Ke ʻAno Loaʻa ʻana o ke Aʻa o Newton Raphson
Pendahuluan
ʻO ke ʻano hana Newton-Raphson kahi ʻano hana helu kūpono no ka loaʻa ʻana o nā hopena kokoke i nā kaulike nonlinear. Ua hoʻolauna mua ʻia e Isaac Newton a ma hope ua hoʻoponopono ʻia e Joseph Raphson. I ka makemakika a me ka helu ʻana, ʻo ke ʻano hana Newton-Raphson kahi ʻano hana iterative i hoʻohana ʻia e loaʻa ai nā aʻa o kahi hana maoli.
E hoʻomau i ka heluhelu ʻana i kēia ʻatikala e hoʻomaopopo i nā loina kumu o ke ʻano Newton-Raphson, kāna mau ʻanuʻu kikoʻī, kona hoʻohana ʻana i nā hihia like ʻole, a me kona mau pono a me nā hemahema.
Nā Kumumanaʻo Kumu o ke ʻAno Newton-Raphson
ʻO ke kumu, ʻo ke ʻano hana Newton-Raphson ka manaʻo e kuhi i nā aʻa o ka hoohalike `f(x) = 0`. Hoʻomaka kēia ʻano hana me kahi kuhi mua o `x0`. Mai kēia wahi, loaʻa kahi kuhi maikaʻi aʻe o nā aʻa me ka hoʻohana ʻana i ka derivative o ka hana.
Ma ke ʻano makemakika, ua hōʻike ʻia ke ʻano Newton-Raphson e ke ʻano penei:
\[ x_{n+1} = x_n – \frac{f(x_n)}{f'(x_n)} \]
Ma hea:
– ʻO \( x_{n+1} \) ke kiko i manaʻo ʻia aʻe.
– ʻO \( x_n \) ke kiko i manaʻo ʻia i kēia manawa.
– ʻO \( f(x_n) \) ka waiwai o ka hana ma \( x_n \).
– ʻO \( f'(x_n) \) ka waiwai o ka derivative o ka hana ma \( x_n \).
Hoʻokumu ʻia ke ʻano hana ma luna o kahi hoʻokokoke linear o kahi hana paʻakikī, kahi e lawe ʻia ai kēia hoʻokokoke linear ma ke ʻano he laina tangent ma ke kiko hoʻokokoke o kēia manawa. A laila hāʻawi kēia laina tangent i kahi intercept-x e lilo i hoʻokokoke maikaʻi aʻe o ke aʻa i ka hana hou aʻe.
Nā ʻanuʻu Newton-Raphson
Eia nā ʻanuʻu nui o ke ʻano hana Newton-Raphson:
1. E koho i kahi Kuhi mua: E hoʻomaka me kahi waiwai mua \( x_0 \). ʻO ka waiwai mua i koho ʻia e hoʻopilikia nui i ka hui ʻana o kēia ʻano hana.
2. Loiloi i nā Hana a me kā lākou Derivatives: E helu i ka waiwai hana a me ka waiwai derivative hana ma ke kiko \( x_n \).
3. E helu i ka Kuhi aʻe: E hoʻohana i ke ʻano hana Newton-Raphson e loaʻa ai ka waiwai i kuhi ʻia aʻe \( x_{n+1} \).
4. E nānā no ka Convergence: E nānā inā kokoke ka waiwai i manaʻo ʻia o \( x_{n+1} \) i ke kumu maoli ma o ka hoʻohana ʻana i kahi pae hoʻōki, e like me:
– He liʻiliʻi ka loli maoli ma waena o ʻelua mau hana hou \( |x_{n+1} – x_n| \).
– He liʻiliʻi ka waiwai hana ma ke kiko kokoke i ka zero \( |f(x_{n+1})| \) .
5. E hana hou: Inā ʻaʻole i hoʻokō ʻia nā pae hoʻōki, e hoʻi i ka hana 2 ma ke pani ʻana iā \( x_n \) me \( x_{n+1} \).
Hoʻomau kēia kaʻina hana hou a hiki i ka loaʻa ʻana o kahi hopena pololei.
Nā Laʻana o nā Hoʻohana ʻana o Newton-Raphson
E hoʻopili kākou i kēia ʻano hana i kahi laʻana kikoʻī. Manaʻo mākou e makemake mākou e ʻimi i nā aʻa o ka hoohalike \( f(x) = x^2 – 2 \).
KaʻAnuʻu Hana 1: Kuhi mua
Manaʻo mākou e hoʻomaka me \( x_0 = 1 \).
KaʻAnuʻu Hana 2: Loiloi i ka Hana a me kāna mau Derivatives
ʻO ka hana \( f(x) = x^2 – 2 \) a me ka derivative o ka hana \( f'(x) = 2x \).
Ka loiloi ma \( x_0 = 1 \):
– \( f(x_0) = 1^2 – 2 = -1 \)
– \( f'(x_0) = 2 \times 1 = 2 \)
KaʻAnuʻu Hana 3: E helu i ka Manaʻo Aʻe
Ke hoʻohana nei i ke ʻano hana Newton-Raphson:
\[ x_{1} = 1 – \frac{-1}{2} = 1 + 0.5 = 1.5 \]
KaʻAnuʻu Hana 4: E Nānā i ka Hui ʻana
E nānā i ka loli piha a me ka waiwai hana:
– \( |x_1 – x_0| = |1.5 – 1| = 0.5 \)
– \( |f(1.5)| = |1.5^2 – 2| = |2.25 – 2| = 0.25 \)
Ke hoʻomau nei mākou i ka hana aʻe no ka mea ʻaʻole i hoʻokō ʻia nā kūlana.
KaʻAnuʻu Hana 5: E hana hou
Ka loiloi ma \( x_1 = 1.5 \):
– \( f(x_1) = 1.5^2 – 2 = 0.25 \)
– \( f'(x_1) = 2 \times 1.5 = 3 \)
Ke hoʻohana hou nei i ke ʻano Newton-Raphson:
\[ x_2 = 1.5 – \frac{0.25}{3} = 1.5 – 0.0833 = 1.4167 \]
E nānā i ka loli piha a me ka waiwai hana:
– \( |x_2 – x_1| = |1.4167 – 1.5| = 0.0833 \)
– \( |f(1.4167)| = |1.4167^2 – 2| \approx 0.0069 \)
ʻOiai ʻaʻole i lawa ka hui ʻana o ka hana hou ʻana, hoʻomau mākou a hiki i ka hoʻokō ʻia ʻana o nā pae hoʻōki.
E hoʻomau ʻia kēia kaʻina hana a hiki i ka loaʻa ʻana o ka hui ʻana.
Nā Pōmaikaʻi a me nā Pōʻino o ke ʻAno Newton-Raphson
ʻOi aku
1. Ka wikiwiki o ka hui ʻana: He wikiwiki o ka hui ʻana o quadratic ko ke ʻano hana Newton-Raphson, ʻo ia hoʻi, he liʻiliʻi loa ka helu o nā hana hou e pono ai e hoʻokokoke aku i ke aʻa i hoʻohālikelike ʻia me nā ʻano hana ʻē aʻe e like me ke ʻano bisection a i ʻole ke ʻano secant.
2. Pololei: ʻOi aku ka pololei o kēia ʻano hana i ka loaʻa ʻana o nā aʻa inā kokoke ka kuhi mua i ke aʻa maoli.
3. Hoʻohana ākea: Hiki ke hoʻopili ʻia i nā ʻano hana like ʻole, ʻo ka polynomial a me ka non-polynomial.
hapa
1. Hilinaʻi ma luna o nā Waiwai Mua: ʻO ka hopena hope loa e hilinaʻi nui ʻia ma luna o ka waiwai i manaʻo mua ʻia. Inā mamao loa ka manaʻo mai ke kumu, hiki ke hāʻule ke ʻano hana a i ʻole e koi i nā hana hou he nui.
2. Pono e ʻike ʻia ka Derivative: Pono kēia ʻano hana i ka helu ʻana i ka derivative o ka hana, hiki ke paʻakikī a kūpono ʻole paha no kekahi mau hana paʻakikī.
3. ʻAʻole Paʻa: ʻAʻole hui mau kēia ʻano hana. Aia kekahi mau kūlana kūikawā e hiki ai i kēia ʻano hana ke hāʻule, e like me ka loaʻa ʻana o kahi kiko koʻikoʻi o ka hana a i ʻole kahi loli koʻikoʻi i ka derivative.
Ka hopena
He mea hana ikaika ke ʻano hana Newton-Raphson i ka helu helu e hiki ai iā mākou ke loaʻa koke a pololei i nā aʻa o kahi hoʻohālikelike nonlinear. Eia nō naʻe, e like me nā ʻano hana helu āpau, he mau palena a me nā kūlana kahi e hana maikaʻi ʻole ai. ʻO ka hoʻomaopopo piha ʻana i nā hana a me nā derivatives, a me ke koho ʻana i nā waiwai mua kūpono, he mea nui ia i ka hoʻohana pono ʻana i kēia ʻano hana.
Me ka hoʻomaopopo pono a me ka hoʻopili ʻana, hiki i ke ʻano Newton-Raphson ke lilo i hopena kūpono i nā pilikia ʻimi aʻa like ʻole i ka makemakika a me ka ʻepekema kamepiula.
