Gauss: misalai na mafita da kuma lokuta na musamman

Gauss hanya, kuma ake kira Hanyar stepwise kawar da ba a sani ba canji, mai suna bayan da shahararren masanin kimiyyar Jamus KF Gauss, yayin da har yanzu rai samu da unofficial suna "Sarkin lissafi." Duk da haka, wannan hanya da aka sani tun kafin haihuwa da al'adun Turai, har ma a cikin na karni. BC. e. Ancient Sin malamai sun yi amfani da shi a cikin rubuce-rubucen.

Gauss ne mai classic hanyar warware tsarin na mikakke algebraic lissafai (Slough). Shi ne manufa domin sauri bayani da iyaka size matrices.

A Hanyar kanta kunshi biyu motsa: gaba da baya. Direct shakka kira da jerin nuna SLAE triangular form, watau sifili darajar karkashin babban diagonal. Karyata unshi da m Ganowar da canji, ya nuna kowane m ta hanyar da suka gabata.

Koyi to tambaya a yi, Gauss ke kawai isa ya san ainihin dokoki na multiplication, Bugu da kari kuma subtraction na lambobin.

Domin nuna algorithm domin warware mikakke tsarin da wannan hanya, za mu bayyana wani misali.

Saboda haka, za a iya warware ta amfani da Gauss:

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

Muna bukatar na biyu da kuma na uku Lines zuwa rabu da m x. Don wannan mu ƙara zuwa gare shi na farko ta tara da -2, kuma -4, bi da bi. mu samu:

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

Yanzu da 2nd line ninka ta 5 da kuma ƙara da shi zuwa uku:

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

Mun kawo tsarin mu zuwa triangular form. Yanzu mun kawo daga baya. Mun fara da na karshe line:
-3z = -18,
z = 6.

Na biyu line:
2y + 3z = 0
2y + 18 = 0
2y = -18,
y = -9

A farko line:
x + 2y + 4z = 3
x-18 + 24 = 3
x = 18-24 + 3
x = -3

Musanya da dabi'u na canji a cikin na asali bayanai, mu tabbatar da correctness na yanke shawara.

Wannan misali za a iya warware da yawa wani substitutions, amma amsar ya kamata su zama guda.

Yana haka ya faru da cewa manyan abubuwa na farko jere ana shirya tare da ma kananan dabi'u. Yana ba da ban tsoro, amma wajen dagula lissafin. The bayani ne ga Gauss da pivoting a kan wani shafi. Its ainihi shi ne kamar haka: na farko line na matsakaicin nemi modulo kashi, da shafi a cikin abin da aka located, canji wuraren da 1st shafi, cewa shi ne mu iyakar kashi zama na farko kashi na babban diagonal. Next ne mai misali lissafi tsari. Idan dole, da hanya ya musanya ginshiƙai a wasu wuraren za a iya maimaita.

Wani version na hanyar ne Hanyar Gauss Gauss-Jordan.

Ana amfani da warware mikakke tsarin square, a lokacin da kishiya matrix na matrix da daraja (yawan nonzero Lines).

Jigon wannan hanya shi ne cewa asalin tsarin da aka canza kama da canje-canje a cikin ainihi matrix da kara binciken canji.

A algorithm ne cewa shi:

1. A tsarin na lissafai ne, kamar yadda a cikin Hanyar Gauss, a triangular form.

2. Kowace line ne zuwa kashi takamaiman adadin a cikin irin wannan hanya da cewa naúrar ya juya a kan babban diagonal.

3. A karshe line aka yawaita da wani yawan, kuma mu cire daga penultimate haka kamar yadda ba a samu a babban diagonal 0.

4. Mataki 3 an maimaita sequentially ga duk layuka har ƙarshe ba form naúrar matrix.