Reaching the superlinear convergence phase of the CG method

Authors:
O. Axelsson;J. Karátson
Affiliations:
-;-
Venue:
Journal of Computational and Applied Mathematics
Year:
2014

Citing 6
Cited 0

On the eigenvalue distribution of a class of preconditioning methods

Numerische Mathematik
The rate of convergence of conjugate gradients

Numerische Mathematik
Iterative solution methods

Iterative solution methods
Iterative Methods for Sparse Linear Systems

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Superlinearly convergent CG methods via equivalent preconditioning for nonsymmetric elliptic operators

Numerische Mathematik
Mesh Independent Superlinear PCG Rates Via Compact-Equivalent Operators

SIAM Journal on Numerical Analysis

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Abstract

The rate of convergence of the conjugate gradient method takes place in essentially three phases, with respectively a sublinear, a linear and a superlinear rate. The paper examines when the superlinear phase is reached. To do this, two methods are used. One is based on the K-condition number, thereby separating the eigenvalues in three sets: small and large outliers and intermediate eigenvalues. The other is based on annihilating polynomials for the eigenvalues and, assuming various analytical distributions of them, thereby using certain refined estimates. The results are illustrated for some typical distributions of eigenvalues and with some numerical tests.