BIT Numerical Mathematics

, Volume 34, Issue 2, pp 313–317 | Cite as

A note on best conditioned preconditioners

  • X. -Q. Jin
Scientific Notes

Abstract

We discuss the solution of Hermitian positive definite systemsAx=b by the preconditioned conjugate gradient method with a preconditionerM. In general, the smaller the condition numberκ(M −1/2 AM −1/2 ) is, the faster the convergence rate will be. For a given unitary matrixQ, letM Q = {Q*Λ N Q | Λ n is ann-by-n complex diagonal matrix} andM Q + ={Q*Λ n Q | Λ n is ann-by-n positive definite diagonal matrix}. The preconditionerM b that minimizesκ(M −1/2 AM −1/2 ) overM Q + is called the best conditioned preconditioner for the matrixA overM Q + . We prove that ifQAQ* has Young's Property A, thenM b is nothing new but the minimizer of ‖MA F overM Q . Here ‖ · ‖ F denotes the Frobenius norm. Some applications are also given here.

AMS subject classification

65F10 65F15 65F35 

Key words

Toeplitz matrix circulant matrix best conditioned preconditioner preconditioned conjugate gradient method 

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Copyright information

© the BIT Foundation 1994

Authors and Affiliations

  • X. -Q. Jin
    • 1
  1. 1.Faculty of Science and TechnologyUniversity of MacauMacau

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