Two classes of multisecant methods for nonlinear acceleration

Haw Ren Fang, Yousef Saad

Research output: Contribution to journalArticlepeer-review

99 Scopus citations

Abstract

Many applications in science and engineering lead to models that require solving large-scale fixed point problems, or equivalently, systems of nonlinear equations. Several successful techniques for handling such problems are based on quasi-Newton methods that implicitly update the approximate Jacobian or inverse Jacobian to satisfy a certain secant condition. We present two classes of multisecant methods which allow to take into account a variable number of secant equations at each iteration. The first is the Broyden-like class, of which Broyden's family is a subclass, and Anderson mixing is a particular member. The second class is that of the nonlinear Eirola-Nevanlinna-type methods. This work was motivated by a problem in electronic structure calculations, whereby a fixed point iteration, known as the self-consistent field (SCF) iteration, is accelerated by various strategies termed 'mixing'.

Original languageEnglish (US)
Pages (from-to)197-221
Number of pages25
JournalNumerical Linear Algebra with Applications
Volume16
Issue number3
DOIs
StatePublished - Mar 1 2009

Keywords

  • Anderson mixing
  • Broyden's methods
  • Fixed point problems
  • Quasi-Newton methods
  • Self-consistent field (SCF) iteration

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