A technique for improving the computation of functions of triangular matrices

ABSTRACT

We propose a simple technique that, if combined with algorithms for computing functions of triangular matrices, can make them more efficient. Basically, such a technique consists in a specific scaling similarity transformation that reduces the departure from normality of a triangular matrix, thus decreasing its norm and in general its function condition number. It can easily be extended to non-triangular matrices, provided that it is combined with algorithms involving a prior Schur decomposition. Situations where the technique should be used or not will be discussed in detail. Special attention is devoted to particular algorithms like the inverse scaling and squaring to the matrix logarithm or inverse cosine and the scaling and squaring to the matrix exponential. The advantages of our proposal are supported by theoretical results and illustrated with numerical experiments, involving matrices of small, medium and large size.

KEYWORDS:

• Schur decomposition
• matrix functions
• matrix logarithm
• matrix square roots
• matrix inverse cosine
• fractional powers of a matrix

AMS SUBJECT CLASSIFICATIONS:

• 15A15
• 15A16
• 15A60

Acknowledgements

The authors would like to thank Pedro Miraldo, from University of Lisbon, for useful discussions on the numerical experiments.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The work of João R. Cardoso was partially supported by the Centre for Mathematics of the University of Coimbra [grant number UIDB/00324/2020], funded by the Portuguese Government through FCT/MCTES.