2011/05/02 by Ilse C. F. Ipsen, Dean Lee, Ipsen, Ilse C. F. +1 · 2 citations
Computer Science · Decision Sciences · Mathematics · #15A15 #15A18 #15A42 #15A90 #65F40 #FOS: Mathematics #Mathematical Approximation and Integration #Matrix Theory and Algorithms #Numerical Analysis (math.NA) #Probabilistic and Robust Engineering Design
paper · pdf · doi:10.48550/arxiv.1105.0437
openalex publication_date 2011/05/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
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.