Mokhoa oa ho Iteration oa ho Fumana Metso
Ho lipalo tse sebediswang, fisiks, boenjiniere le saense ya dikhomphutha, bothata ba "ho fumana metso" bo hlaha kgafetsa haholo. Motso ke boleng ba \(x\) bo etsang hore mosebetsi e be lefela, ke hore, tharollo ya equation:
\[
f(x)=0
\]
Hase di-equation tsohle tse nang le ditharollo tse ka hlahiswang ka diforomo tse kwetsweng, jwalo ka di-equation tsa quadratic. Bakeng sa maemo a mangata a lefatshe la nnete—jwalo ka di-equation tse rarahaneng tse seng tsa mola—re hloka mekgwa ya dipalo. E nngwe ya mekgwa ya bohlokwa ka ho fetisisa ke mokgwa wa ho pheta-pheta, mokgwa o hlahisang letoto la ditharollo tse hakanyetswang tse atamelang motso ka ho pheta-pheta.
Sengoloa sena se tšohla likhopolo tsa motheo tsa mekhoa ea ho pheta-pheta, maemo a ho kopana ha tsona, le mekhoa e meng e sebelisoang khafetsa ea ho pheta-pheta bakeng sa ho fumana metso.
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1. Khopolo ea Motheo ea Mokhoa oa ho Iteration
Mokhoa oa ho pheta-pheta o sebetsa ka ho hakanya qalong \(x_0\), ebe o e ntlafatsa butle-butle ho fumana tatellano:
\[
x_0, x_1, x_2, \matheba, x_n
\]
ka litebello:
\[
x_n \ho isa ho \alpha
\]
moo \(\alpha\) e leng motso wa nnete wa equation \(f(x)=0\).
Ka kakaretso, mokhoa oa ho pheta-pheta o fetola bothata \(f(x)=0\) hore e be sebopeho se lekanang:
\[
x = g(x)
\]
Ebe phetoho e etsoa:
\[
x_{n+1} = g(x_n)
\]
Haeba ts'ebetso ena e kopana, ntlha e tsitsitseng ea \(g(x)\) ke tharollo ea motso oa equation ea pele.
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2. Kopano: Phetoho e Atleha Neng?
Hase mesebetsi yohle \(g(x)\) e hlahisang diphetolelo tse tsitsitseng. E le hore phetoho \(x_{n+1}=g(x_n)\) e kopane le motso \(\alpha\), maemo a akaretsang a sebediswang hangata ke:
1. \(g(\alpha)=\alpha\) (motso ke ntlha e tsitsitseng)
2. \(|g'(\alpha)| < 1\) (ho kobeha ha sebaka) Temoho ya \(|g'(\alpha)| < 1\) ke: haufi le tharollo, tshebetso \(g\) ha e "tebile haholo", kahoo phetoho ka nngwe e tlisa boleng ba \(x_n\) haufi, eseng ho feta moo. Kopano le yona e angwa ke kgakanyo ya pele. Mekhoa e mmedi e tshwanang e ka atleha kapa ya hloleha ho itshetlehile ka \(x_0\). --- 3. Mokgwa wa ho arola ka bobedi e le phetapheto e bonolo Le hoja hangata e arolwa ka thoko, mokgwa wa ho arola ka bobedi o ka bonwa e le mokgwa o matla haholo wa phetapheto. Maemo ke: tshebetso \(f(x)\) e tswela pele karolong \([a,b]\) mme ho na le phetoho ya letshwao: \[ f(a)\cdot f(b) < 0 \] Ke hore, ho na le motso pakeng tsa \(a\) le \(b\). Algorithm: 1. Bala ntlha e bohareng \(c=\frac{a+b}{2}\) 2. Fumana sebaka se ka tlase se ntseng se koahetse motso (ho latela phetoho ea letšoao) 3. Pheta ho fihlela mamello e fihletsoe Molemo oa mokhoa ona: e tla kopana haeba boemo ba phetoho ea letšoao bo fihlelloa. Bothata: kopano e lieha hobane phoso e fokotseha hoo e ka bang ka halofo ka pheta-pheto e 'ngoe le e 'ngoe (pheta-pheto e otlolohileng). --- 4. Mokhoa oa Pheta-pheto ea Lintlha tse Tiileng Ena ke mokhoa o tobileng ka ho fetisisa oa pheta-pheto: \[ x_{n+1} = g(x_n) \] Mehato: 1. Fetola \(f(x)=0\) ho \(x=g(x)\) 2. Khetha khakanyo ea pele \(x_0\) 3. Pheta-pheto ho fihlela \(|x_{n+1}-x_n|\) kapa \(|f(x_n)|\) e le nyane ho feta mamello Molemo ke bonolo. Leha ho le joalo, mokhoa ona o tebile haholo khethong ea \(g(x)\). Bakeng sa equation e tšoanang, ho na le litsela tse ngata tsa ho ngola \(x=g(x)\), empa ke tse ling tsa tsona feela tse kopanang.
Mohlala, haeba re batla ho fumana metso ea \(f(x)=x^3-2x-5\), re ka ngola: - \(x = \sqrt[3]{2x+5}\) e le hore \(g(x)=\sqrt[3]{2x+5}\) Ebe re pheta \(x_{n+1}=\sqrt[3]{2x_n+5}\). Katleho ea pheta-pheto e itšetlehile ka hore na \(|g'(x)|<1\) e potoloha motso. --- 5. Mokhoa oa Newton-Raphson: Phetoho e Potlakileng e Thehiloeng ho Derivative Mokhoa oa Newton-Raphson ke o mong oa mekhoa e tsebahalang haholo hobane ho kopana ha oona hangata ho potlakile haholo. Foromo ea pheta-pheto ke: \[ x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \] Tlhaloso: ho \(x_n\), re haha tangent ho mosebetsi \(f(x)\). Ho kopana ha tangent le axis ea \(x\) ho sebelisoa e le khakanyo e latelang. Melemo: - Ho kopana ha quadratic (ka potlako haholo) haeba e le haufi le motso le \(f'(\alpha)\neq 0\). Mathata: - E hloka derivative ea \(f'(x)\). - E ka hloleha haeba khakanyo ea pele e le mpe, kapa haeba \(f'(x_n)\) e le haufi le lefela, e leng se etsang hore mohato oa ho pheta-pheta o se tsitse. Mokhoa ona o sebelisoa haholo ho ntlafatsong, ho etsa mohlala oa fisiks, le ho sebelisa k'homphieutha ea boenjiniere ka lebaka la bokhoni ba oona ha maemo a le matle. --- 6. Mokhoa oa Secant: Mokhoa o Mong oa Newton o se nang Li-Derivatives Haeba li-derivatives li le thata ho li bala, mokhoa oa secant o fana ka tumellano. Khopolo e ka sehloohong ke ho hakanya derivative ka dipharologanyo tse fokolang: \[ f'(x_n)\approx \frac{f(x_n)-f(x_{n-1})}{x_n-x_{n-1}} \] Kahoo foromo ya iteration ke: \[ x_{n+1}=x_n - f(x_n)\,\frac{x_n-x_{n-1}}{f(x_n)-f(x_{n-1})} \] Mokhoa ona o hloka diqapetso tse pedi tsa pele: \(x_0\) le \(x_1\). Lebelo la yona la ho kopana ka kakaretso le betere ho feta karohano e bonolo ya karohano le ntlha e tsitsitseng, le hoja hangata e lieha hanyane ho feta Newton. Leha ho le jwalo, hobane ha e hloke di-derivatives, secant hangata e sebetsa haholoanyane.
--- 7. Mekhoa ea ho Emisa Palong ea lipalo, ho pheta-pheta ho lokela ho emisoa ha ho nepahetse ka ho lekaneng kapa haeba ho belaelloa hore ha ho kopane. Mekhoa e akaretsang: 1. Phoso e nyane ea ho pheta-pheta: \[ |x_{n+1}-x_n|<\varepsilon \] 2. Boleng ba mosebetsi bo haufi le lefela: \[ |f(x_n)|<\varepsilon \] 3. Moeli o moholo oa ho pheta-pheta ho thibela li-loops tse sa feleng: \[ n \le n_{\max} \] Khetho ea mamello \(\varepsilon\) e itšetlehile ka litlhoko: li-simulation tsa boenjiniere li ka hloka mamello e tiileng, ha lipalo tse thata li sa phutholoha. --- 8. Papiso e Khutšoanyane ea Mekhoa ea ho Pheta-pheta Ka kakaretso: - Ho arola ka bobeli: ho tsitsitse ka ho fetisisa, ho tiisitsoe hore ho tla kopana (phetoho ea pontšo e fanoeng), empa e lieha. - Ntlha e tsitsitseng: e bonolo haholo, empa ho kopana ha ho tiisetsoe kamehla. - Newton-Raphson: e potlakile haholo, empa e hloka lintho tse tsoang ho eona 'me e ela hloko likhakanyo tsa pele. - Secant: ha ho hlokahale di-derivative, di potlakile haholo, empa di ka ba tse sa tsitsang jwalo ka karohano ya dipalo. Ha e le hantle, kgetho ya mokgwa e itshetlehile hodima mofuta wa mosebetsi, ho fumaneha ha di-derivative, tlhoko ya lebelo le botsitso. --- Qetello Mekhoa ya phetapheto ke motheo wa ho fumana metso ya dipalo bakeng sa di-equation tse seng tsa mola. Ka ho aha tatellano ya dikgakanyo tse ntjhafaditsweng ka phetapheto, re ka atamela tharollo ha mekgwa ya tlhahlobo e sa fumanehe. Ho utlwisisa kopano, kgetho ya ho hakanya ha pele, le tekanyetso ya ho emisa di bohlokwa bakeng sa phetapheto ho hlahisa metso e nepahetseng le e sebetsang hantle. Ditshebedisong tsa lefatshe la nnete, leano le kopaneng le atisa ho sebediswa: ho qala ka mokgwa o tsitsitseng jwalo ka karohano ya dipalo ho "notlela" karohano ya motso, ebe o fetohela ho Newton kapa karohano ho potlakisa kopano. Sena se fihlella tekano pakeng tsa ho tshepahala le lebelo—dikarolo tse pedi tsa bohlokwa haholo k'homphieutheng ya dipalo. --- Haeba o lakatsa, nka eketsa mohlala wa mohato ka mohato (wa dipalo) wa mekgwa efe kapa efe e ka hodimo ho etsa hore sengoloa se be se tiileng haholoanyane.
