Tridiagonalization of systems of coupled linear differential equations with variable coefficients by a Lanczos-like method

Abstract

We show constructively that, under certain regularity assumptions, any system of coupled linear differential equations with variable coefficients can be tridiagonalized by a time-dependent Lanczos-like method. The proof we present formally establishes the convergence of the so-called *-Lanczos algorithm and yields a full characterization of algorithmic breakdowns.

From there, the solution of the original differential system is available in a finite and treatable number of scalar integral equations. This is a key piece in evaluating the elusive ordered exponential function both formally and numerically. (C) 2021 Elsevier Inc.

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Keywords

• Tridiagonalization Matrix differential equations
• Lanczos algorithm
• Time-ordered exponential
• Tridiagonal matrices
• Distributions