Preconditioners for the conjugate gradient algorithm using Gram–Schmidt and least squares methods

Abstract

This paper is devoted to the study of some preconditioners for the conjugate gradient algorithm used to solve large sparse linear and symmetric positive definite systems. The construction of a preconditioner based on the Gram–Schmidt orthogonalization process and the least squares method is presented. Some results on the condition number of the preconditioned system are provided. Finally, numerical comparisons are given for different preconditioners.

Keywords:

• Conjugate gradient method
• Preconditioner
• Gram–Schmidt orthogonalization
• Least squares
• Condition number