GAUSS: imizekelo izisombululo namatyala ezizodwa

indlela GAUSS, ebizwa kwakhona indlela kupheliswe stepwise eziguquguqukayo ezingaziwayo, ogama emva sisazinzulu abadumileyo German kf GAUSS, ngoxa wayesaphila wafumana isihloko sencwadi "Kumkani lemathematika." Nangona kunjalo, le ndlela iye yaziwa elide phambi kokuba ekuzalweni kwimpucuko yaseYurophu, nkqu kwinkulungwane I. BC. e. Abaphengululi yamandulo Chinese belisebenzisa kwiincwadi zakhe.

Lo yindlela classic yokusombulula kweenkqubo quadratic yomgama aljibra (Slough). Luhle isicombululo ekhawulezayo ukuya matrices yemida.

Indlela ngokwayo iqulathe ehamba ezimbini: phambili reverse. indlela ngqo ebizwa ngokulandelelana eboniswe ifomu elingunxantathu SLAE, okt zero ixabiso phantsi idiagonal engundoqo. Uyazisola kubandakanya okufunyanisiweyo engqinelana kweziganga, evakalisa-bume nganye ngokusebenzisa yangaphambili.

Funda ukuba zisebenza ku umkhuba, Lo nje ngokwaneleyo ukuze ukwazi imigaqo esisiseko-phindo, ukudibanisa nokuthabatha amanani.

Ukuze ubonise algorithm ukuxazulula iinkqubo linear ngale ndlela, sichaza umzekelo omnye.

Ngoko ke, ingalungiswa kusetyenziswa GAUSS:

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

Kufuneka imigca yesibini neyesithathu ukuba lokuyigxotha le x bume. Ukuze oku kongeza kuye kuqala iphinda phindiweyo ngo -2, no -4, ngokulandelelana kwazo. sifumana:

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

Ngoku ke line-2 phinda ngo-5 kwaye uyifaka lesithathu:

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

Sayibuyisela inkqubo yethu ukuba uhlobo ngunxantathu. Ngoku sifeza umva. Siqala kumgca wokugqibela:
-3z = -18,
z = 6.

Umgca wesibini:
2y + 3z = 0
2y + 18 = 0
2y = -18,
y = -9

Umgca lokuqala:
x + 2y + 4z = 3
x-18 + 24 = 3
x = 18-24 + 3
x = -3

Kufakwe amanani yezahlukileyo kwi idatha yoqobo, siya ukuqinisekisa ukuchaneka sigqibo.

Lo mzekelo ingasombululwa okuninzi nayiphi na labadlali, kodwa impendulo bekumele ukuba enye.

Kwenzeka ke ukuba izinto ezikhokelela kumqolo wokuqala zicwangciswe kunye neenqobo ezincinane kakhulu. Asiyonto Kwakunzima, kodwa kunokuba nzima izibalo. Isisombululo ukuba GAUSS kunye pivoting kumhlathi. umongo wayo ngolu hlobo lulandelayo: umgca wokuqala ubuninzi wafuna isiqalelo modulo, umhlathi apho emi kuyo, iindawo ukutshintsha ne kumhlathi 1, oko kukuthi element wethu eliphezulu iba Into yokuqala idiagonal engundoqo. Okulandelayo yinkqubo ubalo eqhelekileyo. Ukuba kuyimfuneko, inkqubo utshintsha kwiikholamu kwezinye iindawo ingaphinda-phindwa.

Enye inguqulelo ye indlela yokwenza yi indlela GAUSS GAUSS-Jordan.

Isetyenziswa zokusombulula iinkqubo yomgama square, xa eguqulweyo ngokubaxwa i matrix of isizalo kunye nesikhundla (inani imigca nonzero).

Undoqo kule ndlela kukuba inkqubo yokuqala utshintshwa yi notshintsho matrix uzichaze nge ezinye eziguquguqukayo ziphumo.

I algorithm ukuba:

1. Le nkqubo kwizibalo kukuthi, njengokuba kwi indlela GAUSS, ifomu ngunxantathu.

2. umgca nganye yahlulwe inani elithile esinjalo ngendlela iyunithi iye wajika idiagonal main.

3. Umgca lokugqibela liphindaphindwe inani elithile yaye sisuswe ongamphambi kokugqibela koKuhlala ukwenzela ukuba bafumane engundoqo oxwesileyo 0.

4. Inyathelo 3 iyaphindwa ngokulandelelana kuzo zonke imiqolo de ekugqibeleni ezingeyo iyunithi matrix.