Tikanga Whakakore Gaussian: He Kupu Whakataki Hōhonu
Ko te tikanga whakakore Gaussian tētahi o ngā tikanga tino taketake, tino whānuitia hoki e whakamahia ana i roto i te ara taurangi rārangi hei whakaoti rapanga i ngā pūnaha whārite rārangi. I tapaina i muri i te tohunga pāngarau rongonui a Carl Friedrich Gauss, nāna i takoha nui ki ngā momo manga maha o te pāngarau. I roto i tēnei tuhinga, ka tūhuratia e mātou ngā ariā taketake, ngā tukanga, me ngā tauira tono o te tikanga whakakore Gaussian.
Hītori me te Papamuri
Ko Carl Friedrich Gauss, i noho i te mutunga o te rautau 18 me te tīmatanga o te rautau 19, e kiia ana ko ia tētahi o ngā tohunga pāngarau nui rawa atu o ngā wā katoa. Ko te tikanga whakakore e mōhiotia nei ko tōna ingoa i mua noa atu i te whānautanga o Gauss, engari ko tana takoha nui ko te whakapai ake me te whakatairanga i taua tikanga.
Te Hiranga o te Tikanga Whakakore Gaussian
I roto i te pāngarau me te pūtaiao rorohiko, he raruraru noa te whakaoti rapanga i ngā pūnaha whārite rārangi. Ko te āhua whānui o tētahi pūnaha whārite rārangi:
\[
a_{11}x_1 + a_{12}x_2 + … + a_{1n}x_n = b_1
\]
\[
a_{21}x_1 + a_{22}x_2 + … + a_{2n}x_n = b_2
\]
\[
...
\]
\[
a_{m1}x_1 + a_{m2}x_2 + … + a_{mn}x_n = b_m
\]
Ko te whāinga o te tikanga whakakore Gaussian he huri i tēnei pūnaha ki tētahi āhua māmā ake kia ngāwari ai te whakaoti mā te whakamahi i te whakakapinga whakamuri.
Te Tukanga Whakakorenga Gaussian
Ngā Hipanga Taketake
E rua ngā wāhanga matua o te tukanga whakakorenga Gaussian: te whakakorenga whakamua me te whakakapinga whakamuri.
1. Whakakorenga Whakamua
Ko te whāinga o tēnei taahiraa he huri i te pūnaha whārite hei matihiko tapatoru o runga. Ka tutuki tēnei mā te mahi i ngā mahi rarangi taketake, tae atu ki:
– Whakawhitiwhitinga rārangi-rua.
– Whakareatia he rarangi ki tētahi tau ehara i te kore.
– Tāpirihia, tangohia rānei ngā taurangi mai i tētahi rarangi ki tētahi atu.
Me kī he pūnaha whārite rārangi tā tātou kei roto i te āhua matihiko \(Ax = b\), ko \(A\) te matihiko tauwehenga, ko \(x\) te ira taurangi, ā, ko \(b\) te ira pumau. Ko ngā mahi i roto i te whakakorenga whakamua ko:
1. Tīpakohia he huānga pivot, e tīmata ana i te nuinga o te wā mai i \(a_{11}\).
2. Whakamahia te huānga pivot hei muku (kia kore) te huānga i raro iho i taua pou kotahi.
3. Whakahokia tēnei tukanga mō te huānga pivot e whai ake nei i raro i te rarangi hauroki.
Hei tauira, me titiro tātou ki tētahi pūnaha me ngā whārite e toru:
\[
a_{11}x_1 + a_{12}x_2 + a_{13}x_3 = b_1
\]
\[
a_{21}x_1 + a_{22}x_2 + a_{23}x_3 = b_2
\]
\[
a_{31}x_1 + a_{32}x_2 + a_{33}x_3 = b_3
\]
Ka tīmata tātou me te pivot \(a_{11}\), ka mahi i ngā mahi hei tango i a \(a_{21}\) me \(a_{31}\).
2. Whakakapinga Whakamuri
I muri i te whakakorenga whakamua, ka whiwhi tātou i tētahi pūnaha whārite e tohuhia ana e te matihiko o runga. Hei tauira:
\[
u_{11}x_1 + u_{12}x_2 + u_{13}x_3 = d_1
\]
\[
u_{22}x_2 + u_{23}x_3 = d_2
\]
\[
u_{33}x_3 = d_3
\]
I tēnei wāhanga, ka mahia te whakakapinga whakamuri mai i raro ki runga:
1. Mō \(x_3\): \(x_3 = d_3 / u_{33}\).
2. Mō \(x_2\): \(x_2 = (d_2 – u_{23}x_3) / u_{22}\).
3. Mō \(x_1\): \(x_1 = (d_1 – u_{12}x_2 – u_{13}x_3) / u_{11}\).
Ngā Tauira Taupānga
Hei whakamārama i te whakamārama i runga ake nei, me tango he tauira tuturu.
Me kī kei a tātou te pūnaha whārite rārangi e whai ake nei:
\[
2x + 3y + z = 1
\]
\[
4x + y – 2z = -2
\]
\[
3x + 2y + 3z = 7
\]
Kua tuhia ki te āhua matihiko:
\[
\begin{pmatrix}
2 me te 3 me te 1
4 me te 1 me te -2
3 me te 2 me te 3
\end{pmatrix}
\begin{pmatrix}
x \\
y \\
z \\
\end{pmatrix}
=
\begin{pmatrix}
1
-2
7
\end{pmatrix}
\]
1. Whakakorenga Whakamua:
– Tīpakohia te huānga pivot \(2\), te huānga tuatahi o te rarangi tuatahi.
– Waihangahia he kore huānga i raro i te huānga pivot tuatahi:
– Rārangi 2: \(4 – 2(2) = 0\)
– Rārangi 3: \(3 – \frac{3}{2}(2) = 0\)
– Ko ngā hua i muri i te pokanga:
\[
\begin{pmatrix}
2 me te 3 me te 1
0 me te -5 me te -4
0 me \frac{1}{2} me \frac{7}{2} \\
\end{pmatrix}
=
\begin{pmatrix}
1
-2
7
\end{pmatrix}
\]
2. Whakakapinga Whakamuri:
Tīmata mai i te huānga o raro, ka piki haere mārire ki te kimi i ngā uara taurangi.
– \(z = 1\)
– \(y = \frac{-19}{10}\)
– \(x = \frac{31}{10}\)
Ngā Painga me ngā Here
He maha ngā painga o te tikanga whakakorenga Gaussian. Ko ētahi o ēnei:
– Te Whaihuatanga: Ka taea te whakamahi ki ngā pūnaha he nui ake ngā taurangi.
– Taumata Rorohiko: He iti ake te iti o te whai huatanga o te rorohiko i ngā mahi taketake.
– Ka taea te whakamahi i roto i ngā āhuatanga rerekē: I roto i ngā āhua matihiko iti me ngā āhua matihiko nui.
Heoi anō, he iti anō hoki ngā herenga o tēnei tikanga. Hei tauira, i ngā āhuatanga e tata ana te takitahi o te matihiko, e iti ana rānei te whakatau, ka raru nui pea ngā hapa whakaawhiwhi. Me āta whakamahi i te whakamārama tau mō tēnei mea.
Whakamutunga
He taputapu kaha te tikanga whakakorenga Gaussian mō te whakaoti rapanga i ngā pūnaha whārite rārangi, i roto i te pāngarau ariā me ngā tono mahi puta noa i te whānuitanga o ngā mara. Mai i te tātari hangarau ki te ōhanga me te tatauranga, kua waiho mai e Gauss he taonga tuku iho pumau o ngā tikanga i roto i te pūtaiao. Ko te mārama ki ngā mātāpono taketake me tā rātou tono i roto i ngā horopaki o te ao tūturu he mea nui mō te hunga e hiahia ana ki te mōhio ki te pāngarau rārangi me ōna tono.
