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 Lanczos-like algorithm and yields a full characterization of algorithmic breakdowns. From there, the solution of the original differential system is available in closed form. This is a key piece in evaluating the elusive ordered exponential function both formally and numerically.

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• Elementary Integral Series for Heun Functions. With an Application to Black-Hole Perturbation Theory
• A Lanczos-like method for non-autonomous linear ordinary differential equations
• A Fréchet Lie group on distributions
• Computations of general Heun functions from their integral series representations
• Exact solutions for the time-evolution of quantum spin systems under arbitrary waveforms using algebraic graph theory