Generalized Fractional Algebraic Linear System Solvers

• Published in: Journal of scientific computing 2022-04, Vol.91 (1), p.25, Article 25
• Main Authors: Antoine, X., Lorin, E.
• Format: Article
• Language: English
• Subjects: Algebra; Algorithms; Analysis of PDEs; Approximation; Computational Mathematics and Numerical Analysis; Conservation laws; Eigenvalues; Hyperbolic systems; Linear systems; Mathematical analysis; Mathematical and Computational Engineering; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Methods; Numerical Analysis; Partial differential equations; Theoretical
• ISSN: 0885-7474; 1573-7691
• DOI: 10.1007/s10915-022-01785-z

This paper is devoted to the numerical computation of algebraic linear systems involving several matrix power functions; that is finding x solution to ∑ α ∈ R A α x = b . These systems will be referred to as Generalized Fractional Algebraic Linear Systems (GFALS). In this paper, we derive several gradient methods for solving these very computationally complex problems, which themselves require the solution to intermiediate standard Fractional Algebraic Linear Systems (FALS) A α x = b , with α ∈ R + . The latter usually require the solution to many classical linear systems A x = b . We also show that in some cases, the solution to a GFALS problem can be obtained as the solution to a first-order hyperbolic system of conservation laws. We also discuss the connections between this PDE-approach and gradient-type methods. The convergence analysis is addressed and some numerical experiments are proposed to illustrate and compare the methods which are proposed in this paper.