Determinant Approximations

TL;DR

This paper introduces a sequence of determinant and logarithm approximations for complex matrices, providing error bounds and demonstrating efficiency in lattice simulations of nuclear matter.

Contribution

It develops new determinant approximation methods based on matrix expansions applicable to non-Hermitian matrices, extending classical inequalities and improving computational efficiency.

Findings

Efficient determinant approximations for non-Hermitian matrices.

Block diagonal approximations extend Fischer's and Hadamard's inequalities.

Sparse inverse approximation accuracy improves with more matrix elements.

Abstract

A sequence of approximations for the determinant and its logarithm of a complex matrixis derived, along with relative error bounds. The determinant approximations are derived from expansions of det(X)=exp(trace(log(X))), and they apply to non-Hermitian matrices. Examples illustrate that these determinant approximations are efficient for lattice simulations of finite temperature nuclear matter, and that they use significantly less space than Gaussian elimination. The first approximation in the sequence is a block diagonal approximation; it represents an extension of Fischer's and Hadamard's inequalities to non-Hermitian matrices. In the special case of Hermitian positive-definite matrices, block diagonal approximations can be competitive with sparse inverse approximations. At last, a different representation of sparse inverse approximations is given and it is shown that their accuracy increases as more matrix elements are included.