Multiprecision Algorithms for Computing the Matrix Logarithm

Fasi, Massimiliano; Higham, Nicholas J.

doi:10.1137/17m1129866

Cited by 26 publications

(23 citation statements)

References 32 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Finally, though we derive the values of θ m in (3.5) for half, single, and double precisions, θ m can be evaluated for any arbitrary precision. Algorithm 5.1 can be extended to be a multiprecision algorithm as in [6] since the function ρ m (3.2) has an explicit expression that is easy to be handled by optimization software.…”

Section: Numerical Experimentsmentioning

confidence: 99%

A Truncated Taylor Series Algorithm for Computing the Action of Trigonometric and Hyperbolic Matrix Functions

Al-Mohy¹

2018

SIAM J. Sci. Comput.

View full text Add to dashboard Cite

A new algorithm is derived for computing the actions f (tA)B and f (tA 1/2 )B, where f is cosine, sinc, sine, hyperbolic cosine, hyperbolic sinc, or hyperbolic sine function. A is an n × n matrix and B is n×n 0 with n 0 n. A 1/2 denotes any matrix square root of A and it is never required to be computed. The algorithm offers six independent output options given t, A, B, and a tolerance. For each option, actions of a pair of trigonometric or hyperbolic matrix functions are simultaneously computed. The algorithm scales the matrix A down by a positive integer s, approximates f (s −1 tA)B by a truncated Taylor series, and finally uses the recurrences of the Chebyshev polynomials of the first and second kind to recover f (tA)B. The selection of the scaling parameter and the degree of Taylor polynomial are based on a forward error analysis and a sequence of the form A k 1/k in such a way the overall computational cost of the algorithm is optimized. Shifting is used where applicable as a preprocessing step to reduce the scaling parameter. The algorithm works for any matrix A and its computational cost is dominated by the formation of products of A with n × n 0 matrices that could take advantage of the implementation of level-3 BLAS. Our numerical experiments show that the new algorithm behaves in a forward stable fashion and in most problems outperforms the existing algorithms in terms of CPU time, computational cost, and accuracy.

show abstract

Section: Numerical Experimentsmentioning

confidence: 99%

A Truncated Taylor Series Algorithm for Computing the Action of Trigonometric and Hyperbolic Matrix Functions

Al-Mohy¹

2018

SIAM J. Sci. Comput.

View full text Add to dashboard Cite

show abstract

“…Later, most of the developed algorithms basically used the inverse scaling and squaring method with Padé approximants to dense or triangular matrices; see [29][30][31][32][33][34][35]. Nevertheless, some new algorithms are based on other methods, among which the following can be highlighted:…”

Section: Introduction and Notationmentioning

confidence: 99%

An Improved Taylor Algorithm for Computing the Matrix Logarithm

et al. 2021

View full text Add to dashboard Cite

The most popular method for computing the matrix logarithm is a combination of the inverse scaling and squaring method in conjunction with a Padé approximation, sometimes accompanied by the Schur decomposition. In this work, we present a Taylor series algorithm, based on the free-transformation approach of the inverse scaling and squaring technique, that uses recent matrix polynomial formulas for evaluating the Taylor approximation of the matrix logarithm more efficiently than the Paterson–Stockmeyer method. Two MATLAB implementations of this algorithm, related to relative forward or backward error analysis, were developed and compared with different state-of-the art MATLAB functions. Numerical tests showed that the new implementations are generally more accurate than the previously available codes, with an intermediate execution time among all the codes in comparison.

show abstract

“…This occurs, for instance, in discretized applications, possibly in a multiresolution framework, or in inexactly weighted linear and nonlinear least‐squares, where the product itself is obtained by applying an iterative procedure . The second is the increasing importance of computations in multiprecision arithmetic on the new generations of high‐performance computers (see References 2‐8 and the many references therein), in which the use of varying levels of floating point precision is a key ingredient for obtaining state‐of‐the‐art energy‐efficient computer architectures. In both cases, using inexact matrix‐vector products (while controlling their inexactness) within the method of conjugate gradients (CG) of Hestenes and Stiefel 9 is a natural option.…”

Section: Introductionmentioning

confidence: 99%

Minimizing convex quadratics with variable precision conjugate gradients

Gratton

Simon

Titley-Péloquin

et al. 2020

Numerical Linear Algebra App

View full text Add to dashboard Cite

We investigate the method of conjugate gradients, exploiting inaccurate matrix-vector products, for the solution of convex quadratic optimization problems. Theoretical performance bounds are derived, and the necessary quantities occurring in the theoretical bounds estimated, leading to a practical algorithm. Numerical experiments suggest that this approach has significant potential, including in the steadily more important context of multiprecision computations.

show abstract

Multiprecision Algorithms for Computing the Matrix Logarithm

Cited by 26 publications

References 32 publications

A Truncated Taylor Series Algorithm for Computing the Action of Trigonometric and Hyperbolic Matrix Functions

A Truncated Taylor Series Algorithm for Computing the Action of Trigonometric and Hyperbolic Matrix Functions

An Improved Taylor Algorithm for Computing the Matrix Logarithm

Minimizing convex quadratics with variable precision conjugate gradients

Contact Info

Product

Resources

About