Quasi-Newton Methods, Motivation and Theory

Abstract

This paper is an attempt to motivate and justify quasi-Newton methods as useful modifications of Newton's method for general and gradient nonlinear systems of equations. References are given to ample numerical justification; here we give an overview of many of the important theoretical results and each is accompanied by sufficient discussion to make the results and henc6 the methods plausible.

Two classes of multisecant methods for nonlinear acceleration

“…Dennis and Moré [15] showed that the Broyden update can also be obtained by minimizing E(J k+1 ) = J k+1 − J k 2 F with respect to terms of J k+1 , subject to the secant condition (4).…”

Consistent tangent matrices for substepping schemes

“…The Jacobian is Equations (A.14) and (A.15) can be obtained using Sherman and Morrison's lemma [20]. Note that these equations only involve matrix products.…”

Semilocal analysis of equations with smooth operators

“…Consideration of these algorithms led to the research on so-called quasi-Newton methods, surveyed in [2,3]. Potra-Ptk [17,18,19] used nondiscrete induction to obtain convergence and error bounds for the Newton, multistep Newton, and generalized regular falsi methods.…”