Recurrence relations for the harmonic mean Newton's method in Banach spaces (Q2917659)
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scientific article; zbMATH DE number 6088909
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | Recurrence relations for the harmonic mean Newton's method in Banach spaces |
scientific article; zbMATH DE number 6088909 |
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1 October 2012
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Newton Method
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nonlinear equations
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Banach space
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0.9358851
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0.90603083
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0.89391834
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0.87913954
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0.87202716
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0.86950415
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Recurrence relations for the harmonic mean Newton's method in Banach spaces (English)
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To solve the nonlinear problem \(F(x)=0\) in a Banach space, the authors consider a modification of the Newton method: NEWLINE\[NEWLINE\begin{aligned}NEWLINEx_{n+1} &= y_n-\frac12H(x_n,y_n)(y_n-x_n); \\ NEWLINEy_n&=x_n-\Gamma_nF(x_n);\\ NEWLINEH(x_n,y_n)&=\bar\Gamma_n[F'(y_n)-F'(x_n)].NEWLINE\end{aligned}NEWLINE\]NEWLINEHere \(\Gamma_n=F'(x_n)^{-1}\) and \(\bar\Gamma_n=F'(y_n)^{-1}\). Upon establishing recurrence relations for the proposed method they are able to prove the convergence of this method and derive the convergence rate. A numerical example is used to illustrate the performance of the method.
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