Let , be a sufficiently differentiable mapping, , , and the Jacobi matrix at . The author considers the following multipoint method for the approximate computation of a zero of :
He shows that (+) under appropiate conditions converges locally with order three to a simple zero of . 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 (and then .
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 is singular – could be dealt with, and under what conditions (+) may converge globally.