Gauss: atụ nke ngwọta na pụrụ iche ikpe

Gauss usoro, na-akpọ usoro nke stepwise mkpochapu amaghị variables, aha ya bụ mgbe a ma ama na German ọkà mmụta sayensị KF Gauss, mgbe ha ka nọ ndụ natara unofficial aha "Eze nke mgbakọ na mwepụ." Otú ọ dị, usoro a a mara ogologo oge tupu ọmụmụ nke European mmepeanya, ọbụna m na narị afọ. BC. e. Ancient Chinese ọkà mmụta ji ya na ihe odide ya.

Gauss bụ a kpochapụwo ụzọ idozi usoro linear algebraic arụmarụ (Slough). Ọ bụ ezigbo maka a ngwa ngwa ngwọta ole na size matrices.

The usoro onwe ya mejupụtara abụọ Nkea: atụ na laa azu. Direct N'ezie a na-akpọ usoro gosiri SLAE zita ụdị, ntụgharị efu uru n'okpuru isi diagonal. Retraction na-agụnye na-agbanwe agbanwe Inweta nke variables, na-ekwupụta na onye ọ bụla na agbanwe site gara aga.

Ịmụta itinye na omume, Gauss bụ nnọọ iji mara ihe ndị bụ isi iwu nke multiplication, mgbakwunye na mwepu nke nọmba.

Iji gosi na algọridim maka idozi linear usoro site usoro a, anyị na-akọwa otu ihe atụ.

Ya mere, a ga-edozi iji Gauss:

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

Anyị kwesịrị abụọ na nke atọ edoghi tufuo nke agbanwe x. A anyị na-atụkwasịrị ya mbụ ọtụtụ site -2, na -4, karị. anyị na-enweta:

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

Ugbu a 2nd akara ọtụtụ site 5 na tinye ya nke atọ:

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

Anyị na-ada anyị na usoro na a zita ụdị. Ugbu a anyị na-arụ reverse. Anyị na-amalite na n'ahịrị ikpeazụ:
-3z = -18,
z = 6.

Nke abụọ bụ:
2y + 3z = 0
2y + 18 = 0
2y = -18,
y = -9

The akpa akara:
x + 2y + 4z = 3
x-18 + 24 = 3
x = 18-24 + 3
x = -3

N'ọnọdụ ụkpụrụ nke variables na mbụ data, anyị nyochaa correctness mkpebi ikpe.

Ihe atụ a ga-edozi a otutu ihe ọ bụla ọzọ substitutions, ma azịza kwesịrị ịbụ otu ihe ahụ.

Ọ ka na-eme na ndị na-eduga na ihe nke mbụ n'usoro mere ndokwa na kwa obere ụkpụrụ. Ọ bụghị egwu, kama sikwuoro mgbawa. Ihe ngwọta bụ Gauss na pivoting on a kọlụm. Ya kachasi mkpa bụ ka ndị a: akpa akara nke kacha chọọ modulo mmewere, na kọlụm nke ọ na-emi odude, mgbanwe ebe na 1st kọlụm, nke ahụ bụ anyị kacha mmewere na-aghọ ndị mbụ mmewere nke isi diagonal. Next bụ a ọkọlọtọ ngụkọta oge usoro. Ọ bụrụ na ọ dị mkpa, usoro na-agbanwe ogidi n'ebe ụfọdụ nwere ike ugboro ugboro.

Ọzọ version nke usoro bụ usoro nke Gauss Gauss-Jordan.

Ọ na-eji maka idozi linear systems square, mgbe inverse matriks nke matriks na ọkwá (nọmba nke nonzero edoghi).

Ihe kachasi mkpa usoro a bụ na ndị mbụ usoro bụ gbanwee site na mgbanwe ndị na-amata na matriks na a n'ihu Inweta variables.

The algọridim bụ na ọ na:

1. The usoro nke arụmarụ bụ, dị ka usoro nke Gauss, a zita ụdị.

2. Onye ọ bụla akara ekewa a kpọmkwem ọnụ ọgụgụ dị otú ahụ n'ụzọ na unit emewo na isi diagonal.

3. The ikpeazụ akara na-uba site a ụfọdụ ọnụ ọgụgụ na subtracted si penultimate ka ọ ghara inwe na isi diagonal 0.

4. Nzọụkwụ 3 na-ugboro ugboro sequentially niile ahịrị ruo mgbe emecha na-etolite na na unit matriks.