Gauss: Mienzaniso mhinduro uye anokosha zviitiko

Gauss nzira, kunonziwo nzira stepwise kubviswa zvisingazivikanwi variables, mazita akakurumbira musayendisiti German KF Gauss, vachiri vapenyu vagamuchira ndibatsireiwo rokuti "Mambo masvomhu." Zvisinei, nzira iyi yave kuzivikanwa refu asati aberekwa European ibudirire, kunyange ini remakore. BC. e. Chinese Nyanzvi Ancient vakashandisa nayo mune zvaakanyora.

Gauss ari chaiwo nzira kugadzirisa mugadziriro Linear algebraic equations (Slough). Zviri rakanakira kukurumidza kugadzirisa shoma kukura matrices.

Nzira pacharo inoumbwa maviri runoita: mberi uye harungashandurwi. Direct kosi inonzi kutevedzana wakaratidzwa SLAE mativi chimiro, kureva razero ukoshi pasi huru diagonal. Yokururamisa kunosanganisira kupindirana kuwanikwa variables, kuratidza shanduka nerimwe kuburikidza yapfuura.

Dzidza kushandisa tsika, Gauss sezvachiri zvakakwana kuziva inokosha mitemo chokuwanziridza, Kuwedzera uye kubvisa nhamba.

Kuti aratidze kuti algorithm kuti nekugadzirisa Linear enyika nenzira iyi, isu kutsanangura imwe muenzaniso.

Saka, kugadziriswa kushandisa Gauss:

X + 2y + 4z = 3
2x + 6y + 11z = 6
4x-2y-2z = -6

Tinoda pemitsara wechipiri newechitatu kubvisa ari shanduka x. To ichi isu uchawedzera kwaari pakutanga kane -2, uye -4, ukuwo. isu tiwane:

X + 2y + 4z = 3
2y + 3z = 0
-10y-18z = -18

Zvino 2nd mutsetse kuwanda ne 5 uye kuwedzera rechitatu:

X + 2y + 4z = 3
2y + 3z = 0
-3z = -18

We vakauya Hurongwa yedu kuti mativi chimiro. Zvino isu tiite neizvi. Tinotanga yekupedzisira mutsetse:
-3z = -18,
Z = 6.

Wechipiri mutsetse:
2y + 3z = 0
2y + 18 = 0
2y = -18,
Y = -9

The mutsetse wokutanga:
X + 2y + 4z = 3
x-18 + 24 = 3
x = 18-24 + 3
x = -3

Dzinomiririra tsika pakati variables chepakutanga data, isu iratidzwe yakarurama rokusarudza.

Muenzaniso uyu zvinogona kugadziriswa yakawanda chimwe vokutsiva, asi mhinduro iri kunge zvakafanana.

It saka zvinoitika kuti aitungamirira zvinhu mutsara wokutanga vanosarudzirwa anokoshesa chiduku. Hazvina kutyisa, asi zvinoomesa Masvomhu. Mhinduro iri Gauss pamwe pivoting pamusoro wechikamu. Chaizvoizvo wayo sezvinotevera: mutsara kufadza rokutanga vakatsvaka modulo maatomu akafanana, mbiru umo riri, shanduko nzvimbo 1st mbiru, kuti ndiye yedu kunonyanyisa maatomu akafanana anova chinhu chikuru diagonal rokutanga. Next chinhu mwero kukarukureta kwacho. Kana zvakakodzera, muitiro achinje mbiru mune dzimwe nzvimbo zvinogona kudzokororwa.

Chimwe shanduro nzira ndiyo nzira Gauss Gauss-Jordan.

Zvinonzi inoshandiswa kugadzirisa Linear hurongwa square, kana Ivan Kuchin chizvaro pakati chizvaro uye chinzvimbo (nhamba nonzero mitsetse).

Musimboti Nzira ichi ndechokuti yepakutanga gadziriro Shandurwai nokuchinja nokuzivikanwa chizvaro ane mberi kuwanikwa variables.

The algorithm ndechokuti:

1. urongwa equations ndiko, sezvo iri nzira Gauss, munhu mativi chimiro.

2. mutsetse Mumwe rakakamurwa kuva chaiyo nhamba nenzira kuti chikwata rwauya pamusoro huru diagonal.

3. wokupedzisira mutsetse zvikawanda imwe nhamba uye subtracted kubva Penultimate kuitira kuti musi huru diagonal 0.

4. Step 3 inodzokororwa sequentially kuti mitsara zvose kusvikira pakupedzisira kwete kuumba chikwata chizvaro.