Open AccessParallel Newton–Chebyshev polynomial preconditioners for the conjugate gradient method
Author(s) -
Bergamaschi Luca,
Martinez Calomardo Angeles
Publication year - 2021
Publication title -
computational and mathematical methods
Language(s) - English
Resource type - Journals
ISSN - 2577-7408
DOI - 10.1002/cmm4.1153
Subject(s) - conjugate gradient method , conjugate residual method , derivation of the conjugate gradient method , chebyshev polynomials , mathematics , convergence (economics) , eigenvalues and eigenvectors , polynomial , chebyshev filter , chebyshev nodes , matrix (chemical analysis) , chebyshev iteration , connection (principal bundle) , chebyshev equation , mathematical optimization , mathematical analysis , computer science , orthogonal polynomials , gradient descent , geometry , classical orthogonal polynomials , physics , materials science , quantum mechanics , machine learning , artificial neural network , economics , composite material , economic growth
In this note, we exploit polynomial preconditioners for the conjugate gradient method to solve large symmetric positive definite linear systems in a parallel environment. We put in connection a specialized Newton method to solve the matrix equation X −1 = A and the Chebyshev polynomials for preconditioning. We propose a simple modification of one parameter which avoids clustering of extremal eigenvalues in order to speed‐up convergence. We provide results on very large matrices (up to 8.6 billion unknowns) in a parallel environment showing the efficiency of the proposed class of preconditioners.