Fine spectral estimates with applications to the optimally fast solution of large FDE linear systems

TL;DR

This paper develops advanced spectral estimation techniques for matrices from fractional differential equations, enabling faster solutions of large linear systems with improved accuracy over existing methods.

Contribution

It introduces novel spectral bounds for complex matrix sequences from distributed order FDEs, enhancing the design of efficient numerical algorithms for large linear systems.

Findings

New spectral bounds improve existing estimates

Enhanced accuracy in eigenvalue localization

Numerical algorithms achieve faster convergence

Abstract

In the present note we consider a type of matrices stemming in the context of the numerical approximation of distributed order fractional differential equations (FDEs): from one side they could look standard, since they are, real, symmetric and positive definite. On the other hand they present specific difficulties which prevent the successful use of classical tools. In particular the associated matrix-sequence, with respect to the matrix-size, is ill-conditioned and it is such that a generating function does not exists, but we face the problem of dealing with a sequence of generating functions with an intricate expression. Nevertheless, we obtain a real interval where the smallest eigenvalue belongs, showing also its asymptotic behavior. We observe that the new bounds improve those already present in the literature and give a more accurate spectral information, which are in fact used in the design of fast numerical algorithms for the associated large linear systems, approximating the given distributed order FDEs. Very satisfactory numerical results are presented and critically discussed, while a section with conclusions and open problems ends the current note.