On multiplicative structure in Quasi-Newton methods for nonlinear equations

We address the problem how additive and multiplicative structure in the derivatives can be exploited for the construction of Quasi-Newton approximations in smooth nonlinear equations. We derive a model algorithm and show its convergence properties based on a Broyden-like update rule. As a consequence of the use of exact multiplicative parts the convergence factor of the q-linear convergence rate is monotonically decreasing with the norm of the multiplicative part at the solution. Moreover, q-superlinear convergence can be shown, if certain compactness properties are valid, and q-quadratic convergence is obtained, if the multiplicative part vanishes at the solutionAvailable from TIB Hannover: RR 1843(92-22) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman