A New Sparsity Preserving Quasi-Newton Update for Solving Nonlinear Equations

A new quasi-Newton update is proposed based on a least relative change in the updated matrix, rather than on a least absolute change as is normally the case for quasi-Newton methods. The method corresponds to a quasi-Newton method weighted in a scale corresponding to a row scaling. The new method retains sparsity without having to include it as an affine transformation within the derivation. Unlike Broyden’s and Schubert’s methods, the method is scale-invariant. Comparisons are made on a set of problems that show promising results for the new method.