Nzira yekuwana midzi yeNewton Raphson
Pendauluan
Nzira yeNewton-Raphson inzira inoshanda yekuverenga nhamba yekuwana mhinduro dzinofungidzirwa dzemaequation asiri emutsara. Yakatanga kuunzwa naIsaac Newton uye yakazogadziriswa naJoseph Raphson. Mumasvomhu nemakombiyuta, nzira yeNewton-Raphson inzira inodzokororwa inoshandiswa kuwana midzi yebasa chairo.
Ramba uchiverenga chinyorwa chino kuti unzwisise misimboti yekutanga yenzira yeNewton-Raphson, matanho ayo akadzama, mashandisirwo ayo muzviitiko zvakasiyana-siyana, uye zvakanakira nezvayakaipira.
Nheyo dzekutanga dzeNewton-Raphson Method
Chaizvoizvo, nzira yeNewton-Raphson ine chinangwa chekufungidzira midzi ye equation `f(x) = 0`. Nzira iyi inotanga nekufungidzira kwekutanga kwe `x0`. Kubva pano, kufungidzira kuri nani kwemidzi kunowanikwa uchishandisa derivative yebasa.
Pamasvomhu, nzira yeNewton-Raphson inoratidzwa nefomura inotevera:
\[ x_{n+1} = x_n – \frac{f(x_n)}{f'(x_n)} \]
Di mana:
– \( x_{n+1} \) ndiyo poindi inotevera inofungidzirwa.
– \( x_n \) ndiyo poindi inofungidzirwa parizvino.
– \( f(x_n) \) ndiko kukosha kwebasa riri pa \( x_n \).
– \( f'(x_n) \) ndiko kukosha kwechinhu chinobva pabasa riri pa \( x_n \).
Fomura iyi yakavakirwa pakuenzanisa kwakadzama kwebasa rakaoma, uko kufananidza uku kwakadzama kunotorwa semutsetse wetangent panzvimbo yekufananidza iripo. Mutsetse uyu wetangent unozopa x-intercept iyo ichave iri nani pakuenzanisa mudzi mukudzokorora kunotevera.
Matanho aNewton-Raphson
Matanho makuru ekushandisa nzira yeNewton-Raphson ndeaya anotevera:
1. Sarudza Kufungidzira Kwekutanga: Tanga nemutengo wekutanga \( x_0 \). Mutengo wekutanga wakasarudzwa uchakanganisa zvakanyanya kubatana kwenzira iyi.
2. Ongorora Mashandiro neZvinobva Maari: Verenga kukosha kwebasa uye kukosha kwezvinobva pabasa panzvimbo \( x_n \).
3. Verenga Kufungidzira Kunotevera: Shandisa fomura yeNewton-Raphson kuti uwane kukosha kunotevera \( x_{n+1} \).
4. Tarisa kuti pane Kubatana: Tarisa kana kukosha kwakafungidzirwa kwe \( x_{n+1} \) kuri pedyo zvakakwana nemudzi chaiwo uchishandisa nzira yekumira, yakaita se:
– Kuchinja kwakakwana pakati pema "iterations" maviri \( |x_{n+1} – x_n| \) kudiki.
– Kukosha kwebasa riri pedyo ne zero \( |f(x_{n+1})| \) idiki.
5. Dzokorora: Kana zvinodiwa pakumisa zvikasazadzikiswa, dzokera kudanho rechipiri nekutsiva \( x_n \) na \( x_{n+1} \).
Maitiro aya anoenderera mberi kusvika mhinduro chaiyo yawanikwa.
Mienzaniso yekushandiswa kwaNewton-Raphson
Ngatishandisei nzira iyi pamuenzaniso chaiwo. Ngatitii tinoda kuwana midzi ye equation \( f(x) = x^2 – 2 \).
Danho 1: Kufungidzira Kwekutanga
Ngatitii tatanga na \( x_0 = 1 \).
Danho rechipiri: Ongorora Basa uye Zvibereko Zvaro
Basa \( f(x) = x^2 – 2 \) uye chinobva pabasa \( f'(x) = 2x \).
Kuongorora pa \( x_0 = 1 \):
– \( f(x_0) = 1^2 – 2 = -1 \)
– \( f'(x_0) = 2 \kawa 1 = 2 \)
Danho rechitatu: Verenga Kufungidzira Kunotevera
Kushandisa fomura yaNewton-Raphson:
\[ x_{1} = 1 – \frac{-1}{2} = 1 + 0.5 = 1.5 \]
Danho rechina: Tarisa Kubatana
Tarisa shanduko yakakwana uye kukosha kwebasa:
– \( |x_1 – x_0| = |1.5 – 1| = 0.5 \)
– \( |f(1.5)| = |1.5^2 – 2| = |2.25 – 2| = 0.25 \)
Tinoenderera mberi nekudzokorora kunotevera nekuti zvinodiwa hazvina kuzadzikiswa.
Danho rechishanu: Dzokorora
Kuongorora pa \( x_1 = 1.5 \):
– \( f(x_1) = 1.5^2 – 2 = 0.25 \)
– \( f'(x_1) = 2 \kawa 1.5 = 3 \)
Kushandisa fomura yaNewton-Raphson zvakare:
\[ x_2 = 1.5 – \frac{0.25}{3} = 1.5 – 0.0833 = 1.4167 \]
Tarisa shanduko yakakwana uye kukosha kwebasa:
– \( |x_2 – x_1| = |1.4167 – 1.5| = 0.0833 \)
– \( |f(1.4167)| = |1.4167^2 – 2| \inenge 0.0069 \)
Sezvo kudzokorora kusina kukwana, tinoenderera mberi kusvika zvinodiwa zvekumira zvazadzikiswa.
Maitiro aya acharamba achienderera mberi kusvika zvasangana.
Zvakanakira uye Zvakaipira zveNewton-Raphson Method
Kelebihan
1. Kumhanya kweKusangana: Nzira yeNewton-Raphson ine kumhanya kwekubatana kwemativi mana, zvichireva kuti huwandu hwekudzokorora hunodiwa kuti usvike pamudzi hudiki zvikuru kana tichienzanisa nedzimwe nzira dzakadai senzira yekubvisa zvidimbu zviviri kana nzira yekubvisa zvidimbu zviviri.
2. Kururama: Nzira iyi inowanzova yakarurama pakutsvaga midzi kana fungidziro yekutanga iri pedyo nemidzi chaiyo.
3. Kushandiswa Kwakakura: Inogona kushandiswa kumhando dzakasiyana dzemabasa, ese ari maviri epolynomial neasiri epolynomial.
Kushaya
1. Kuvimba neMavambo Ekutanga: Mhedzisiro yekupedzisira inonyanya kutsamira pakuvamba kwekufungidzira. Kana kufungidzira kuri kure nemudzi, nzira yacho inogona kukundikana kana kuda kudzokorora kakawanda.
2. Derivative Inofanira Kuzivikanwa: Nzira iyi inoda kuverenga derivative yebasa, izvo zvinogona kuoma kana kusashanda kune mamwe mabasa akaomarara.
3. Hazvina Simba: Nzira iyi haigari ichienderana. Kune mamwe mamiriro ezvinhu akakosha ayo nzira iyi inogona kutadza, akadai sekunge basa racho rine chinhu chakakosha kana shanduko huru mu derivative.
Mhedziso
Nzira yeNewton-Raphson chishandiso chine simba mukuverenga nhamba chinotibvumira kuwana nekukurumidza uye nemazvo midzi ye equation isina mutsara. Zvisinei, senzira dzese dzekuverenga nhamba, ine miganhu nemamiriro ezvinhu apo ingasashande zvakanaka. Kunzwisisa kwakakwana mabasa uye ma derivatives, pamwe nekusarudza ma principal values akakodzera, ndizvo zvakakosha pakushandisa nzira iyi zvinobudirira.
Nekunzwisisa uye kushandisa kwakakodzera, nzira yeNewton-Raphson inogona kuva mhinduro inoshanda kumatambudziko akasiyana-siyana ekutsvaga midzi mumasvomhu nesainzi yemakombiyuta.
