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**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.

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.

Reviewer: B. Döring (Düsseldorf)

### MSC:

65H10 | Numerical computation of solutions to systems of equations |

### Keywords:

systems of nonlinear equations; multipoint method of order three; generalized Bloch equation; convergence; comparison of methods; rootfinding; Newton method; coupled algebraic equations; affine-invariant Newton technique; hybrid method### Citations:

Zbl 0172.18703
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\textit{H. H. H. Homeier}, J. Comput. Appl. Math. 169, No. 1, 161--169 (2004; Zbl 1059.65044)

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### References:

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