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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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.
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1. 2. 1. 0.
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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.
2 I( B. i! [- U0 T
1.5 2.5 1.135767284302283 0.
% I4 g9 K. l0 q, b4 J; m
1.5 2.5 3.301732715697718 0.
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1.5 3. 1.499999999999999 0.
9 x- ~3 b5 d% `' A5 Z3 f
1.5 3. 3.000000000000004 0.
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2. 1. 1. 0.
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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.
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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.
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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.
1 b" {) e8 v& U% A* N X$ q* d* u
2.5 2. 4.350781059358212 0.
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2.5 2.5 1.25 0.
4 X/ |! C) e5 u; s! f! G
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.
8 m7 q: s5 I% i5 h% g* Z" N& S
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.
2 }; x0 d% N7 O0 `& O& G* h" C* M
2.5 4.5 5.368874449714605 0.
& q: F2 |! \$ |0 x/ J
2.5 5. 2.5 0.
5 F: `( {& S0 m9 K4 e( ^0 ?2 b
2.5 5. 4.999999999999998 0.
4 U5 M: g4 k( }# K) |$ G
2.5 5.5 3.292157134434847 0.
6 b, x* c% ^, s
2.5 5.5 4.176592865565161 0.
, F& g/ T9 L ~! \- {3 S
3. 1.5 1.499999999999999 0.
5 R( S! `' ?( n- M# u% a& R1 D) }
3. 1.5 3.000000000000004 0.
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3. 2. 1.347582530373997 0.
6 v2 H$ i- F" [- ?% E
3. 2. 4.452417469626003 0.
2 V8 N. g8 f. s2 e: [
3. 2.5 1.39451558049041 0.
3 a- [# A' ^, p \5 ^1 }
3. 2.5 5.378211692236861 -5.684341886080802e-014
- i+ w i1 u' F6 D
3. 3. 1.5 0.
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3. 3. 6.000000000000001 0.
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3. 3.5 1.641371232518948 0.
1 ~& J5 H: j2 \1 B" U7 |/ b
3. 3.5 6.397090305942587 0.
6 y0 \5 b# x+ f' ?
3. 4. 1.814243197655509 -2.842170943040401e-014
' w6 w; v9 v9 ]5 E( p) M
3. 4. 6.614328230915922 0.
0 B1 K. K6 n e( i# \6 q4 F
3. 4.5 2.021373795560996 0.
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3. 4.5 6.678626204439006 0.
' M: I' l1 I) v3 G7 K2 W' {+ P6 B
3. 5. 2.271534568604568 0.
7 q* Y9 |6 J$ B" [) ` W
3. 5. 6.603465431395431 0.
. x3 J1 C# F+ [, a, m% z
3. 5.5 2.583225948048685 0.
8 ]2 t. `% |6 t2 h1 |. x! S" I
3. 5.5 6.387362287245431 0.
9 J: }$ S, F& w& m) D: T
3. 6. 3. 0.
7 t( T ]" ]# ^
3. 6. 6.000000000000004 0.
5 S& p$ m4 W. Q, J9 z' f
3. 6.5 3.692013773129524 0.
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3. 6.5 5.281670437396789 0.
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3.5 2. 1.611932882193128 0.
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3.5 2. 4.342612572352325 0.
' E. I/ v! d \4 w" a
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.
4 D* Q# M+ k; m9 G, _
3.5 3. 6.397090305942591 0.
& e4 N# g$ L6 l) l, i4 { m
3.5 3.5 1.75 0.
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3.5 3.5 7. 0.
5 D- I8 F4 A8 {0 O, O
3.5 4. 1.889112461544294 0.
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3.5 4. 7.410887538455708 0.
: Y2 U3 f+ n4 S
3.5 4.5 2.055177184113076 0.
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3.5 4.5 7.663572815886926 0.
- r! Y( @' x: s' k, K, `
3.5 5. 2.249335186315033 0.
; K( y0 B2 a3 T: A
3.5 5. 7.78007657839085 0.
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3.5 5.5 2.476297487447864 0.
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3.5 5.5 7.773702512552139 0.
+ ?; Y+ |% Q# D% I, E# D6 @! @" e& D" g
3.5 6. 2.745300211529267 0.
" e7 P* O1 y6 w0 F- W7 {/ A6 j4 o; {
3.5 6. 7.649436630576002 0.
7 I7 X* e! b& V, ]( Y: y' k
3.5 6.5 3.073844239487252 0.
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3.5 6.5 7.401155760512754 0.
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3.5 7. 3.5 0.
2 ]' o% d& @% H5 }" [0 ~# S
3.5 7. 7. 0.
6 T2 z# `2 t3 H
3.5 7.5 4.146248634662254 0.
$ R; f. a! p3 o( g# O# K
3.5 7.5 6.331024092610477 0.
" I5 K# x& C# ^4 H- h7 E1 f% B
4. 2. 2.000000000000001 0.
7 r' R# Q; S( |6 J% a
4. 2. 3.999999999999997 0.
2 Q# h2 O; ~) [- ~4 }
4. 2.5 1.804522914534426 0.
7 B( o6 J6 O* E$ N7 Q" I
4. 2.5 5.541630931619421 0.
, w% h; v7 s9 n7 b( n* J
4. 3. 1.814243197655509 0.
7 p/ b2 F; Y# x8 h1 o1 Y
4. 3. 6.614328230915922 0.
+ {9 U( K! D" Y6 U! i
4. 3.5 1.889112461544296 0.
, _* f; X( a$ B/ u2 X
4. 3.5 7.410887538455708 0.
]0 d+ ~/ f) ~7 g( c: q
4. 4. 2. 0.
$ v2 u* \& s$ `1 d9 J+ y
4. 4. 8. 0.
3 b/ H8 q1 f& {
4. 4.5 2.137407852977751 0.
0 _3 g' b" U6 o9 E) E* G$ c. }- _
4. 4.5 8.421415676434013 0.
, i# B! Q0 a% V
4. 5. 2.298437881283576 0.
% [4 S3 N J! M7 \
4. 5. 8.701562118716424 0.
5 `& z0 d7 r7 i* v! A
4. 5.5 2.483445700017514 0.
& A' Q- v/ A; z
4. 5.5 8.858659563140384 0.
/ F c a& T: _: N6 n* V
4. 6. 2.695165060747996 0.
, J9 y V$ O7 z4 H, l
4. 6. 8.904834939252009 0.
) {2 |/ h* O$ R
4. 6.5 2.938916502606828 0.
6 h) ]. T5 o2 i1 n- s; P- o
4. 6.5 8.846797783107462 0.
% e. `- ?) U+ i6 V! v
4. 7. 3.223865764386256 0.
: @, S$ S& B# }2 G# X
4. 7. 8.685225144704656 0.
% D* x- s) ^% s
4. 7.5 3.566387492405245 0.
+ }5 c0 j4 H; B% w. T
4. 7.5 8.411873377159974 0.
, _0 n, c' Z. T, A) I& z, f
4. 8. 4.000000000000002 0.
% ?& J; b: N% U4 |8 E
4. 8. 7.999999999999995 0.
, o1 [4 g3 r1 N: j, H- ? L/ B+ ~
4. 8.5 4.618832614156824 0.
8 Y0 O/ L6 F2 o N8 O. r* ~8 s( z, P
4. 8.5 7.361167385843178 0.
: p! v! }, H+ I
4.5 2.5 2.095411264571111 0.
& p3 P, i6 }/ N7 [. F* k h+ }
4.5 2.5 5.368874449714603 0.
( P- x0 y, T4 l4 P4 V! Z) x
4.5 3. 2.021373795560996 0.
+ e* B) ?# y' p) H: y7 c
4.5 3. 6.678626204439004 0.
/ c. K. u0 G8 x2 S" Q4 q
4.5 3.5 2.055177184113076 0.
9 w7 c% H# o' M# `4 X7 C
4.5 3.5 7.663572815886925 0.
- d" l. f* @( Y* p/ d+ Z7 T& Q
4.5 4. 2.137407852977751 0.
9 }: Q0 t$ R! G( q: j
4.5 4. 8.421415676434013 0.
+ U, ~" D6 R3 D" ]. O4 Z
4.5 4.5 2.25 0.
- r4 E( ]+ L& Z) o3 I- f/ L
4.5 4.5 9. 0.
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4.5 5. 2.386074153449669 0.
' S, a, i6 V$ k! z# S$ g
4.5 5. 9.429715320234545 0.
2 M4 H/ G2 G0 \9 i# h0 n
4.5 5.5 2.543211996792689 0.
7 }3 U; A6 P! L7 M0 R
4.5 5.5 9.731788003207315 0.
M% O R- _8 P# |1 Y
4.5 6. 2.721364796483262 0.
$ q0 W- I! M" k5 X: E( b3 M
4.5 6. 9.92149234637388 0.
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4.5 6.5 2.922185008548723 0.
1 C8 O) {5 u' E8 q1 s* q( W3 i
4.5 7. 3.149000780229209 0.
# k h2 Z" L) A+ J% D
4.5 7.5 3.407301852906852 0.
% q: E7 Q! q! }5 P& N' H2 I% t* j
4.5 7.5 9.905198147093154 0.
% a2 ~& q. M. A+ R
4.5 8. 3.705986018674346 0.
; F( z5 y# o+ e7 e
4.5 8. 9.714013981325651 0.
5 s+ M* \% A' z% n6 L
4.5 8.5 4.060312832145664 0.
4 g) q0 y# x( Y) c! o
4.5 8.5 9.420456398623564 0.
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4.5 9. 4.5 0.
" {# _# L/ }/ {& Q# b+ v3 A
4.5 9. 9.000000000000005 0.
7 p5 [+ x$ B0 h7 f" m; E) [
4.5 9.5 5.100326474258905 0.
8 V; N. }! k8 ^! U
4.5 9.5 8.381816382883958 0.
- r' j2 n! P. _: e3 ?3 \/ V& a- N; [
4.5 10. 6.402193416458818 0.
* C- A) i9 d( A
4.5 10. 7.028841066299776 4.547473508864641e-013
6 r7 e* t+ W: {& n4 W( k
5. 2.5 2.499999999999997 0.
1 J' R6 g5 Y5 O) O j9 ~( f
5. 2.5 4.999999999999998 0.
6 [" f! K6 L2 s% Q& z
5. 3. 2.271534568604568 0.
0 n# }0 ^6 y( R; \; X
5. 3. 6.603465431395431 0.
. Y5 ~% g9 I8 B5 K! P
5. 3.5 2.249335186315031 0.
3 n" l) |/ p v9 ^# E6 c
5. 3.5 7.78007657839085 0.
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5. 4. 2.298437881283576 0.
[+ l3 X- F2 w) H& M& x: D
5. 4. 8.701562118716424 0.
7 t8 Y" [: U4 m( D
5. 4.5 2.386074153449668 0.
2 h U6 @+ V% @' P
5. 4.5 9.429715320234545 0.
$ L. S) ?3 m- y. G1 T% w: x
5. 5. 2.5 0.
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5. 5. 9.999999999999993 0.
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5. 5.5 2.635001373375082 0.
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5. 6. 2.789031160980821 0.
# K7 F( P. Z0 j2 v! h N" q* Q$ }
5. 6.5 2.961827475384206 0.
/ x1 N. D K$ i/ c
5. 7. 3.154404956432853 0.
, o7 J7 _) Y' }. e- c" b% F0 T8 g
5. 7.5 3.368956325934996 0.
* b/ r! g2 ]% n" `, r# @" r
5. 8. 3.609045829068848 0.
+ O8 j' u6 |* n# v* ^8 j
5. 8.5 3.880150804346176 0.
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5. 9. 4.190822529142221 0.
0 h8 A% S. i6 \( H* {5 K
5. 9.5 4.555265557190405 0.
/ B& v& }" |( a+ t/ ?% a0 b
5. 10. 4.999999999999997 0.
" C1 M7 w4 W1 z: d
5. 10. 10. 0.
; p* G+ H2 Z- h
5.5 2.5 3.292157134434847 0.
' F2 o4 ~) m: {# d8 x
5.5 2.5 4.176592865565167 0.
0 Q6 K P. }9 R+ y2 |/ W
5.5 3. 2.583225948048685 0.
- n3 T z5 T* M3 ~( J2 }
5.5 3. 6.387362287245431 0.
7 ?* `* k8 k: c2 S0 u5 Y+ L
5.5 3.5 2.476297487447864 0.
! `" p+ L/ E O0 Q X9 q! a6 S
5.5 3.5 7.773702512552133 0.
6 H2 M, R2 z- R3 w
5.5 4. 2.483445700017513 0.
5 e% o2 w/ u* o0 {. l5 }) Q
5.5 4. 8.858659563140384 0.
0 a) w ~7 V# ]. j# T* H9 X
5.5 4.5 2.543211996792689 0.
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5.5 4.5 9.731788003207313 0.
- s# t \ q; e9 D1 }
5.5 5. 2.635001373375081 0.
6 w# ~; K- s5 B0 b: ]- Q
5.5 5.5 2.749999999999999 0.
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5.5 6. 2.884119329429114 0.
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5.5 6.5 3.035604845963815 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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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. 5.5 2.884119329429112 0.
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6. 6.5 3.133381054885423 0.
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6. 7. 3.282742465037893 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. 9.5 4.280465505524228 0.
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6. 10. 4.543069137209136 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. 8.846797783107464 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 6.5 3.25 0.
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6.5 7.5 3.530313399868478 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. 8.685225144704656 0.
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7. 4.5 3.14900078022921 0.
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7. 5.5 3.204083523261021 0.
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7. 8. 3.778224923088592 0.
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7. 10. 4.498670372630065 0.
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7.5 3.5 4.146248634662252 0.
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7.5 3.5 6.331024092610478 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 3.407301852906851 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 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. 4. 8. 0.
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8. 4.5 9.714013981325657 0.
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8. 5. 3.609045829068849 0.
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8. 5.5 3.595369279178607 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. 8. 4.000000000000003 0.
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8. 8.5 4.131333821659408 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.
9 q' a3 `/ J8 T- S, i# b) f
8.5 5. 3.880150804346178 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 6.5 3.868693727795108 0.
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8.5 7. 3.937578943177748 0.
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8.5 7.5 4.026409271112575 0.
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8.5 8. 4.13133382165941 0.
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8.5 8.5 4.25 0.
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8.5 9. 4.380969715903502 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. 4.190822529142222 0.
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9. 5.5 4.074041860443781 0.
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9. 6. 4.042747591121993 0.
+ A' W9 Q. C8 Z" f7 P% X, U% u
9. 6.5 4.060441442861232 -4.547473508864641e-013
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9. 7. 4.110354368226151 0.
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9. 7.5 4.183546741471229 0.
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9. 8. 4.274815705955503 0.
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9. 8.5 4.380969715903502 0.
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9. 9. 4.5 0.
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9. 9.5 4.630645298521483 0.
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9. 10. 4.772148306899336 0.
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9.5 4.5 5.100326474258897 0.
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9.5 4.5 8.381816382883978 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. 4.430014593640445 0.
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9.5 8.5 4.523405444098199 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. 10. 0.
4 I" V% j* L4 a4 `+ M
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. 9.5 4.880354401607461 0.
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6 i$ p6 L" [6 u1 R5 N; @
330.
' j$ O3 j: l5 l7 z
& ~0 b$ p7 {. Q7 _% ^) f+ m3 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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