On a relationship between Quasi-Newton and dominant eigenvalue methods for the numerical solution of non-linear equations

Abstract

In order to compare Quasi-Newton methods with the dominant eigenvalue (DE) method of convergence acceleration, the latter is modified so that an acceleration step is taken at every iteration and the method of calculation of the required coefficients can be suitably chosen. A relationship is derived between the class of rank-one Quasi-Newton methods (QN1) and modified DE methods such that each QN1 method corresponds to a particular modified DE method. However, it was not possible to find a QN1 formula which would correspond to the DE method applied at every iteration. Such a formula would appear to require information in advance, from iteration yet to be done. The advantages of using the equivalent modified DE method instead of Broyden's QN1 method are examined. A new algorithm of QN1 methods is proposed, based on the equivalent modified DE method, which was at least one order of magnitude faster, for Broyden's mehtod[1] on an example problem, than the conventional implementation.