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求方程的全部解

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forcal

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电梯直达

1^#

发表于 2011-1-9 11:24 |只看该作者 |正序浏览

|招呼Ta 关注Ta

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。
参考：http://www.forcal.net/sysm/forcal9/fchtm/fcopt.htm

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;

f(x)=x^6-5*x^5+3*x^4+x^3-7*x^2+7*x-20;) H5 a$ K* m7 R- a5 \6 T7 b
fcopt::isolve[HFor("f")];. [' S9 b9 |0 i F

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-1.402463030422577 3.552713678800501e-015
4.333755446919994 -2.984279490192421e-013
2.

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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forcal

45 主题	3 听众	282 积分

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	难过 2012-8-27 18:22

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发表于 2011-1-11 06:30 |只看该作者 |招呼Ta 关注Ta

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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forcal

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发表于 2011-1-9 12:08 |只看该作者 |招呼Ta 关注Ta

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;

!using("fcopt");% J' ~- Z' Q2 X1 n6 T
f(x,y,z)=(x+y+z)*(x*y+x*z+y*z)-10*x*y*z;
# K6 g) m ~5 m; E% N
init(:x,y,i)={2 _- g( H% ?: B0 L
i=0,
* g# S9 m2 m7 }9 Z7 I
x=1,while{x<=10,
( |! d$ F! v) ^$ d7 {$ C0 A# y
y=1,while{y<=10,\\" A3 D2 ]9 i7 n% C$ [\\" t5 m
i=i+isolve[HFor("f"), optrange,1,10, optthis,2, optpara,x,y],' L) N3 |7 p& z/ u; Q, b
y=y+0.5
% x: {$ }+ _- [0 e5 x$ N* T3 v! O0 F
},0 R: w0 }5 W* }! _
x=x+0.5
/ o8 @, L; Q9 e' U) L/ t
},
: G; }: A4 }6 t) {* h8 ^
i2 _. l+ y% \( h1 |9 W! {! ]
};) y& f; e- Q% p, n6 A1 C; N1 ]# Z

!using("fcopt");% J' ~- Z' Q2 X1 n6 T

f(x,y,z)=(x+y+z)*(x*y+x*z+y*z)-10*x*y*z; 
# K6 g) m  ~5 m; E% N
init(:x,y,i)={2 _- g( H% ?: B0 L

i=0, 
* g# S9 m2 m7 }9 Z7 I
  x=1,while{x<=10, 
( |! d$ F! v) ^$ d7 {$ C0 A# y
    y=1,while{y<=10,\\" A3 D2 ]9 i7 n% C$ [\\" t5 m

i=i+isolve[HFor("f"), optrange,1,10, optthis,2, optpara,x,y],' L) N3 |7 p& z/ u; Q, b

y=y+0.5 
% x: {$ }+ _- [0 e5 x$ N* T3 v! O0 F
    },0 R: w0 }5 W* }! _

x=x+0.5 
/ o8 @, L; Q9 e' U) L/ t
  }, 
: G; }: A4 }6 t) {* h8 ^
  i2 _. l+ y% \( h1 |9 W! {! ]

};) y& f; e- Q% p, n6 A1 C; N1 ]# Z

1.                      1.                      2.                      0.
1.                      1.5                      2.226208734813002       0.
1.                      2.                      1.                      0.
1.                      2.                      1.999999999999999       0.
1.5                      1.                      2.226208734813          0.
1.5                      1.5                      3.                      0.
1.5                      2.                      3.30716411545796       0.
1.5                      2.5                      1.135767284302283       0.
1.5                      2.5                      3.301732715697718       0.
1.5                      3.                      1.499999999999999       0.
1.5                      3.                      3.000000000000004       0.
2.                      1.                      1.                      0.
2.                      1.                      1.999999999999996       0.
2.                      1.5                      3.30716411545796       0.
2.                      2.                      1.                      0.
2.                      2.                      4.000000000000001       0.
2.                      2.5                      1.149218940641788       0.
2.                      2.5                      4.350781059358212       0.
2.                      3.                      1.347582530373998       0.
2.                      3.                      4.452417469626003       0.
2.                      3.5                      1.611932882193128       0.
2.                      3.5                      4.342612572352325       0.
2.                      4.                      1.999999999999999       0.
2.                      4.                      3.999999999999999       0.
2.5                      1.5                      1.135767284302283       0.
2.5                      1.5                      3.301732715697717       0.
2.5                      2.                      1.149218940641788       0.
2.5                      2.                      4.350781059358212       0.
2.5                      2.5                      1.25                   0.
2.5                      2.5                      4.999999999999998       0.
2.5                      3.                      1.39451558049041       0.
2.5                      3.                      5.378211692236864       5.684341886080802e-014
2.5                      3.5                      1.577202478216426       0.
2.5                      3.5                      5.547797521783575       0.
2.5                      4.                      1.804522914534426       0.
2.5                      4.                      5.541630931619419       0.
2.5                      4.5                      2.09541126457111       0.
2.5                      4.5                      5.368874449714605       0.
2.5                      5.                      2.5                      0.
2.5                      5.                      4.999999999999998       0.
2.5                      5.5                      3.292157134434847       0.
2.5                      5.5                      4.176592865565161       0.
3.                      1.5                      1.499999999999999       0.
3.                      1.5                      3.000000000000004       0.
3.                      2.                      1.347582530373997       0.
3.                      2.                      4.452417469626003       0.
3.                      2.5                      1.39451558049041       0.
3.                      2.5                      5.378211692236861       -5.684341886080802e-014
3.                      3.                      1.5                      0.
3.                      3.                      6.000000000000001       0.
3.                      3.5                      1.641371232518948       0.
3.                      3.5                      6.397090305942587       0.
3.                      4.                      1.814243197655509       -2.842170943040401e-014
3.                      4.                      6.614328230915922       0.
3.                      4.5                      2.021373795560996       0.
3.                      4.5                      6.678626204439006       0.
3.                      5.                      2.271534568604568       0.
3.                      5.                      6.603465431395431       0.
3.                      5.5                      2.583225948048685       0.
3.                      5.5                      6.387362287245431       0.
3.                      6.                      3.                      0.
3.                      6.                      6.000000000000004       0.
3.                      6.5                      3.692013773129524       0.
3.                      6.5                      5.281670437396789       0.
3.5                      2.                      1.611932882193128       0.
3.5                      2.                      4.342612572352325       0.
3.5                      2.5                      1.577202478216427       0.
3.5                      2.5                      5.547797521783575       0.
3.5                      3.                      1.641371232518947       0.
3.5                      3.                      6.397090305942591       0.
3.5                      3.5                      1.75                   0.
3.5                      3.5                      7.                      0.
3.5                      4.                      1.889112461544294       0.
3.5                      4.                      7.410887538455708       0.
3.5                      4.5                      2.055177184113076       0.
3.5                      4.5                      7.663572815886926       0.
3.5                      5.                      2.249335186315033       0.
3.5                      5.                      7.78007657839085       0.
3.5                      5.5                      2.476297487447864       0.
3.5                      5.5                      7.773702512552139       0.
3.5                      6.                      2.745300211529267       0.
3.5                      6.                      7.649436630576002       0.
3.5                      6.5                      3.073844239487252       0.
3.5                      6.5                      7.401155760512754       0.
3.5                      7.                      3.5                      0.
3.5                      7.                      7.                      0.
3.5                      7.5                      4.146248634662254       0.
3.5                      7.5                      6.331024092610477       0.
4.                      2.                      2.000000000000001       0.
4.                      2.                      3.999999999999997       0.
4.                      2.5                      1.804522914534426       0.
4.                      2.5                      5.541630931619421       0.
4.                      3.                      1.814243197655509       0.
4.                      3.                      6.614328230915922       0.
4.                      3.5                      1.889112461544296       0.
4.                      3.5                      7.410887538455708       0.
4.                      4.                      2.                      0.
4.                      4.                      8.                      0.
4.                      4.5                      2.137407852977751       0.
4.                      4.5                      8.421415676434013       0.
4.                      5.                      2.298437881283576       0.
4.                      5.                      8.701562118716424       0.
4.                      5.5                      2.483445700017514       0.
4.                      5.5                      8.858659563140384       0.
4.                      6.                      2.695165060747996       0.
4.                      6.                      8.904834939252009       0.
4.                      6.5                      2.938916502606828       0.
4.                      6.5                      8.846797783107462       0.
4.                      7.                      3.223865764386256       0.
4.                      7.                      8.685225144704656       0.
4.                      7.5                      3.566387492405245       0.
4.                      7.5                      8.411873377159974       0.
4.                      8.                      4.000000000000002       0.
4.                      8.                      7.999999999999995       0.
4.                      8.5                      4.618832614156824       0.
4.                      8.5                      7.361167385843178       0.
4.5                      2.5                      2.095411264571111       0.
4.5                      2.5                      5.368874449714603       0.
4.5                      3.                      2.021373795560996       0.
4.5                      3.                      6.678626204439004       0.
4.5                      3.5                      2.055177184113076       0.
4.5                      3.5                      7.663572815886925       0.
4.5                      4.                      2.137407852977751       0.
4.5                      4.                      8.421415676434013       0.
4.5                      4.5                      2.25                   0.
4.5                      4.5                      9.                      0.
4.5                      5.                      2.386074153449669       0.
4.5                      5.                      9.429715320234545       0.
4.5                      5.5                      2.543211996792689       0.
4.5                      5.5                      9.731788003207315       0.
4.5                      6.                      2.721364796483262       0.
4.5                      6.                      9.92149234637388       0.
4.5                      6.5                      2.922185008548723       0.
4.5                      7.                      3.149000780229209       0.
4.5                      7.5                      3.407301852906852       0.
4.5                      7.5                      9.905198147093154       0.
4.5                      8.                      3.705986018674346       0.
4.5                      8.                      9.714013981325651       0.
4.5                      8.5                      4.060312832145664       0.
4.5                      8.5                      9.420456398623564       0.
4.5                      9.                      4.5                      0.
4.5                      9.                      9.000000000000005       0.
4.5                      9.5                      5.100326474258905       0.
4.5                      9.5                      8.381816382883958       0.
4.5                      10.                      6.402193416458818       0.
4.5                      10.                      7.028841066299776       4.547473508864641e-013
5.                      2.5                      2.499999999999997       0.
5.                      2.5                      4.999999999999998       0.
5.                      3.                      2.271534568604568       0.
5.                      3.                      6.603465431395431       0.
5.                      3.5                      2.249335186315031       0.
5.                      3.5                      7.78007657839085       0.
5.                      4.                      2.298437881283576       0.
5.                      4.                      8.701562118716424       0.
5.                      4.5                      2.386074153449668       0.
5.                      4.5                      9.429715320234545       0.
5.                      5.                      2.5                      0.
5.                      5.                      9.999999999999993       0.
5.                      5.5                      2.635001373375082       0.
5.                      6.                      2.789031160980821       0.
5.                      6.5                      2.961827475384206       0.
5.                      7.                      3.154404956432853       0.
5.                      7.5                      3.368956325934996       0.
5.                      8.                      3.609045829068848       0.
5.                      8.5                      3.880150804346176       0.
5.                      9.                      4.190822529142221       0.
5.                      9.5                      4.555265557190405       0.
5.                      10.                      4.999999999999997       0.
5.                      10.                      10.                      0.
5.5                      2.5                      3.292157134434847       0.
5.5                      2.5                      4.176592865565167       0.
5.5                      3.                      2.583225948048685       0.
5.5                      3.                      6.387362287245431       0.
5.5                      3.5                      2.476297487447864       0.
5.5                      3.5                      7.773702512552133       0.
5.5                      4.                      2.483445700017513       0.
5.5                      4.                      8.858659563140384       0.
5.5                      4.5                      2.543211996792689       0.
5.5                      4.5                      9.731788003207313       0.
5.5                      5.                      2.635001373375081       0.
5.5                      5.5                      2.749999999999999       0.
5.5                      6.                      2.884119329429114       0.
5.5                      6.5                      3.035604845963815       0.
5.5                      7.                      3.204083523261024       0.
5.5                      7.5                      3.390173520212255       0.
5.5                      8.                      3.595369279178611       0.
5.5                      8.5                      3.822111065390602       0.
5.5                      9.                      4.074041860443781       0.
5.5                      9.5                      4.356541541407054       0.
5.5                      10.                      4.677786777828594       0.
6.                      3.                      3.                      0.
6.                      3.                      5.999999999999999       0.
6.                      3.5                      2.745300211529266       0.
6.                      3.5                      7.649436630575996       0.
6.                      4.                      2.695165060747994       0.
6.                      4.                      8.904834939252005       0.
6.                      4.5                      2.721364796483262       0.
6.                      4.5                      9.92149234637388       0.
6.                      5.                      2.78903116098082       0.
6.                      5.5                      2.884119329429112       0.
6.                      6.                      3.                      0.
6.                      6.5                      3.133381054885423       0.
6.                      7.                      3.282742465037893       0.
6.                      7.5                      3.447656821925362       2.273736754432321e-013
6.                      8.                      3.628486395311015       0.
6.                      8.5                      3.826272163735232       0.
6.                      9.                      4.042747591121993       0.
6.                      9.5                      4.280465505524228       0.
6.                      10.                      4.543069137209136       0.
6.5                      3.                      3.692013773129525       0.
6.5                      3.                      5.281670437396786       0.
6.5                      3.5                      3.073844239487251       0.
6.5                      3.5                      7.401155760512746       0.
6.5                      4.                      2.938916502606827       0.
6.5                      4.                      8.846797783107464       0.
6.5                      4.5                      2.922185008548724       0.
6.5                      5.                      2.961827475384206       0.
6.5                      5.5                      3.035604845963813       0.
6.5                      6.                      3.133381054885422       0.
6.5                      6.5                      3.25                   0.
6.5                      7.                      3.382753896487015       0.
6.5                      7.5                      3.530313399868478       0.
6.5                      8.                      3.692229149874506       0.
6.5                      8.5                      3.868693727795107       0.
6.5                      9.                      4.060441442861232       0.
6.5                      9.5                      4.268736592110795       0.
6.5                      10.                      4.495435882583067       0.
7.                      3.5                      3.500000000000001       0.
7.                      3.5                      7.                      0.
7.                      4.                      3.223865764386255       0.
7.                      4.                      8.685225144704656       0.
7.                      4.5                      3.14900078022921       0.
7.                      5.                      3.154404956432853       0.
7.                      5.5                      3.204083523261021       0.
7.                      6.                      3.282742465037893       0.
7.                      6.5                      3.382753896487015       0.
7.                      7.                      3.5                      0.
7.                      7.5                      3.632214429249618       0.
7.                      8.                      3.778224923088592       0.
7.                      8.5                      3.93757894317775       0.
7.                      9.                      4.110354368226149       0.
7.                      9.5                      4.297071556823225       0.
7.                      10.                      4.498670372630065       0.
7.5                      3.5                      4.146248634662252       0.
7.5                      3.5                      6.331024092610478       0.
7.5                      4.                      3.566387492405242       0.
7.5                      4.                      8.411873377159969       0.
7.5                      4.5                      3.407301852906851       0.
7.5                      4.5                      9.905198147093145       0.
7.5                      5.                      3.368956325934993       0.
7.5                      5.5                      3.390173520212255       0.
7.5                      6.                      3.447656821925362       0.
7.5                      6.5                      3.530313399868477       0.
7.5                      7.                      3.632214429249618       0.
7.5                      7.5                      3.750000000000003       0.
7.5                      8.                      3.881745401569135       0.
7.5                      8.5                      4.026409271112575       0.
7.5                      9.                      4.183546741471229       0.
7.5                      9.5                      4.353157178625922       0.
7.5                      10.                      4.535607994138771       0.
8.                      4.                      4.000000000000002       0.
8.                      4.                      8.                      0.
8.                      4.5                      3.705986018674347       0.
8.                      4.5                      9.714013981325657       0.
8.                      5.                      3.609045829068849       0.
8.                      5.5                      3.595369279178607       0.
8.                      6.                      3.628486395311016       0.
8.                      6.5                      3.692229149874504       0.
8.                      7.                      3.77822492308859       0.
8.                      7.5                      3.881745401569135       0.
8.                      8.                      4.000000000000003       0.
8.                      8.5                      4.131333821659408       0.
8.                      9.                      4.274815705955503       0.
8.                      9.5                      4.430014593640446       0.
8.                      10.                      4.596875762567152       0.
8.5                      4.                      4.618832614156823       0.
8.5                      4.                      7.361167385843172       0.
8.5                      4.5                      4.060312832145666       0.
8.5                      4.5                      9.420456398623564       0.
8.5                      5.                      3.880150804346178       0.
8.5                      5.5                      3.822111065390603       0.
8.5                      6.                      3.826272163735231       0.
8.5                      6.5                      3.868693727795108       0.
8.5                      7.                      3.937578943177748       0.
8.5                      7.5                      4.026409271112575       0.
8.5                      8.                      4.13133382165941       0.
8.5                      8.5                      4.25                   0.
8.5                      9.                      4.380969715903502       0.
8.5                      9.5                      4.523405444098199       0.
8.5                      10.                      4.676893219133843       0.
9.                      4.5                      4.499999999999997       0.
9.                      4.5                      9.000000000000009       0.
9.                      5.                      4.190822529142222       0.
9.                      5.5                      4.074041860443781       0.
9.                      6.                      4.042747591121993       0.
9.                      6.5                      4.060441442861232       -4.547473508864641e-013
9.                      7.                      4.110354368226151       0.
9.                      7.5                      4.183546741471229       0.
9.                      8.                      4.274815705955503       0.
9.                      8.5                      4.380969715903502       0.
9.                      9.                      4.5                      0.
9.                      9.5                      4.630645298521483       0.
9.                      10.                      4.772148306899336       0.
9.5                      4.5                      5.100326474258897       0.
9.5                      4.5                      8.381816382883978       0.
9.5                      5.                      4.555265557190406       0.
9.5                      5.5                      4.356541541407055       0.
9.5                      6.                      4.280465505524228       0.
9.5                      6.5                      4.268736592110795       0.
9.5                      7.                      4.297071556823221       0.
9.5                      7.5                      4.353157178625922       0.
9.5                      8.                      4.430014593640445       0.
9.5                      8.5                      4.523405444098199       0.
9.5                      9.                      4.630645298521484       0.
9.5                      9.5                      4.75                   0.
9.5                      10.                      4.880354401607461       0.
10.                      4.5                      6.402193416458861       0.
10.                      4.5                      7.028841066299777       4.547473508864641e-013
10.                      5.                      4.999999999999997       0.
10.                      5.                      10.                      0.
10.                      5.5                      4.67778677782859       0.
10.                      6.                      4.543069137209136       0.
10.                      6.5                      4.495435882583067       0.
10.                      7.                      4.498670372630065       0.
10.                      7.5                      4.535607994138771       0.
10.                      8.                      4.596875762567152       0.
10.                      8.5                      4.676893219133843       0.
10.                      9.                      4.772148306899336       0.
10.                      9.5                      4.880354401607461       0.
10.                      10.                      5.                      0.
330.

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forcal

45 主题	3 听众	282 积分

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	难过 2012-8-27 18:22

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[LV.1]初来乍到

4^#

发表于 2011-1-9 11:37 |只看该作者 |招呼Ta 关注Ta

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;

!using("fcopt");
! h8 N! p. w6 X
f(x,y)=(x^2+y^2)^3-36*(x^2-y^2)^2;2 p* \# h. h' v+ p e2 `
isolve[HFor("f"), optthis,1, optpara,1.1];0 U+ A% c/ R& }/ i3 h1 \

1.1                      -5.413569621185505       0.
1.1                      -1.52029233271966       0.
1.1                      -0.8671650244626143    8.881784197001252e-016
1.1                      0.8671650244626143       8.881784197001252e-016
1.1                      1.52029233271966       0.
1.1                      5.413569621185504       0.
6.

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forcal

45 主题	3 听众	282 积分

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TA的每日心情

	难过 2012-8-27 18:22

签到天数: 1 天

[LV.1]初来乍到

3^#

发表于 2011-1-9 11:35 |只看该作者 |招呼Ta 关注Ta

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;

!using("fcopt");
5 E- R% _. T4 q# a- d
f(x,y)=(x^2+y^2)^3-36*(x^2-y^2)^2;
( u. m4 [* t# A$ F$ _; N9 m
isolve[HFor("f"),optpara,1];: E+ R2 O3 W% b5 C3 y, Z: r1 k

-5.530393535267971       1.                      -3.637978807091713e-012
-1.32936186218296       1.                      -7.105427357601002e-015
-0.8047014256478491    1.                      -1.776356839400251e-015
0.8047014256478491       1.                      -1.776356839400251e-015
1.32936186218296       1.                      -7.105427357601002e-015
5.530393535267971       1.                      -3.637978807091713e-012
6.