A modified Newton method with cubic convergence: the multivariate case
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., \textit{W. E. Bosarge, jun.} and \textit{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.
scientific article; zbMATH DE number 2091432
scientific article