Towards a mixed-precision ADI method for Lyapunov equations

TL;DR

This paper explores a mixed-precision approach to the low-rank Lyapunov ADI method, reducing computational costs while maintaining accuracy for solving Lyapunov equations in control systems.

Contribution

It introduces a mixed-precision implementation of the LR-ADI method, demonstrating its effectiveness on descriptor systems and potential for computational efficiency.

Findings

Single-precision accumulation yields residuals close to double-precision.

Mixed-precision ADI is competitive in computing system norms.

Empirical tests confirm reduced precision can maintain solution quality.

Abstract

We apply mixed-precision to the low-rank Lyapunov ADI (LR-ADI) by performing certain aspects of the algorithm in a lower working precision. Namely, we accumulate the overall solution, solve the linear systems comprising the ADI iteration, and store the inner low-rank factors of the residuals in various combinations of IEEE 754 single and double precision. We empirically test our implementation on Lyapunov equations arising from first- and second-order descriptor systems. For the first-order examples, accumulating the solution in single-precision yields an almost-as-small residual as for the double-precision solution. For certain applications, like computing the H2 norm of a descriptor system, low- or mixed-precision variants of the ADI can be quite competitive