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

Let m,n, F: m n be a sufficiently differentiable mapping, x 0 ,x n 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:

Forn=1,2,:F n +F n ' c n =0c n ,F n +F ' (x n +1 2c n )d n =0d n ,x n+1 :=x n +d n ·(+)

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.

65H10Systems of nonlinear equations (numerical methods)