Mokhoa oa ho Felisa Gaussian: Selelekela se Tebileng
Mokhoa oa ho felisa oa Gaussian ke o mong oa mekhoa ea motheo le e sebelisoang haholo ho algebra e otlolohileng bakeng sa ho rarolla litsamaiso tsa li-equation tse otlolohileng. O reheletsoe ka setsebi se seholo sa lipalo Carl Friedrich Gauss, ea entseng menehelo e kholo makaleng a mangata a lipalo. Sehloohong sena, re tla hlahloba likhopolo tsa motheo, mekhoa le mehlala ea ts'ebeliso ea mokhoa oa ho felisa oa Gaussian.
Nalane le Semelo
Carl Friedrich Gauss, ea phetseng qetellong ea lekholo la bo18 la lilemo le mathoasong a lekholo la bo19 la lilemo, o nkoa e le e mong oa litsebi tse kholo ka ho fetisisa tsa lipalo tsa nako eohle. Mokhoa oa ho felisa o tsejoang ka lebitso la hae o bile teng khale pele Gauss a tsoaloa, empa monehelo oa hae o moholo e bile ho ntlafatseng le ho etsa hore e tume.
Bohlokoa ba Mokhoa oa ho Felisa oa Gaussian
Lipalong le mahlaleng a khomphutha, ho rarolla litsamaiso tsa li-equation tse otlolohileng ke bothata bo tloaelehileng. Sistimi ea li-equation tse otlolohileng e na le sebopeho se akaretsang:
\[
a_{11}x_1 + a_{12}x_2 + … + a_{1n}x_n = b_1
\]
\[
a_{21}x_1 + a_{22}x_2 + … + a_{2n}x_n = b_2
\]
\[
...
\]
\[
a_{m1}x_1 + a_{m2}x_2 + … + a_{mn}x_n = b_m
\]
Mokhoa oa ho felisa oa Gaussian o ikemiselitse ho fetola sistimi ena hore e be sebopeho se bonolo e le hore e ka rarolloa habonolo ka ho sebelisa mokhoa oa ho fetola morao.
Mokhoa oa ho Felisa Gaussian
Mehato ea Motheo
Mokhoa oa ho felisa Gaussian o kenyelletsa mekhahlelo e 'meli e meholo: ho tlosa ka pele le ho nkela sebaka ka morao.
1. Ho Felisoa Pele
Sepheo sa mohato ona ke ho fetola tsamaiso ea li-equation hore e be matrix e ka holimo ea khutlotharo. Sena se finyelloa ka ho etsa mesebetsi ea mela ea motheo, e kenyeletsang:
– Phapanyetsano ea mela e 'meli.
– Atisa mola ka nomoro e seng lefela.
– Eketsa kapa tlosa dipalo tse ngata ho tloha moleng o mong ho ya ho o mong.
A re re re na le sistimi ea li-equation tse otlolohileng ka sebopeho sa matrix \(Ax = b\), moo \(A\) e leng matrix ea coefficient, \(x\) e le vector e feto-fetohang, 'me \(b\) e le vector e sa fetoheng. Mehato ea ho felisa ka pele ke:
1. Khetha karolo ea pivot, hangata e qalang ho tloha ho \(a_{11}\).
2. Sebelisa karolo ea pivot ho hlakola (etsa lefela) karolo e ka tlase ho eona kholomong e tšoanang.
3. Pheta mokhoa ona bakeng sa karolo e latelang ea pivot ka tlase ho mola o otlolohileng.
Mohlala, ha re shebeng sistimi e nang le li-equation tse tharo:
\[
a_{11}x_1 + a_{12}x_2 + a_{13}x_3 = b_1
\]
\[
a_{21}x_1 + a_{22}x_2 + a_{23}x_3 = b_2
\]
\[
a_{31}x_1 + a_{32}x_2 + a_{33}x_3 = b_3
\]
Re qala ka pivot \(a_{11}\), re etsa mesebetsi ya ho tlosa \(a_{21}\) le \(a_{31}\).
2. Phetolo e Khutlelang Morao
Kamora ho tlosa ka pele, re fumana sistimi ea li-equation e emeloang ke matrix e kaholimo. Mohlala:
\[
u_{11}x_1 + u_{12}x_2 + u_{13}x_3 = d_1
\]
\[
u_{22}x_2 + u_{23}x_3 = d_2
\]
\[
u_{33}x_3 = d_3
\]
Mothating ona, ho nkeloa sebaka ka morao ho etsoa ho tloha tlase ho ea holimo:
1. Bakeng sa \(x_3\): \(x_3 = d_3 / u_{33}\).
2. Bakeng sa \(x_2\): \(x_2 = (d_2 – u_{23}x_3) / u_{22}\).
3. Bakeng sa \(x_1\): \(x_1 = (d_1 – u_{12}x_2 – u_{13}x_3) / u_{11}\).
Mehlala ea Ts'ebeliso
Ho hlakisa tlhaloso e ka holimo, ha re nke mohlala o tiileng.
A re re re na le tsamaiso e latelang ea li-equation tse otlolohileng:
\[
2x + 3y + z = 1
\]
\[
4x + y – 2z = -2
\]
\[
3x + 2y + 3z = 7
\]
E ngotsoe ka mokhoa oa matrix:
\[
\begin{pmatrix}
2 & 3 & 1 \\
4 le 1 le -2 \\
3 & 2 & 3 \\
\end{pmatrix}
\begin{pmatrix}
x \\
y \\
z \\
\end{pmatrix}
=
\begin{pmatrix}
1 \\
-2 \\
7 \\
\end{pmatrix}
\]
1. Ho Felisoa Pele:
– Khetha karolo ea pivot \(2\), karolo ea pele ea mola oa pele.
– Etsa likarolo tse lefela ka tlase ho karolo ea pele ea pivot:
– Mola wa 2: \(4 – 2(2) = 0\)
– Mola wa 3: \(3 – \frac{3}{2}(2) = 0\)
– Liphetho kamora opereishene ke:
\[
\begin{pmatrix}
2 & 3 & 1 \\
0 le -5 le -4 \\
0 & \frac{1}{2} & \frac{7}{2} \\
\end{pmatrix}
=
\begin{pmatrix}
1 \\
-2 \\
7 \\
\end{pmatrix}
\]
2. Ho kenya letsoho ka morao:
Qala ho tloha karolong e ka tlase 'me u sebetse ho ea holimo ho fumana boleng bo fetohang butle-butle.
– \(z = 1\)
– \(y = \frac{-19}{10}\)
– \(x = \frac{31}{10}\)
Melemo le Meeli
Mokhoa oa ho felisa Gaussian o na le melemo e mengata. Tsena li kenyelletsa:
– Tšebeliso: E ka sebelisoa lits'ebetsong tse nang le palo e kholo ea li-variable.
– Boemo ba Khomphutha: Bokgoni ba khomphutha bo theko e tlase haholo ha bo bapiswa le mesebetsi ya motheo.
– E ka sebelisoa maemong a fapaneng: Ka bobeli ka mefuta e menyenyane le e meholo ea matrix.
Leha ho le jwalo, mokhoa ona o boetse o na le mefokolo. Mohlala, maemong ao matrix e batlang e le bonngoe kapa e nang le ntlha e nyane haholo, diphoso tsa round-off e ka ba bothata bo boholo. Tšebeliso e hlokolosi ea tlhaloso ea lipalo ea hlokahala ntlheng ena.
Qetello
Mokhoa oa ho felisa oa Gaussian ke sesebelisoa se matla sa ho rarolla litsamaiso tsa li-equation tse otlolohileng, ka bobeli lipalo tsa khopolo-taba le lits'ebetsong tse sebetsang masimong a mangata. Ho tloha tlhahlobong ea boenjiniere ho ea moruong le lipalo-palo, Gauss o re sietse lefa le tšoarellang la mekhoa saenseng. Ho utloisisa melao-motheo ea motheo le ts'ebeliso ea eona maemong a sebele ke senotlolo ho mang kapa mang ea lakatsang ho tseba algebra e otlolohileng le ts'ebeliso ea eona.
