Generalized Rybicki Press algorithm

TL;DR

This paper introduces a more general and numerically stable Rybicki Press algorithm that efficiently inverts and computes determinants of covariance matrices with exponential sums, using banded matrix techniques.

Contribution

It presents a generalized Rybicki Press algorithm with improved numerical stability and linear complexity for inverting and computing determinants of semi-separable matrices.

Findings

Algorithm achieves linear scaling in benchmarks.

Enables stable inversion of covariance matrices with exponential sums.

Provides efficient determinant computation for large matrices.

Abstract

This article discusses a more general and numerically stable Rybicki Press algorithm, which enables inverting and computing determinants of covariance matrices, whose elements are sums of exponentials. The algorithm is true in exact arithmetic and relies on introducing new variables and corresponding equations, thereby converting the matrix into a banded matrix of larger size. Linear complexity banded algorithms for solving linear systems and computing determinants on the larger matrix enable linear complexity algorithms for the initial semi-separable matrix as well. Benchmarks provided illustrate the linear scaling of the algorithm.