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

Let $$m,n\in\mathbb{N}$$, $$F: \mathbb{R}^m\to\mathbb{R}^n$$ be a sufficiently differentiable mapping, $$x_0, x_n\in\mathbb{R}^m$$, $$F_n:= F(x_n)$$, 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., 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 Numerical computation of solutions to systems of equations

Zbl 0172.18703

Maple; GIANT
Full Text:

### References:

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