An Improved Taylor Algorithm for Computing the Matrix Logarithm

Ibáñez, Javier; Sastre, Jorge; Ruiz, Pedro A.; Alonso, Jesús José Beltrán de Heredia; Defez, Emilio

doi:10.3390/math9172018

Cited by 4 publications

(4 citation statements)

References 45 publications

(71 reference statements)

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“…The MATLAB Symbolic Math Toolbox with 256 digits of precision was employed to compute "exactly" the matrix logarithm function using the vpa (variable-precision floating-point arithmetic) function. The battery featured the following three matrix sets, which are practically the same as the ones used and described in [44]:…”

Section: Numerical Testsmentioning

confidence: 99%

“…The size of these was 128 × 128. The matrix logarithm of each matrix was computed "exactly" according to this protocol, as described in [44]:…”

Section: Numerical Testsmentioning

confidence: 99%

“…MATLAB incorporates logm as a built-in function that uses the algorithms described in [33,34] to compute the principal matrix logarithm. Recently, an implementation that used matrix polynomial formulas to efficiently evaluate the Taylor approximation of the matrix logarithm was described in [44].…”

Section: Introductionmentioning

confidence: 99%

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Computing the Matrix Logarithm with the Romberg Integration Method

Ibáñez,

Alonso,

Defez

et al. 2023

Algorithms

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The matrix logarithm function has applicability in many engineering and science fields. Improvements in its calculation, from the point of view of both accuracy and/or execution time, have a direct impact on these disciplines. This paper describes a new numerical algorithm devoted to matrix logarithm computation and using the Romberg integration method, together with the inverse scaling and squaring technique. This novel method was implemented and compared with three different state-of-the-art codes, all based on Padé approximation. The experimental results, under a heterogeneous matrix test battery, showed that the new method was numerically stable, with an elapsed time midway among the other codes, and it generally offered a higher accuracy.

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Section: Numerical Testsmentioning

confidence: 99%

“…The size of these was 128 × 128. The matrix logarithm of each matrix was computed "exactly" according to this protocol, as described in [44]:…”

Section: Numerical Testsmentioning

confidence: 99%

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Computing the Matrix Logarithm with the Romberg Integration Method

Ibáñez,

Alonso,

Defez

et al. 2023

Algorithms

Self Cite

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“…Computation methods for the principal value of the matrix logarithm can be found in several articles using different approaches. In [3], an improvement of the inverse scaling and squaring method is shown; in [4], an alternative formulation of the matrix arithmeticgeometric mean algorithm is used; in [5], multiprecision algorithms are analyzed; in [6], algorithms based on a numerical spectral decomposition or on a numerical Jordan decomposition are used; in [7], algorithms based on the double exponential formula are used; and in [8], an improved Taylor algorithm is applied. However, the closest example to our approach is the paper by N. Hale, N.J. Higham, N.J. and L.N.…”

Section: Introductionmentioning

confidence: 99%

Logarithm of a Non-Singular Complex Matrix via the Dunford–Taylor Integral

Caratelli

Ricci

2022

Axioms

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Using the Dunford–Taylor integral and a representation formula for the resolvent of a non-singular complex matrix, we find the logarithm of a non-singular complex matrix applying the Cauchy’s residue theorem if the matrix eigenvalues are known or a circuit integral extended to a curve surrounding the spectrum. The logarithm function that can be found using this technique is essentially unique. To define a version of the logarithm with multiple values analogous to the one existing in the case of complex variables, we introduce a definition for the argument of a matrix, showing the possibility of finding equations similar to those of the scalar case. In the last section, numerical experiments performed by the first author, using the computer algebra program Mathematica©, confirm the effectiveness of this methodology. They include the logarithm of matrices of the fifth, sixth and seventh order.

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