Simple iteration mokhoa bakeng sa ho rarolla litsamaiso tsa ditekanyo guttate (Slough)

Simple iteration mokhoa, eo hape a bitsoang mokhoa oa ho mo e ka nnang latellanang, - e-tharabololo thuto ea lipalo bakeng sa ho fumana melemo ea boleng e sa tsejoeng ka butle-butle hlakisa eona. Motheo oa mokhoa ona ke hore, e le lebitso la ka fana ka maikutlo a, ba butle-butle ho bontša e mo e ka nnang ea pele ea batho ba e ileng ea latela, ba e-ba le liphello tse ngata e hloekisitsoeng. Mokhoa ona e sebediswa ho fumana boleng ba polygonal le ka mosebetsi e fanoeng, 'me ho rarolla litsamaiso tsa ditekanyo, bobeli guttate le batho bao e seng guttate.

A re ke re boneng hore na mokhoa ona e sebediswa ho ya tharollo ea litsamaiso guttate. tsepamisitsoeng-ntlha iteration-tharabololo e tjena:

1. The netefatso ya maemo a convergence ka popeho ea pele. A convergence Theorem: haeba pele tsamaiso e khopo popeho ke diagonally ka sehloohong (khr, moleng ka mong oa metsoako ea diagonale ka sehloohong e lokela ho ba moholo ka boholo ho feta chelete ea likarolo lehlakoreng diagonals ka boleng feletseng), le mokhoa oa ditoboketso e bonolo - convergent.

2. popeho ea tsamaiso ea pele hase kamehla ho diagonale predominance. Maemong a joalo, tsamaiso ea ka le fetohe. The ditekanyo hore khotsofatsa boemo convergence setseng senyeha, ka khotsofatseng le etsa tlhakantsuke guttate, i.e. atisa, ho tlosa, abel a phuthilweng hammoho ho hlahisa sephetho batlang.

Ha tsamaiso ea o ile a fumana ka e ka sehloohong diagonale na le mabaka a le tšitiso, ea ntan'o ba mahlakoreng a mabeli a abel ena di a phaella ka ho ya ka mokgwa _kea * x, _kea, e leng lokela tsamaisane le matšoao lipontšo tsa thepa ea motheo e diagonale.

3. ho fetola tsamaiso ea sephetho le ho pono ea ntho e tloaelehileng:

^{x, - = β - + α *} x, -

Sena se ka etsoa ka litsela tse ngata, mohlala, ka tsela ena: ho abel pele ho bontša: x ₁ ka e sa tsejoeng tse ling ho tloha vtorogo- x, _2, x, ₃ of tretego- jj Kahoo, re ba sebelisa moralo o balletsweng e:

_{α ij = - (a ij /} e ii)

_kea = b, _kea / a _{ii a}
Etsa bonnete ba hore hape hore tsamaiso sephetho le ea mofuta ho tloaelehile tšoana le boemo ba convergence:

Σ (j = 1) | α _ij | ≤ 1, 'me kea = 1,2, ... n g

4. Qala sebelisoa, ha e le hantle, mokhoa oa ho approximations latellanang.

: x ⁽⁰⁾ - mo e ka nnang lekhetlo la pele, re bontša hore meetly x, ^(1), lateloa ke x, ⁽¹⁾ x tsau ^(2). The moralo kakaretso ya foromo popeho tjena:

x, ^{(N g) = β - + α} *: x (n- ¹⁾

Re ho sebedisa khompyuta, ho fihlela re fihla ho nepahala lakatsa:

_{, max | x, kea (K) -x} kea (K + 1) ≤ ε

Ho joalo, a re ke re sheba liketso, mokhoa oa ho iteration bonolo. mohlala:
Rarolla tsamaiso ea guttate:

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

Sheba hlōla haeba thepa ea motheo e diagonale tsa mojulu ona.

Re bona hore boemo convergence e khotsofatsoa ke abel boraro. Pele le la bobeli fetola, ho abel pele re kenya tse pedi:

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

Ho tlosa ho tswa ho boraro le 'ngoe:

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

Re fetotse tsamaiso pele ka lekanang le:

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

Hona joale re fokotsa tsamaiso ea ho maikutlo a tloaelehile:

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

Re sheba convergence ea ho iterative:

0,0789 + 0,3158 = 0,3947 ≤ 1
0,6429 + 0,2857 = 0,9286 ≤ 1
0.383+ 0.5319 = 0.9149 ≤ 1, i.e. boemo e ba ile ba kopana.

.3947
Ka lekhetlo la pele mo e ka nnang ya x ⁽⁰⁾ = 0.4762
.8511

Ka nkang sebaka makgabane tsena ka abel tsa mofuta o tlwaelehileng, re fumana melao ea boitšoaro tse latelang:

0,08835
x, ⁽¹⁾ = 0.486793
0.446639

Ka nkang sebaka makgabane e ncha, re fumana:

0.215243
x, ⁽²⁾ = 0.405396
0.558336

Re tsoela pele ho bala ho fihlela ho fihlela o fumana haufi le melao ea boitšoaro eo kopana le maemo a qoolloa.

0,18813

x, ⁽⁷⁾ = 0.441091

0.544319

0.188002

x, ⁽⁸⁾ = 0.44164

0.544428

Hlahloba ho nepahala ea sephetho sa:

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

Results fumanoa ka a emeng makgabane ba fumana ka abel pele, ka botlalo khotsofatsa abel.

Joalokaha re ka bona, le iteration mokhoa e bonolo o fana ka sephetho se ka toka e nepahetseng, empa ho rarolla abel ena, re tlameha ho qeta nako e ngata le ho etsa dipalelo boima.