arXiv subjects

Tomohiro Sogabe

Publications and source records attributed to Tomohiro Sogabe.

At least 19 recordsLinked to original sources

arXiv2026

Factorized Krylov subspace methods for solving large Sylvester equations

Krylov subspace methods, such as the Conjugate Gradient (CG) and BiCGSTAB methods, are widely used in scientific computing for solving linear systems. In this study, we propose a new framework for solving large Sylvester equations in a low-rank format by reconstructing matrix-oriented Krylov subspace methods. The framework realizes efficient algorithms that are mathematically equivalent to the matrix-oriented Krylov subspace methods by exploiting the mathematical properties of the Sylvester operator and the low-rank structure of the right-hand side. Specifically, by leveraging these properties, approximate solutions can be expressed in a low-rank factorized form, enabling efficient computation and reduced memory requirements. The effectiveness of our algorithms is demonstrated through numerical experiments.

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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 $A^\alpha \boldsymbol{b}$, 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 $A^{\alpha} \boldsymbol{b}$ 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 $A^\alpha \boldsymbol{b}$ below a prescribed error tolerance. Numerical results demonstrate that the proposed criterion enables the computation of $A^\alpha \boldsymbol{b}$ within prescribed tolerance limits.

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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 $\boldsymbol{A} = \tau \boldsymbol{M}^{-1} \boldsymbol{K}$, 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.

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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 $3.0\times 10^5$.

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arXiv2024

Adaptive projected SOR algorithms for nonnegative quadratic programming

The choice of relaxation parameter in the projected successive overrelaxation (PSOR) method for nonnegative quadratic programming problems is problem-dependent. We present novel adaptive PSOR algorithms that adaptively control the relaxation parameter using the Wolfe conditions. The method and its variants can be applied to various problems without requiring additional assumptions, barring the positive semidefiniteness concerning the matrix that defines the objective function, and the cost for updating the parameter is negligible in the whole iteration. Numerical experiments show that the proposed methods often perform comparably to (or sometimes superior to) the PSOR method with a nearly optimal relaxation parameter.

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arXiv2024

A tensor Alternating Anderson-Richardson method for solving multilinear systems with M-tensors

It is well-known that a multilinear system with a nonsingular M-tensor and a positive right-hand side has a unique positive solution. Tensor splitting methods generalizing the classical iterative methods for linear systems have been proposed for finding the unique positive solution. The Alternating Anderson-Richardson (AAR) method is an effective method to accelerate the classical iterative methods. In this study, we apply the idea of AAR for finding the unique positive solution quickly. We first present a tensor Richardson method based on tensor regular splittings, then apply Anderson acceleration to the tensor Richardson method and derive a tensor Anderson-Richardson method, finally, we periodically employ the tensor Anderson-Richardson method within the tensor Richardson method and propose a tensor AAR method. Numerical experiments show that the proposed method is effective in accelerating tensor splitting methods.

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arXiv2023

Matrix equation representation of the convolution equation and its unique solvability

We consider the convolution equation $F*X=B$, where $F\in\mathbb{R}^{3\times 3}$ and $B\in\mathbb{R}^{m\times n}$ are given, and $X\in\mathbb{R}^{m\times n}$ is to be determined. The convolution equation can be regarded as a linear system with a coefficient matrix of special structure. This fact has led to many studies including efficient numerical algorithms for solving the convolution equation. In this study, we show that the convolution equation can be represented as a generalized Sylvester equation. Furthermore, for some realistic examples arising from image processing, we show that the generalized Sylvester equation can be reduced to a simpler form, and analyze the unique solvability of the convolution equation.

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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.

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arXiv2021

Quantum Algorithms based on the Block-Encoding Framework for Matrix Functions by Contour Integrals

The matrix functions can be defined by Cauchy's integral formula and can be approximated by the linear combination of inverses of shifted matrices using a quadrature formula. In this paper, we show a concrete construction of a framework to implement the linear combination of the inverses on quantum computers and propose a quantum algorithm for matrix functions based on the framework. Compared with the previous study [S. Takahira, A. Ohashi, T. Sogabe, and T.S. Usuda, Quant. Inf. Comput., 20, 1&2, 14--36, (Feb. 2020)] that proposed a quantum algorithm to compute a quantum state for the matrix function based on the circular contour centered at the origin, the quantum algorithm in the present paper can be applied to a more general contour. Moreover, the algorithm is described by the block-encoding framework. Similarly to the previous study, the algorithm can be applied even if the input matrix is not a Hermitian or normal matrix.

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arXiv2021

Quantum algorithm for matrix functions by Cauchy's integral formula

For matrix $A$, vector $\boldsymbol{b}$ and function $f$, the computation of vector $f(A)\boldsymbol{b}$ arises in many scientific computing applications. We consider the problem of obtaining quantum state $\lvert f \rangle$ corresponding to vector $f(A)\boldsymbol{b}$. There is a quantum algorithm to compute state $\lvert f \rangle$ using eigenvalue estimation that uses phase estimation and Hamiltonian simulation $\mathrm{e}^{\mathrm{\bf i} A t}$. However, the algorithm based on eigenvalue estimation needs $\textrm{poly}(1/ε)$ runtime, where $ε$ is the desired accuracy of the output state. Moreover, if matrix $A$ is not Hermitian, $\mathrm{e}^{\mathrm{\bf i} A t}$ is not unitary and we cannot run eigenvalue estimation. In this paper, we propose a quantum algorithm that uses Cauchy's integral formula and the trapezoidal rule as an approach that avoids eigenvalue estimation. We show that the runtime of the algorithm is $\mathrm{poly}(\log(1/ε))$ and the algorithm outputs state $\lvert f \rangle$ even if $A$ is not Hermitian.

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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.

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arXiv2020

A thick-restart Lanczos type method for Hermitian $J$-symmetric eigenvalue problems

A thick-restart Lanczos type algorithm is proposed for Hermitian $J$-symmetric matrices. Since Hermitian $J$-symmetric matrices possess doubly degenerate spectra or doubly multiple eigenvalues with a simple relation between the degenerate eigenvectors, we can improve the convergence of the Lanczos algorithm by restricting the search space of the Krylov subspace to that spanned by one of each pair of the degenerate eigenvector pairs. We show that the Lanczos iteration is compatible with the $J$-symmetry, so that the subspace can be split into two subspaces that are orthogonal to each other. The proposed algorithm searches for eigenvectors in one of the two subspaces without the multiplicity. The other eigenvectors paired to them can be easily reconstructed with the simple relation from the $J$-symmetry. We test our algorithm on randomly generated small dense matrices and a sparse large matrix originating from a quantum field theory.

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arXiv2020

K$ω$ -- Open-source library for the shifted Krylov subspace method of the form $(zI-H)x=b$

We develop K$ω$, an open-source linear algebra library for the shifted Krylov subspace methods. The methods solve a set of shifted linear equations $(z_k I-H)x^{(k)}=b\, (k=0,1,2,...)$ for a given matrix $H$ and a vector $b$, simultaneously. The leading order of the operational cost is the same as that for a single equation. The shift invariance of the Krylov subspace is the mathematical foundation of the shifted Krylov subspace methods. Applications in materials science are presented to demonstrate the advantages of the algorithm over the standard Krylov subspace methods such as the Lanczos method. We introduce benchmark calculations of (i) an excited (optical) spectrum and (ii) intermediate eigenvalues by the contour integral on the complex plane. In combination with the quantum lattice solver $\mathcal{H} Φ$, K$ω$ can realize parallel computation of excitation spectra and intermediate eigenvalues for various quantum lattice models.

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arXiv2019

Modified Strang splitting for semilinear parabolic problems

We consider applying the Strang splitting to semilinear parabolic problems. The key ingredients of the Strang splitting are the decomposition of the equation into several parts and the computation of approximate solutions by combining the time evolution of each split equation. However, when the Dirichlet boundary condition is imposed, order reduction could occur due to the incompatibility of the split equations with the boundary condition. In this paper, to overcome the order reduction, a modified Strang splitting procedure is presented for the one-dimensional semilinear parabolic equation with first-order spatial derivatives, like the Burgers equation.

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arXiv2019

A structure-preserving Fourier pseudo-spectral linearly implicit scheme for the space-fractional nonlinear Schrödinger equation

We propose a Fourier pseudo-spectral scheme for the space-fractional nonlinear Schrödinger equation. The proposed scheme has the following features: it is linearly implicit, it preserves two invariants of the equation, its unique solvability is guaranteed without any restrictions on space and time step sizes. The scheme requires solving a complex symmetric linear system per time step. To solve the system efficiently, we also present a certain variable transformation and preconditioner.

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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.

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arXiv2019

Relation between the T-congruence Sylvester equation and the generalized Sylvester equation

The T-congruence Sylvester equation is the matrix equation $AX+X^{\mathrm{T}}B=C$, where $A\in\mathbb{R}^{m\times n}$, $B\in\mathbb{R}^{n\times m}$, and $C\in\mathbb{R}^{m\times m}$ are given, and $X\in\mathbb{R}^{n\times m}$ is to be determined. Recently, Oozawa et al. discovered a transformation that the matrix equation is equivalent to one of the well-studied matrix equations (the Lyapunov equation); however, the condition of the transformation seems to be too limited because matrices $A$ and $B$ are assumed to be square matrices ($m=n$). In this paper, two transformations are provided for rectangular matrices $A$ and $B$. One of them is an extension of the result of Oozawa et al. for the case $m\ge n$, and the other is a novel transformation for the case $m\le n$.

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arXiv2018

Adaptive SOR methods based on the Wolfe conditions

Because the expense of estimating the optimal value of the relaxation parameter in the successive over-relaxation (SOR) method is usually prohibitive, the parameter is often adaptively controlled. In this paper, new adaptive SOR methods are presented that are applicable to a variety of symmetric positive definite linear systems and do not require additional matrix-vector products when updating the parameter. To this end, we regard the SOR method as an algorithm for minimising a certain objective function, which yields an interpretation of the relaxation parameter as the step size following a certain change of variables. This interpretation enables us to adaptively control the step size based on some line search techniques, such as the Wolfe conditions. Numerical examples demonstrate the favourable behaviour of the proposed methods.

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