arXiv subjects

Fuminori Tatsuoka

Publications and source records attributed to Fuminori Tatsuoka.

7 recordsLinked to original sources

arXiv2026

Error control technique of quadrature-based algorithms for the action of real powers of a Hermitian positive-definite matrix

This study considers quadrature-based algorithms to compute $A^α\boldsymbol{b}$, the action of a real power of a Hermitian positive-definite matrix $A$ on a vector $ \boldsymbol{b}$. In these algorithms, the computation of an integral representation of $A^α \boldsymbol{b}$ is reduced to solving several tens or hundreds of shifted linear systems. Current approaches usually analyze the quadrature discretization error, but rarely take into account the additional error introduced by solving these shifted linear systems with iterative solvers. Here, we bound this error with the residual of the approximated solution of these linear systems. This allows the derivation of a stopping criterion for iterative solvers to keep the error of $A^α\boldsymbol{b}$ below a prescribed error tolerance. Numerical results demonstrate that the proposed criterion enables the computation of $A^α\boldsymbol{b}$ within prescribed tolerance limits.

math.NA↗

arXiv2026

An error control framework for computing the exponential of matrices arising from the finite element discretization

Several methods for computing the action of the matrix exponential $\mathrm{e}^{\boldsymbol{A}} \boldsymbol{b}$ are expressed by substituting $\boldsymbol{A}$ into a rational approximation of the scalar exponential function. The error of such methods can be estimated using the numerical range of $\boldsymbol{A}$, which enables the computation of $\mathrm{e}^{\boldsymbol{A}}\boldsymbol{b}$ with a prescribed accuracy. However, when the input matrix has the structure $\boldsymbol{A} = τ\boldsymbol{M}^{-1} \boldsymbol{K}$, this approach is challenging because computing the bounding box of numerical range is difficult and the numerical range may be too large to construct rational approximations on it. In this paper, focusing on the case where $\boldsymbol{M}$ is a well-conditioned symmetric positive definite matrix, we propose considering the numerical range of a similarity transformed matrix of $\boldsymbol{A}$. The numerical range of transformed matrix is not only numerically computable but can also be theoretically bounded depending on properties of $\boldsymbol{K}$. Numerical experiments confirm that the computations can be performed within the prescribed error tolerance.

math.NA↗

arXiv2025

A preconditioning technique of Gauss--Legendre quadrature for the logarithm of symmetric positive definite matrices

This note considers the computation of the logarithm of symmetric positive definite matrices using the Gauss--Legendre (GL) quadrature. The GL quadrature becomes slow when the condition number of the given matrix is large. In this note, we propose a technique dividing the matrix logarithm into two matrix logarithms, where the condition numbers of the divided logarithm arguments are smaller than that of the original matrix. Although the matrix logarithm needs to be computed twice, each computation can be performed more efficiently, and it potentially reduces the overall computational cost. It is shown that the proposed technique is effective when the condition number of the given matrix is approximately between $130$ and $3.0\times 10^5$.

math.NA↗

arXiv2023

Computing the matrix exponential with the double exponential formula

This paper considers the computation of the matrix exponential $\mathrm{e}^A$ with numerical quadrature. Although several quadrature-based algorithms have been proposed, they focus on (near) Hermitian matrices. In order to deal with non-Hermitian matrices, we use another integral representation including an oscillatory term and consider applying the double exponential (DE) formula specialized to Fourier integrals. The DE formula transforms the given integral into another integral whose interval is infinite, and therefore it is necessary to truncate the infinite interval. In this paper, to utilize the DE formula, we analyze the truncation error and propose two algorithms. The first one approximates $\mathrm{e}^A$ with the fixed mesh size which is a parameter in the DE formula affecting the accuracy. Second one computes $\mathrm{e}^A$ based on the first one with automatic selection of the mesh size depending on the given error tolerance.

math.NA↗

arXiv2021

Computing the matrix fractional power with the double exponential formula

Two quadrature-based algorithms for computing the matrix fractional power $A^α$ are presented in this paper. These algorithms are based on the double exponential (DE) formula, which is well-known for its effectiveness in computing improper integrals as well as in treating nearly arbitrary endpoint singularities. The DE formula transforms a given integral into another integral that is suited for the trapezoidal rule; in this process, the integral interval is transformed to the infinite interval. Therefore, it is necessary to truncate the infinite interval into an appropriate finite interval. In this paper, a truncation method, which is based on a truncation error analysis specialized to the computation of $A^α$, is proposed. Then, two algorithms are presented -- one computes $A^α$ with a fixed number of abscissas, and the other computes $A^α$ adaptively. Subsequently, the convergence rate of the DE formula for Hermitian positive definite matrices is analyzed. The convergence rate analysis shows that the DE formula converges faster than the Gaussian quadrature when $A$ is ill-conditioned and $α$ is a non-unit fraction. Numerical results show that our algorithms achieved the required accuracy and were faster than other algorithms in several situations.

math.NA↗

arXiv2019

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

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 addresses error control.

math.NA↗

arXiv2016

A cost-efficient variant of the incremental Newton iteration for the matrix $p$th root

Incremental Newton (IN) iteration, proposed by Iannazzo, is stable for computing the matrix $p$th root, and its computational cost is $\mathcal{O}(n^3p)$ flops per iteration. In this paper, a cost-efficient variant of IN iteration is presented. The computational cost of the variant well agrees with $\mathcal{O} (n^3 \log p)$ flops per iteration, if $p$ is up to at least 100.

math.NA↗