An important variation of preconditioned conjugate gradient algorithms is inexact preconditioner implemented with inner-outer iterations [G. H. Golub and M. L. Overton, Numerical Analysis, Lecture Notes in Math. 912, Springer, Berlin, New York, 1982], 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 definite 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 behavior 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 effectiveness of this choice and to compare the convergence bound.
[1]
Alston S. Householder,et al.
Some inequalities involving the euclidean condition of a matrix
,
1960
.
[2]
Gene H. Golub,et al.
Inner and Outer Iterations for the Chebyshev Algorithm
,
1998
.
[3]
Louis A. Hageman,et al.
Iterative Solution of Large Linear Systems.
,
1971
.
[4]
G. Golub,et al.
Inexact and preconditioned Uzawa algorithms for saddle point problems
,
1994
.
[5]
Qiang Ye,et al.
Analysis of the finite precision bi-conjugate gradient algorithm for nonsymmetric linear systems
,
2000,
Math. Comput..
[6]
H. V. D. Vorst,et al.
The rate of convergence of Conjugate Gradients
,
1986
.
[7]
G. Golub,et al.
The convergence of inexact Chebyshev and Richardson iterative methods for solving linear systems
,
1988
.