Simple iteration hanya domin magance tsarin na mikakke lissafai (Slough)

Simple iteration hanya, kuma ake kira Hanyar m kimantawa, - wani ilmin lissafi algorithm domin gano da dabi'u na unknown darajar ta sauka a hankali bayyana shi. Jigon wannan hanya shi ne cewa, kamar yadda sunan ya nuna, aka hankali bayyana wani na farko kimantawa na m su, suna mafi zama mai ladabi da sakamakon. Wannan hanya da ake amfani da su sami darajar da m, a wa aiki, da kuma warware tsarin na lissafai, duka biyu mikakke kuma ba mikakke.

Bari mu ga yadda wannan hanya ke aiwatar a cikin bayani da mikakke tsarin. ajali-aya iteration algorithm ne kamar haka:

1. A tabbaci na haduwa da yanayi a cikin na farko matrix. A haduwa Theorem: idan asalin tsarin matrix ne diagonally rinjaye (ie, kowane layi na abubuwa na babban diagonal dole ne mafi girma a girma fiye da Naira Miliyan Xari da abubuwa gefen diagonals a cikakkar darajar), da hanyar da sauki iterations - convergent.

2. A matrix na asali tsarin ne ba ko da yaushe diagonal predominance. A irin haka ne, tsarin za a iya canja. A lissafai cewa gamsar da haduwa yanayin da aka bar m, tare da unsatisfying da kuma yin mikakke haduwa, Ina nufin riɓaɓɓanya, debewa, lissafi folded tare, don samar da ake so sakamakon.

Idan samu tsarin a kan babban diagonal ne m dalilai, sa'an nan zuwa garesu daga wannan lissafi suna kara da cewa tare da sharuddan da siffan _i * x _i, abin da ya kamata yayi daidai da ãyõyi alamun diagonal abubuwa.

3. mayar da sakamakon tsarin al'ada view:

^{x - = β - + α *} x -

Wannan za a iya yi a hanyoyi da dama, misali, kamar haka: na farko lissafi don bayyana x ₁ ta hanyar wasu unknown daga vtorogo- x _2, x ₃ na tretego- da dai sauransu Kamar wancan muka amfani da dabara:

_{α ij = - (a ij /} a ii)

_i = b _i / a _ii
Tabbatar sake cewa sakamakon tsarin na al'ada irin yayi dace da haduwa yanayin:

Σ (j = 1) | α _ij | ≤ 1, da kuma i = 1,2, ... n

4. Fara amfani da, a zahiri, da hanyar da m approximations.

x ⁽⁰⁾ - na farko kimantawa, mun bayyana therethrough x ^(1), ya bi ta hanyar x ⁽¹⁾ x kar ^(2). A general dabara na wani matrix form kamar haka:

x ^{(N) = β - + α} * x (n- ¹⁾

Mun lissafta, har muka isa ake so daidaito:

_{max | x i (k) -x} i (k + 1) ≤ ε

Saboda haka, bari mu duba a yi, da hanyar da sauki iteration. misali:
Warware mikakke tsarin:

4,5x1-1.7x2 + 3.5x3 = 2
3.1x1 + 2.3x2-1.1x3 = 1
1.8x1 + 2.5x2 + 4.7x3 = 4 da daidaito ε = 10 ^-3

Dubi fi idan diagonal abubuwa na koyaushe.

Mun gani cewa haduwa da yanayin da aka bayyana ta da wani uku lissafi. A farko da na biyu fasalin, da farko lissafi mu ƙara biyu:

7,6x1 + 0.6x2 + 2.4x3 = 3

Debewa daga cikin uku daya:

-2,7x1 + 4.2x2 + 1.2x3 = 2

Mun canza asalin tsarin a cikin m:

7,6x1 + 0.6x2 + 2.4x3 = 3
-2,7x1 + 4.2x2 + 1.2x3 = 2
1.8x1 + 2.5x2 + 4.7x3 = 4

Yanzu mun rage tsarin al'ada view:

x1 = 0.3947-0.0789x2-0.3158x3
x2 = 0.4762 + 0.6429x1-0.2857x3
X3 = 0.8511-0.383x1-0.5319x2

Mu duba haduwa da iterative tsari:

0,0789 + 0,3158 = 0,3947 ≤ 1
0,6429 + 0,2857 = 0,9286 ≤ 1
0.383+ 0.5319 = 0.9149 ≤ 1, Ina nufin da yanayin da aka hadu.

.3947
Harufan kimantawa x ⁽⁰⁾ = 0.4762
.8511

Musanya wadannan dabi'u a cikin lissafi na al'ada irin, mu sami wadannan dabi'u:

0,08835
x ⁽¹⁾ = 0.486793
0.446639

Canza sabon dabi'u, mun samu:

0.215243
x ⁽²⁾ = 0.405396
0.558336

Mu ci gaba da yin lissafi har sai ka sami kusanci da dabi'u cewa hadu kayyade yanayi.

0,18813

x ⁽⁷⁾ = 0.441091

0.544319

0.188002

x ⁽⁸⁾ = 0.44164

0.544428

Duba daidaitar sakamakon:

4,5 * 0,1880 -1,7 * 0,441 + 3,5 * 0,544 = 2,0003
3,1 * 0,1880 + 2,3 * 0,441-1.1x * 0,544 = 0,9987
1,8 * 2,5 * 0,1880 + 0,441 + 4,7 * 0,544 = 3,9977

Sakamakon samu yaron da ya samu dabi'u a cikin na asali lissafi, cikakken gamsar lissafi.

Kamar yadda zamu iya gani, da sauki iteration hanyar bada fairly m sakamakon, amma a warware wannan lissafi, muna da ciyar da yawa da lokaci da kuma yin amfani da tsauraran matakan lissafin.