## Abstract

We describe a Lanczos-based algorithm for approximating the product of a rational matrix function with a vector. This algorithm, which we call the Lanczos method for optimal rational matrix function approximation (Lanczos-OR), returns the optimal approximation from a given Krylov subspace in a norm depending on the rational function's denominator, and it can be computed using the information from a slightly larger Krylov subspace. We also provide a low-memory implementation which only requires storing a number of vectors proportional to the denominator degree of the rational function. Finally, we show that Lanczos-OR can be used to derive algorithms for computing other matrix functions, including the matrix sign function and quadrature-based rational function approximations. In many cases, it improves on the approximation quality of prior approaches, including the standard Lanczos method, with little additional computational overhead.

Original language | English (US) |
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Pages (from-to) | 670-692 |

Number of pages | 23 |

Journal | SIAM Journal on Matrix Analysis and Applications |

Volume | 44 |

Issue number | 2 |

DOIs | |

State | Published - Jun 2023 |

## Keywords

- Krylov subspace method
- Lanczos
- low-memory
- matrix function approximation
- optimal approximation

## ASJC Scopus subject areas

- Analysis