Reaching the superlinear convergence phase of the CG method

Axelsson, Owe and Karátson, János (2014) Reaching the superlinear convergence phase of the CG method. JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS, 260. pp. 244-257. ISSN 0377-0427

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Abstract

The rate of convergence of the conjugate gradient method takes place in essen- tially 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 eigenval- ues in three sets: small and large outliers and intermediate eigenvalues. The other is based on annihilating polynomials for the eigenvalues and, assuming various an- alytical distributions of them, thereby using certain refined estimates. The results are illustrated for some typical distributions of eigenvalues and with some numerical tests.

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