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A modified Newton method with cubic convergence: the multivariate case. (English) Zbl 1059.65044
Let $m,n\in\bbfN$, $F: \bbfR^m\to\bbfR^n$ be a sufficiently differentiable mapping, $x_0, x_n\in\bbfR^m$, $F_n:= F(x_n)$, and $F_n':= F'(x_n)$ the Jacobi matrix at $x_n$. The author considers the following multipoint method for the approximate computation of a zero of $F$: $$\text{For }n=1,2,\dots: F_n+ F_n'c_n= 0\Rightarrow c_n,\ F_n+ F'(x_n+\tfrac12 c_n) d_n= 0\Rightarrow d_n,\ x_{n+1}:= x_n+ d_n.\tag +$$ 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 $F(x_n+ c_n)+ F_n'c_n= 0$ (and then $x_{n+1}:= x_n+ c_n+ d_n)$. 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., {\it W. E. Bosarge, jun.} and {\it 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:
65H10Systems of nonlinear equations (numerical methods)
Software:
Maple
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References:
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