Tikanga Kimi Pūtake o Newton Raphson
Pendahuluan
He tikanga tau whai hua te tikanga Newton-Raphson mō te kimi i ngā otinga tata ki ngā whārite kore-raina. I whakaurua tuatahitia e Isaac Newton, ā, i muri mai ka whakatikatikaina e Joseph Raphson. I roto i te pāngarau me te rorohiko, he tikanga tāruarua te tikanga Newton-Raphson e whakamahia ana hei kimi i ngā pūtake o tētahi mahi tūturu.
Pānuihia tonutia tēnei tuhinga kia mārama ai koe ki ngā mātāpono taketake o te tikanga Newton-Raphson, ōna mahi taipitopito, tōna whakamahinga i ngā momo take, me ōna painga me ōna ngoikoretanga.
Ngā Mātāpono Taketake o te Tikanga Newton-Raphson
Ko te whāinga matua o te tikanga Newton-Raphson he whakatau tata i ngā pūtake o te whārite `f(x) = 0`. Ka tīmata tēnei tikanga me te whakatau tata tuatahi o `x0`. Mai i tēnei wā, ka whiwhihia he whakatau tata pai ake mō ngā pūtake mā te whakamahi i te pānga o te mahi.
I roto i te pāngarau, ka whakaatuhia te tikanga Newton-Raphson e te tātai e whai ake nei:
\[ x_{n+1} = x_n – \frac{f(x_n)}{f'(x_n)} \]
Kei hea:
– Ko \( x_{n+1} \) te pūwāhi whakatau tata e whai ake nei.
– Ko \( x_n \) te pūwāhi whakatau tata o nāianei.
– Ko te uara o te mahi i runga i te \( x_n \) ko \( f(x_n) \).
– Ko te uara o te pānga o te mahi i runga i te \( x_n \) ko te \( f'(x_n) \).
Ko te tātai e hangai ana ki tētahi whakatata rārangi o tētahi mahi matatini, e tangohia ai tēnei whakatata rārangi hei rārangi pātata i te pūwāhi whakatata o nāianei. Kātahi ka whakaratohia e tēnei rārangi pātata he taunga-x hei whakatata pai ake mō te pūtake i te mahi tāruarua e whai ake nei.
Ngā kaupae o Newton-Raphson
Anei ngā mahi matua o te tikanga Newton-Raphson:
1. Kōwhiria he Whakatau Tatau Tuatahi: Tīmata ki te uara tīmatanga \( x_0 \). Ka nui te pānga o te uara tīmatanga kua whiriwhiria ki te huihuinga o tēnei tikanga.
2. Arotakehia ngā Mahi me ā rātou Huanga Whakaputa: Tātaihia te uara mahi me te uara huanga whakaputa i te pūwāhi \( x_n \).
3. Tātaihia te Whakatau Tata e Whai Ake Nei: Whakamahia te tātai Newton-Raphson hei tiki i te uara whakatau tata e whai ake nei \( x_{n+1} \).
4. Tirohia mēnā he Ōritenga: Tirohia mēnā he tata te uara whakatau tata o \( x_{n+1} \) ki te pūtake tūturu mā te whakamahi i tētahi paearu whakamutu, pēnei i:
– He iti te huringa tino i waenga i ngā tāruarua e rua \( |x_{n+1} – x_n| \).
– He iti te uara mahi i te pūwāhi tata tata ki te kore \( |f(x_{n+1})| \) .
5. Whakahokia: Ki te kore e tutuki ngā paearu whakamutu, hoki ki te taahiraa 2 mā te whakakapi i te \( x_n \) ki te \( x_{n+1} \).
Ka haere tonu tēnei tukanga tāruarua kia kitea rā anō he otinga tika.
Ngā Tauira o ngā Whakamahinga o Newton-Raphson
Me whakamahi tātou i tēnei tikanga ki tētahi tauira motuhake. Me kī tātou e hiahia ana ki te kimi i ngā pūtake o te whārite \( f(x) = x^2 – 2 \).
Hipanga 1: Te Whakatau Tatau Tuatahi
Me tīmata tātou ki te \( x_0 = 1 \).
Hipanga 2: Arotakehia te Mahi me ōna Pānga
Ko te mahi \( f(x) = x^2 – 2 \) me te taupū o te mahi \( f'(x) = 2x \).
Aromatawai i \( x_0 = 1 \):
– \( f(x_0) = 1^2 – 2 = -1 \)
– \( f'(x_0) = 2 \whakanuia te 1 = 2 \)
Hipanga 3: Tātaihia te Whakatau Tata e Whai Ake Nei
Mā te whakamahi i te tātai Newton-Raphson:
\[ x_{1} = 1 – \frac{-1}{2} = 1 + 0.5 = 1.5 \]
Hipanga 4: Tirohia mēnā he Huinga
Tirohia te huringa tino me te uara mahi:
– \( |x_1 – x_0| = |1.5 – 1| = 0.5 \)
– \( |f(1.5)| = |1.5^2 – 2| = |2.25 – 2| = 0.25 \)
Ka haere tonu tātou ki te mahi e whai ake nei nā te mea kāore anō kia tutuki ngā paearu.
Hipanga 5: Whakahokia
Aromatawai i \( x_1 = 1.5 \):
– \( f(x_1) = 1.5^2 – 2 = 0.25 \)
– \( f'(x_1) = 2 \whakanuia te 1.5 = 3 \)
Mā te whakamahi anō i te tātai Newton-Raphson:
\[ x_2 = 1.5 – \frac{0.25}{3} = 1.5 – 0.0833 = 1.4167 \]
Tirohia te huringa tino me te uara mahi:
– \( |x_2 – x_1| = |1.4167 – 1.5| = 0.0833 \)
– \( |f(1.4167)| = |1.4167^2 – 2| \tata ki te 0.0069 \)
I te mea kāore anō kia tino tutuki te whakatauritenga, ka haere tonu tātou kia tutuki rā anō ngā paearu whakamutu.
Ka haere tonu tēnei tukanga kia tutuki rā anō te whakakotahitanga.
Ngā Painga me ngā Huakore o te Tikanga Newton-Raphson
Te taikaha
1. Tere o te Huinga: He tere o te huinga tapawhā tō te tikanga Newton-Raphson, arā, he iti noa iho te maha o ngā tāruarua e hiahiatia ana hei whakatata atu ki te pūtake ki te whakataurite ki ētahi atu tikanga pēnei i te tikanga wehewehe, te tikanga wehe rānei.
2. Te Tika: He tika ake te tikanga nei mō te kimi i ngā pūtake mena e tata ana te whakatau tata tuatahi ki te pūtake tuturu.
3. Whānuitanga Whakamahinga: Ka taea te whakamahi ki ngā momo mahi rerekē, arā, te pūrau me te kore-pūrau.
Te kore
1. Te Whakawhirinakitanga ki ngā Uara Tīmatanga: He tino whakawhirinaki te hua whakamutunga ki te uara tuatahi i whakatauhia. Mena kei tawhiti te whakatau tata i te pūtake, tera pea ka rahua te tikanga, ka hiahiatia rānei he maha ngā mahi tāruarua.
2. Me Mōhio te Pānga Taurite: Me tatau i te pānga taurite o te mahi i tēnei tikanga, he mea uaua pea, he mea kore rānei e whai hua mō ētahi mahi uaua.
3. Kāore i te Pakari: Kāore tēnei tikanga e hono tonu i ngā wā katoa. Tērā ētahi āhuatanga motuhake ka kore ai e angitu tēnei tikanga, pērā i te mea he pūwāhi nui kei te mahi, he panoni nui rānei kei roto i te tauwehenga.
Whakamutunga
He taputapu kaha te tikanga Newton-Raphson i roto i te rorohiko tau e āhei ai tātou ki te kimi tere me te tika i ngā pūtake o tētahi whārite kore-raina. Heoi, pērā i ngā tikanga tau katoa, he herenga me ngā āhuatanga e kore ai e pai te mahi. Ko te māramatanga hōhonu ki ngā mahi me ngā pānga, me te whiriwhiri i ngā uara tīmatanga e tika ana, he mea nui ki te whakamahi angitu i tēnei tikanga.
Mā te māramatanga tika me te whakamahinga tika, ka taea e te tikanga Newton-Raphson te noho hei otinga whai hua mō ngā raruraru kimi pūtake maha i roto i te pāngarau me te pūtaiao rorohiko.
