Efficient Calculation of Determinants of Symbolic Matrices with Many Variables

TL;DR

This paper compares two algorithms for calculating determinants of symbolic matrices with many variables, showing minor expansion can be faster and proposing optimizations validated through empirical testing.

Contribution

It provides a performance comparison between fraction-free Gaussian elimination and minor expansion for symbolic matrices, introducing optimizations for minor expansion.

Findings

Minor expansion is often faster than Gaussian elimination under a simplified model.

Optimizations significantly improve minor expansion performance.

Empirical results confirm the effectiveness of proposed optimizations.

Abstract

Efficient matrix determinant calculations have been studied since the 19th century. Computers expand the range of determinants that are practically calculable to include matrices with symbolic entries. However, the fastest determinant algorithms for numerical matrices are often not the fastest for symbolic matrices with many variables. We compare the performance of two algorithms, fraction-free Gaussian elimination and minor expansion, on symbolic matrices with many variables. We show that, under a simplified theoretical model, minor expansion is faster in most situations. We then propose optimizations for minor expansion and demonstrate their effectiveness with empirical data.