Automatic preconditioning by limited memory quasi-Newton updating

José Luis Morales*, Jorge Nocedal

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

119 Scopus citations


This paper proposes a preconditioner for the conjugate gradient method (CG) that is designed for solving systems of equations Acursive Greek chi = bi with different right-hand-side vectors or for solving a sequence of slowly varying systems Akcursive Greek chi = bk. The preconditioner has the form of a limited memory quasi-Newton matrix and is generated using information from the CG iteration. The automatic preconditioner does not require explicit knowledge of the coefficient matrix A and is therefore suitable for problems where only products of A times a vector can be computed. Numerical experiments indicate that the preconditioner has most to offer when these matrix-vector products are expensive to compute and when low accuracy in the solution is required. The effectiveness of the preconditioner is tested within a Hessian-free Newton method for optimization and by solving certain linear systems arising in finite element models.

Original languageEnglish (US)
Pages (from-to)1079-1096
Number of pages18
JournalSIAM Journal on Optimization
Issue number4
StatePublished - 2000


  • Conjugate gradient method
  • Hessian-free Newton method
  • Limited memory method
  • Preconditioning
  • Quasi-Newton method

ASJC Scopus subject areas

  • Software
  • Theoretical Computer Science


Dive into the research topics of 'Automatic preconditioning by limited memory quasi-Newton updating'. Together they form a unique fingerprint.

Cite this