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A modified Newton method with cubic convergence: the multivariate case. (English) Zbl 1059.65044

Let $m,n\in ℕ$, $F:{ℝ}^{m}\to {ℝ}^{n}$ be a sufficiently differentiable mapping, ${x}_{0},{x}_{n}\in {ℝ}^{m}$, ${F}_{n}:=F\left({x}_{n}\right)$, and ${F}_{n}^{\text{'}}:={F}^{\text{'}}\left({x}_{n}\right)$ the Jacobi matrix at ${x}_{n}$. The author considers the following multipoint method for the approximate computation of a zero of $F$:

$\text{For}\phantom{\rule{4.pt}{0ex}}n=1,2,\cdots :{F}_{n}+{F}_{n}^{\text{'}}{c}_{n}=0⇒{c}_{n},\phantom{\rule{4pt}{0ex}}{F}_{n}+{F}^{\text{'}}\left({x}_{n}+\frac{1}{2}{c}_{n}\right){d}_{n}=0⇒{d}_{n},\phantom{\rule{4pt}{0ex}}{x}_{n+1}:={x}_{n}+{d}_{n}·\phantom{\rule{2.em}{0ex}}\left(+\right)$

He shows that (+) under appropiate conditions converges locally with order three to a simple zero of $F$. Moreover, he uses two nontrivial examples to compare the computational results of (+) with those of Newton’s method but not, e.g. with the respective results of a similar multipoint method of – under appropiate assumptions – order three which differs from (+) in the second equation which is replaced by $F\left({x}_{n}+{c}_{n}\right)+{F}_{n}^{\text{'}}{c}_{n}=0$ (and then ${x}_{n+1}:={x}_{n}+{c}_{n}+{d}_{n}\right)$.

In this case, the two systems of linear equations in question to be solved in each iteration step have the same coefficient matrix, respectively [Comp., e.g., W. E. Bosarge, jun. and P. L. Falb, J. Optimization Theory Appl. 4, 155–166 (1969; Zbl 0172.18703)]. Finally, the author shortly discusses how certain numerical difficulties – e.g., if the Jadobi matrix at the zero of $F$ is singular – could be dealt with, and under what conditions (+) may converge globally.

##### MSC:
 65H10 Systems of nonlinear equations (numerical methods)