Polynomial preconditioners for conjugate gradient calculations. (English) Zbl 0563.65020

Johnson, Olin G.; Micchelli, Charles A.; Paul, George

SIAM J. Numer. Anal. 20, 362-376 (1983).

The authors explore the computational form of the preconditioned conjugate gradient algorithm for the solution of a system of linear equations \(Ax=b\) where A is an \(n\times n\) symmetric positive matrix. The basic theory of generalized optimal polynomial preconditioners is given. A polynomial preconditioning recurrence formula is obtained. A new family of parametrized conjugate gradient algorithms with mth degree polynomial preconditioner is given. The authors conclude that the algorithm may be easily programmed. This article should be very useful for system programmers.

Reviewer: P.Stavre

MSC:

65F10

Iterative numerical methods for linear systems

Keywords:

preconditioned conjugate gradient algorithm; symmetric positive matrix; recurrence formula