Computing the Weighted Geometric Mean of Two Large-Scale Matrices and Its Inverse Times a Vector

Fasi, Massimiliano; Iannazzo, Bruno

doi:10.1137/16m1073315

Cited by 17 publications

(23 citation statements)

References 33 publications

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“…which is obtained by applying ( ) = (1 − ) /(1 + ) to (1). Integral representations similar to (28) are considered in [2,7]. In [7], under the assumption that max min = 1, the error of the GJ2 is estimated 1 to be…”

Section: Comparison Of the Convergence Speedmentioning

confidence: 99%

“…It is known that uniquely exists when all the eigenvalues of lie in the set { ∈ C : ∉ (−∞, 0]}; see, e.g., [10]. The matrix fractional power arises in several situations of computational science, e.g., fractional differential equations [2,4,15], lattice QCD calculation [3], and computation of the weighted matrix geometric mean [7].…”

Section: Introductionmentioning

confidence: 99%

“…Several computational methods for have been proposed. Examples are the Schur-Padé algorithm [10,11], the Schur logarithmic algorithm [12], and quadrature-based algorithms [2,5,7,8]. Among them, quadrature-based algorithms have two advantages.…”

Section: Introductionmentioning

confidence: 99%

“…In contrast to the Cauchy integral, the integral representation (1) allows us to avoid selecting the integral path and avoid complex arithmetic for real matrices. In [2,5,7], the integral (1) was transformed into other integrals on [−1, 1], and the transformed integrals were computed with the Gaussian quadrature. For example, transformations ( ) = (1 + )/(1 − ) and ( ) = (1 − ) /(1 + ) were considered.…”

Section: Introductionmentioning

confidence: 99%

“…NS_ill, = 0.8 were the extreme eigenvalues of˜as in [7]. When the estimated number of the abscissas was greater than 1000, we set the number of abscissas to 1000.…”

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confidence: 99%

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Algorithms for the computation of the matrix logarithm based on the double exponential formula

Tatsuoka

Sogabe

Miyatake

et al. 2020

Journal of Computational and Applied Mathematics

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We consider the computation of the matrix logarithm by using numerical quadrature. The efficiency of numerical quadrature depends on the integrand and the choice of quadrature formula. The Gauss-Legendre quadrature has been conventionally employed; however, the convergence could be slow for ill-conditioned matrices. This effect may stem from the rapid change of the integrand values. To avoid such situations, we focus on the double exponential formula, which has been developed to address integrands with endpoint singularity. In order to utilize the double exponential formula, we must determine a suitable finite integration interval, which provides the required accuracy and efficiency. In this paper, we present a method for selecting a suitable finite interval based on an error analysis as well as two algorithms, and one of these algorithms is an adaptive quadrature algorithm.First, numerical quadrature can make use of the sparseness of A if A is sparse. It is useful when computing the multiplication of the matrix logarithm and a vector log(A)b (b ∈ R n ), which appears in applications such as [8, 6, 7]1, without computing and storing dense matrices. Conversely, the ISS algorithm and the algorithm based on the AGM iteration include the computation of the matrix square root, which means that the calculation involves dense matrices even if A is sparse. The second reason is that numerical quadrature is potentially more favorable for parallel computers because of independent computation of the integrand on each abscissa.Because the integrand in (1) includes matrix inversion, the computational cost of numerical quadrature depends on the number of evaluations of the integrand. Although numerical quadrature is suitable for parallelization, the quadrature formula should be selected carefully to reduce the computational cost and save computational resources.The method conventionally used to compute (1) is the Gauss-Legendre (GL) quadrature. If the spectral radius of A − I is smaller than 1, the GL quadrature, which can be regarded as a rational approximation of log(A), coincides with the Padé approximants of log(A) at I [5, Thm. 4.3]. Therefore, it is natural to use the GL quadrature to reduce the number of abscissas when A is close to I. However, the convergence of the GL quadrature becomes slow when A is not close to I. For example, the convergence in our experiments became slow when A was ill-conditioned which may be explained by rapid changes in the integrand value when it is closer to the endpoint of the interval.In this paper, we consider the double exponential (DE) formula [15], which can be used to compute integrals with singularities at one (or both) of the endpoints. For this reason, the DE formula may be useful in scenarios in which the GL quadrature does not perform well. However, when using the DE formula, a finite interval needs to be selected because the integrand in (1) is transformed into a corresponding function on the infinite interval. This selection is important, i.e., if the finite interval is too narr...

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Section: Comparison Of the Convergence Speedmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…NS_ill, = 0.8 were the extreme eigenvalues of˜as in [7]. When the estimated number of the abscissas was greater than 1000, we set the number of abscissas to 1000.…”

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confidence: 99%

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