Improved algorithms for computing the matrix logarithm based on the double-exponential formula

Abstract

Tatsuoka et al. proposed an efficient algorithm for computing the matrix logarithm using the double-exponential (DE) formula. Its discretization error decreases exponentially as the number of quadrature nodes grows, and its nested grids allow previously computed values to be reused when the mesh is refined. They also reported that their approach is more robust than Gauss--Legendre quadrature for ill-conditioned matrices. We propose two algorithms based on the DE formula. The first adaptively refines the mesh while reusing previously computed values and requires no matrix norm computations. The second uses a rigorous error bound to select the number of quadrature nodes a priori. Numerical experiments show that both methods attain lower errors than Tatsuoka et al.'s method at comparable elapsed times over the tested ranges.

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BibTeX RIS

BibTeX

@misc{okayama2026improved,
  title = {{Improved algorithms for computing the matrix logarithm based on the double-exponential formula}},
  author = {Tomoaki Okayama and Tsubasa Setogawa},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.25361},
  eprint = {2610.25361},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.25361},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Improved algorithms for computing the matrix logarithm based on the double-exponential formula
AU  - Tomoaki Okayama
AU  - Tsubasa Setogawa
PY  - 2026
DA  - 2026/10/24
PB  - arXiv
DO  - 10.48550/arxiv.2610.25361
UR  - https://arxiv.org/abs/2610.25361
AB  - Tatsuoka et al. proposed an efficient algorithm for computing the matrix logarithm using the double-exponential (DE) formula. Its discretization error decreases exponentially as the number of quadrature nodes grows, and its nested grids allow previously computed values to be reused when the mesh is refined. They also reported that their approach is more robust than Gauss--Legendre quadrature for ill-conditioned matrices. We propose two algorithms based on the DE formula. The first adaptively refines the mesh while reusing previously computed values and requires no matrix norm computations. The second uses a rigorous error bound to select the number of quadrature nodes a priori. Numerical experiments show that both methods attain lower errors than Tatsuoka et al.'s method at comparable elapsed times over the tested ranges.
ER  -

Abstract

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