Inexact Preconditioned Conjugate Gradient Method with Inner-Outer Iteration

An important variation of preconditioned conjugate gradient algorithms is inexact precon-ditioner implemented with inner-outer iterations 5], where the preconditioner is solved by an inner iteration to a prescribed precision. In this paper, we formulate an inexact preconditioned conjugate gradient algorithm for a symmetric positive deenite system and analyze its convergence property. We establish a linear convergence result using a local relation of residual norms. We also analyze the algorithm using a global equation and show that the algorithm may have the superlinear convergence property, when the inner iteration is solved to high accuracy. The analysis is in agreement with observed numerical behaviour of the algorithm. In particular, it suggests a heuristic choice of the stopping threshold for the inner iteration. Numerical examples are given to show the eeectiveness of this choice and to compare the convergence bound.