& j3 z' Z, x& k* a# d3 `
4.5 4. 2.137407852977751 0.
6 i( S s! ^) Q. ^/ Y7 X
4.5 4. 8.421415676434013 0.
0 t$ O4 U' F# n: b: G8 D
4.5 4.5 2.25 0.
2 Y- M6 V0 f3 n& G1 H$ n2 d' V
4.5 4.5 9. 0.
! J& f2 u9 X1 u- u0 h Q
4.5 5. 2.386074153449669 0.
) e8 M3 ?$ l$ W3 a9 @0 S
4.5 5. 9.429715320234545 0.
6 _2 Y8 w$ I$ l/ h1 q
4.5 5.5 2.543211996792689 0.
: N) m: z+ g) F" W! J$ E
4.5 5.5 9.731788003207315 0.
7 W% z: r9 V! H
4.5 6. 2.721364796483262 0.
) g! A* k g, }6 o+ P: D3 o" Y6 Q2 x
4.5 6. 9.92149234637388 0.
5 L+ e* _; k, r) W# e
4.5 6.5 2.922185008548723 0.
! |5 o8 [3 g F' f' ^
4.5 7. 3.149000780229209 0.
I2 ?8 F# e- z
4.5 7.5 3.407301852906852 0.
2 A0 g1 c, T" {9 L; y
4.5 7.5 9.905198147093154 0.
, A, X5 S- |4 {; S0 N. V
4.5 8. 3.705986018674346 0.
9 T* G* t$ g3 M2 H5 _
4.5 8. 9.714013981325651 0.
: W1 v6 C" x- @) h3 w' B% G
4.5 8.5 4.060312832145664 0.
* B% G& p. z, l# J9 U6 i9 |0 V( j
4.5 8.5 9.420456398623564 0.
7 P; _3 r1 n( H
4.5 9. 4.5 0.
9 `, k: g& \# I' c" h0 w* y7 @
4.5 9. 9.000000000000005 0.
9 r5 d$ F8 v$ f* v4 R% J3 w
4.5 9.5 5.100326474258905 0.
- C# v% x* A L) \* r- O; n6 w
4.5 9.5 8.381816382883958 0.
" v V4 D: O3 u' t1 i) b# V
4.5 10. 6.402193416458818 0.
4 R4 y& t$ @& F8 I' p# e* r
4.5 10. 7.028841066299776 4.547473508864641e-013
h# c+ y- Z8 b4 P9 T) s
5. 2.5 2.499999999999997 0.
}: ?' P7 ~% q+ e: r/ N# x
5. 2.5 4.999999999999998 0.
! q% ?. ^7 z- C, C/ y: c
5. 3. 2.271534568604568 0.
# z- Q5 \4 A, @) e
5. 3. 6.603465431395431 0.
' m$ i5 m. e* s# z c7 z* Z+ f
5. 3.5 2.249335186315031 0.
7 p) } W2 P% A" Q% b
5. 3.5 7.78007657839085 0.
( X8 j, X. \- u, w9 Z
5. 4. 2.298437881283576 0.
7 r @$ Y: l! b. ?
5. 4. 8.701562118716424 0.
) F1 t% c0 v2 q' ], _
5. 4.5 2.386074153449668 0.
@0 v' r' y. ~
5. 4.5 9.429715320234545 0.
b4 N1 K! u. W4 C
5. 5. 2.5 0.
+ `7 F1 w) x! G( X7 W
5. 5. 9.999999999999993 0.
& o% s w3 O7 ~3 @
5. 5.5 2.635001373375082 0.
# W) T; B& ~ Z4 I
5. 6. 2.789031160980821 0.
3 q }1 u+ k* p6 {9 J
5. 6.5 2.961827475384206 0.
$ R1 |) i9 I0 u4 |( g9 m2 U2 H( Y/ a
5. 7. 3.154404956432853 0.
) H7 f% C- @) {+ i! r% C
5. 7.5 3.368956325934996 0.
7 E: m% }% X- |
5. 8. 3.609045829068848 0.
# ~) _ S" Q* d" k# t9 K& o9 p
5. 8.5 3.880150804346176 0.
$ f( J2 |) F6 I# f3 W* ~
5. 9. 4.190822529142221 0.
: w" k' B" m' B
5. 9.5 4.555265557190405 0.
: O0 O: |' P% l8 X8 z: J( o, ~' x4 K
5. 10. 4.999999999999997 0.
% O5 J L1 c1 q1 t" v9 Y o. V* t
5. 10. 10. 0.
4 H7 y7 D5 b+ O3 P! m
5.5 2.5 3.292157134434847 0.
9 I& p+ P" g7 h% h% q8 ]( C
5.5 2.5 4.176592865565167 0.
3 [ ]* R/ @- q1 W8 {2 n
5.5 3. 2.583225948048685 0.
( `3 W( r, I7 H4 m6 H- n+ \- [
5.5 3. 6.387362287245431 0.
- p3 Z6 b& p! l2 E3 d" d- f
5.5 3.5 2.476297487447864 0.
& P9 P1 s3 v/ m" r ^8 |) A
5.5 3.5 7.773702512552133 0.
# [' N8 Y9 _9 `: P
5.5 4. 2.483445700017513 0.
) K- d7 a$ \, b* Z8 L/ C
5.5 4. 8.858659563140384 0.
" `- l; H3 q3 I0 V+ U& _/ t/ C
5.5 4.5 2.543211996792689 0.
; _# r4 `' m$ a( f# j5 \
5.5 4.5 9.731788003207313 0.
7 k) w/ C, B r* T* n$ ?
5.5 5. 2.635001373375081 0.
, o- o' l5 J& O/ ~) U
5.5 5.5 2.749999999999999 0.
) u: j# e' M+ D) V7 {8 p8 X: ~
5.5 6. 2.884119329429114 0.
8 j+ V& u9 {& b; q& r# |
5.5 6.5 3.035604845963815 0.
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5.5 7. 3.204083523261024 0.
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5.5 7.5 3.390173520212255 0.
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5.5 8. 3.595369279178611 0.
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5.5 8.5 3.822111065390602 0.
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5.5 9. 4.074041860443781 0.
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5.5 10. 4.677786777828594 0.
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6. 3.5 2.745300211529266 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. 8.904834939252005 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. 8.5 3.826272163735232 0.
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6. 10. 4.543069137209136 0.
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6.5 3. 3.692013773129525 0.
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6.5 3. 5.281670437396786 0.
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6.5 3.5 3.073844239487251 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 4.5 2.922185008548724 0.
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6.5 5. 2.961827475384206 0.
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6.5 5.5 3.035604845963813 0.
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6.5 6. 3.133381054885422 0.
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6.5 6.5 3.25 0.
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6.5 7. 3.382753896487015 0.
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6.5 7.5 3.530313399868478 0.
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6.5 8. 3.692229149874506 0.
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6.5 8.5 3.868693727795107 0.
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6.5 9. 4.060441442861232 0.
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6.5 9.5 4.268736592110795 0.
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6.5 10. 4.495435882583067 0.
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7. 3.5 3.500000000000001 0.
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7. 4.5 3.14900078022921 0.
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7. 5. 3.154404956432853 0.
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7. 5.5 3.204083523261021 0.
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7. 6.5 3.382753896487015 0.
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7. 7. 3.5 0.
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7. 8. 3.778224923088592 0.
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7. 8.5 3.93757894317775 0.
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7. 9.5 4.297071556823225 0.
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7.5 4. 3.566387492405242 0.
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7.5 4. 8.411873377159969 0.
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7.5 4.5 9.905198147093145 0.
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7.5 5. 3.368956325934993 0.
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7.5 5.5 3.390173520212255 0.
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7.5 6. 3.447656821925362 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 8. 3.881745401569135 0.
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7.5 8.5 4.026409271112575 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. 4. 4.000000000000002 0.
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8. 5. 3.609045829068849 0.
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8. 6. 3.628486395311016 0.
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8. 6.5 3.692229149874504 0.
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8. 7. 3.77822492308859 0.
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8. 7.5 3.881745401569135 0.
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8. 9. 4.274815705955503 0.
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8. 9.5 4.430014593640446 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 9.5 4.523405444098199 0.
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8.5 10. 4.676893219133843 0.
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9. 4.5 4.499999999999997 0.
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9. 4.5 9.000000000000009 0.
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9. 5.5 4.074041860443781 0.
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9. 6. 4.042747591121993 0.
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9. 6.5 4.060441442861232 -4.547473508864641e-013
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9. 7.5 4.183546741471229 0.
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9.5 5. 4.555265557190406 0.
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9.5 5.5 4.356541541407055 0.
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9.5 6. 4.280465505524228 0.
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9.5 6.5 4.268736592110795 0.
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9.5 7. 4.297071556823221 0.
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9.5 7.5 4.353157178625922 0.
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9.5 8.5 4.523405444098199 0.
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9.5 9. 4.630645298521484 0.
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9.5 9.5 4.75 0.
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9.5 10. 4.880354401607461 0.
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10. 4.5 6.402193416458861 0.
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10. 4.5 7.028841066299777 4.547473508864641e-013
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10. 5. 4.999999999999997 0.
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10. 5.5 4.67778677782859 0.
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10. 6. 4.543069137209136 0.
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10. 6.5 4.495435882583067 0.
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10. 7. 4.498670372630065 0.
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10. 7.5 4.535607994138771 0.
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10. 8. 4.596875762567152 0.
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10. 8.5 4.676893219133843 0.
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10. 9. 4.772148306899336 0.
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10. 10. 5. 0.
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330.
$ `# L* ^/ I4 `6 l- p+ k
! d) a3 L P! n. y9 W
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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