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16-12-2019 | Original Paper | Issue 4/2020

Numerical Algorithms 4/2020

On prescribing the convergence behavior of the conjugate gradient algorithm

Journal:
Numerical Algorithms > Issue 4/2020
Author:
Gérard Meurant
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Abstract

The conjugate gradient (CG) algorithm is the most frequently used iterative method for solving linear systems Ax = b with a symmetric positive definite (SPD) matrix. In this paper we construct real symmetric positive definite matrices A of order n and real right-hand sides b for which the CG algorithm has a prescribed residual norm convergence curve. We also consider prescribing as well the A-norms of the error. We completely characterize the tridiagonal matrices constructed by the Lanczos algorithm and their inverses in terms of the CG residual norms and A-norms of the error. This also gives expressions and lower bounds for the 2 norm of the error. Finally, we study the problem of prescribing both the CG residual norms and the eigenvalues of A. We show that this is not always possible. Our constructions are illustrated by numerical examples.

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