Faculty Profiles - SOGABE, Tomohiro

Papers - SOGABE, Tomohiro

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Shifted LOPBiCG(ℓ for Solving Nonsymmetric Shifted Linear Systems

Zhao R.J., Sogabe T., Kemmochi T., Zhang S.L.

Numerical Linear Algebra with Applications Vol. 32 ( 5 ) 2025.10

Language：English Publishing type：Research paper (scientific journal) Publisher：Numerical Linear Algebra with Applications

The shifted LOPBiCG method can circumvent a problem of the optimal choice for the initial seed system through a seed-switching technique when solving shifted linear systems, thanks to its natural collinearity of residual vectors. However, LOPBiCG, employing the same one-degree accelerating polynomial as that in the Bi-CGSTAB method, may not perform well for linear systems whose real coefficient matrix has complex eigenvalues with relatively large imaginary parts. Here, we utilize an (Formula presented.) -degree accelerating polynomial to extend LOPBiCG to LOPBiCG((Formula presented.)) and apply LOPBiCG((Formula presented.)) to solve shifted linear systems, which is referred to as shifted LOPBiCG((Formula presented.)). Since the natural collinearity for LOPBiCG((Formula presented.)) is also satisfied, we can incorporate the seed-switching technique into shifted LOPBiCG((Formula presented.)). Numerical experiments demonstrate that our method effectively solves linear systems with a complex spectrum and is free from the issue associated with choosing the initial seed when solving shifted linear systems.

DOI： 10.1002/nla.70033

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

Fuminori Tatsuoka, Tomohiro Sogabe, Tomoya Kemmochi, Shao-Liang Zhang

Applied Mathematics Letters Vol. 167 page： 109552 - 109552 2025.8

GPBi-CG revisited: a hybrid of the CGS method and the GPBi-CG method for nonsymmetric linear systems

Sogabe, T; Zhang, SL

JAPAN JOURNAL OF INDUSTRIAL AND APPLIED MATHEMATICS 2025.7

Language：English Publishing type：Research paper (scientific journal) Publisher：Japan Journal of Industrial and Applied Mathematics

Toward the end of the 20th century, S.-L. Zhang constructed the so-called Zhang’s framework that successfully incorporated the CGS and Bi-CGSTAB methods and subsequently introduced the GPBi-CG method derived from the framework. While the GPBi-CG method often converges faster than the CGS and Bi-CGSTAB methods, there are still cases where the CGS method performs the best. This observation has motivated us to revisit the framework to find a hybrid algorithm, combining the GPBi-CG and CGS methods. In this paper, we introduce the hybrid algorithm, show its efficiency in some numerical experiments, and discuss a possible reason behind its effectiveness.

DOI： 10.1007/s13160-025-00716-3

Variational Quantum Algorithm for Solving Second-Order Linear Differential Equations

Sugaya, R; Sogabe, T; Kemmochi, T; Zhang, SL

QUANTUM INFORMATION & COMPUTATION Vol. 25 ( 3 ) page： 232 - 247 2025.6

Language：English Publishing type：Research paper (scientific journal) Publisher：Quantum Information and Computation

Variational quantum algorithms (VQAs) have attracted attention as quantum algorithms employing on noisy intermediate-scale quantum devices. VQAs for solving the Poisson equation were recently proposed, as this method transforms the linear equation with a symmetric positive definite matrix obtained by discretizing the Poisson equation into a minimization problem. In this study, we propose a VQA for second-order linear differential equations including the Poisson equation on the basis of the referenced study, where the coefficient matrix obtained through discretization is a non-symmetric matrix. Furthermore, in our VQAs, we succeeded in decomposing the coefficient matrices arising from the periodic and the Dirichlet boundary condition into linear combinations of measurement operators and unitary matrices.

DOI： 10.2478/qic-2025-0012

A tensor Alternating Anderson-Richardson method for solving multilinear systems with <i>M</i>-tensors Reviewed

Niu, J; Du, L; Sogabe, T; Zhang, SL

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS Vol. 461 2025.6

Language：English Publishing type：Research paper (scientific journal) Publisher：Journal of Computational and Applied Mathematics

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.

DOI： 10.1016/j.cam.2024.116419

電磁界解析の複素対称行列向け積型反復法 Reviewed

杉本振一郎，曽我部知広，張紹良，荻野正雄，武居周

電気学会論文誌 Vol. 145 ( 2 ) page： 87 - 97 2025.2

Language：Japanese Publishing type：Research paper (scientific journal) Publisher：IEEJ Transactions on Industry Applications

This paper deals with time-harmonic problems with complex symmetric matrices in the electromagnetic field analysis. To solve such problems more stably and quickly, the Product-type Krylov Subspace Methods based on the COCG and COCR methods are proposed, and verified in finite element analyses of the time-harmonic eddy current and high-frequency electromagnetic field problems. As a result, in some cases, the number of iterations and the computational time have been reduced.

DOI： 10.1541/ieejias.145.87

Computing the matrix exponential with the double exponential formula Reviewed

Fuminori Tatsuoka, Tomohiro Sogabe, Tomoya Kemmochi, Shao Liang Zhang

Special Matrices Vol. 12 ( 1 ) 2024.10

Language：English Publishing type：Research paper (scientific journal)

This article considers the computation of the matrix exponential eA 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 article, to utilize the DE formula, we analyze the truncation error and propose two algorithms. The first one approximates eA with the fixed mesh size, which is a parameter in the DE formula affecting the accuracy. The second one computes eA based on the first one with automatic selection of the mesh size depending on the given error tolerance.

DOI： 10.1515/spma-2024-0013

Adaptive projected SOR algorithms for nonnegative quadratic programming Reviewed

Miyatake, Y; Sogabe, T

JAPAN JOURNAL OF INDUSTRIAL AND APPLIED MATHEMATICS 2024.10

Language：English Publishing type：Research paper (scientific journal) Publisher：Japan Journal of Industrial and Applied Mathematics

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.

DOI： 10.1007/s13160-024-00672-4

Matrix equation representation of the convolution equation and its unique solvability Reviewed

Yuki Satake, Tomohiro Sogabe, Tomoya Kemmochi, Shao-Liang Zhang

Special Matrices Vol. 12 ( 1 ) 2024.6

Language：English Publishing type：Research paper (scientific journal) Publisher：Walter de Gruyter GmbH

Abstract

We consider the convolution equation F * X = B F* X=B , where F ∈ R 3 × 3 F\in { { \mathbb{R } } }^{3\times 3} and B ∈ R m × n B\in { { \mathbb{R } } }^{m\times n} are given and X ∈ R m × n 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 we analyze the unique solvability of the convolution equation.

DOI： 10.1515/spma-2024-0001

Other Link： https://www.degruyter.com/document/doi/10.1515/spma-2024-0001/pdf

Shifted LOPBiCG: A locally orthogonal product-type method for solving nonsymmetric shifted linear systems based on Bi-CGSTAB Reviewed

Zhao, RJ; Sogabe, T; Kemmochi, T; Zhang, SL

NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS Vol. 31 ( 2 ) 2024.3

Language：English Publishing type：Research paper (scientific journal) Publisher：Numerical Linear Algebra with Applications

When solving shifted linear systems using shifted Krylov subspace methods, selecting a seed system is necessary, and an unsuitable seed may result in many shifted systems being unsolved. To avoid this problem, a seed-switching technique has been proposed to help switch the seed system to another linear system as a new seed system without losing the dimension of the constructed Krylov subspace. Nevertheless, this technique requires collinear residual vectors when applying Krylov subspace methods to the seed and shifted systems. Since the product-type shifted Krylov subspace methods cannot provide such collinearity, these methods cannot use this technique. In this article, we propose a variant of the shifted BiCGstab method, which possesses the collinearity of residuals, and apply the seed-switching technique to it. Some numerical experiments show that the problem of choosing the initial seed system is circumvented.

DOI： 10.1002/nla.2538

Tensor product-type methods for solving Sylvester tensor equations Reviewed

Niu Jing, Sogabe Tomohiro, Du Lei, Kemmochi Tomoya, Zhang Shao-Liang

APPLIED MATHEMATICS AND COMPUTATION Vol. 457 2023.11

Language：English Publishing type：Research paper (scientific journal) Publisher：Applied Mathematics and Computation

The tensor biconjugate gradient (TBiCG) method has recently been proposed for solving Sylvester tensor equations. The TBiCG method is based on the BiCG method that may exhibit irregular convergence behavior. To overcome the limitations, product-type methods, such as BiCGSTAB and GPBiCG, have been proposed. In this study, we apply the idea of product-type methods to solve Sylvester tensor equations and propose tensor GPBiCG and BiCGSTAB methods. Furthermore, we consider preconditioned algorithms of the tensor GPBiCG and BiCGSTAB methods using the nearest Kronecker product preconditioner. Numerical experiments illustrate that the proposed methods are competitive with some existing methods.

DOI： 10.1016/j.amc.2023.128155

Quantum algorithms based on the block-encoding framework for matrix functions by contour integrals Reviewed International journal

Souichi Takahira, Asuka Ohashi, Tomohiro Sogabe, Tsuyoshi S. Usuda

Quantum Information and Computation Vol. 22 ( 11&12 ) page： 965 - 979 2022.8

Publishing type：Research paper (scientific journal) Publisher：Rinton Press

he 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 propose a quantum algorithm for matrix functions based on a procedure to implement the linear combination of the inverses on quantum computers. Compared with the previous study [S. Takahira, A. Ohashi, T. Sogabe, and T.S. Usuda, Quant. Inf. Comput., \textbf{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. This is an advantage compared with quantum singular value transformation.

DOI： 10.26421/qic22.11-12-4

Recent development for computing singular values of a generalized tensor sum Reviewed

Ohashi Asuka, Sogabe Tomohiro

Journal of Advanced Simulation in Science and Engineering Vol. 9 ( 1 ) page： 136 - 149 2022

Numerical Algorithms for Computing an Arbitrary Singular Value of a Tensor Sum Reviewed Open Access

Ohashi Asuka, Sogabe Tomohiro

AXIOMS Vol. 10 ( 3 ) 2021.9

Language：Japanese Publishing type：Research paper (scientific journal) Publisher：Axioms

We consider computing an arbitrary singular value of a tensor sum: (Formula presented). We focus on the shift-and-invert Lanczos method, which solves a shift-and-invert eigenvalue problem of (TT T − ˜σ2 Iℓmn)−1, where ˜σ is set to a scalar value close to the desired singular value. The desired singular value is computed by the maximum eigenvalue of the eigenvalue problem. This shift-and-invert Lanczos method needs to solve large-scale linear systems with the coefficient matrix TT T − ˜σ2 Iℓmn. The preconditioned conjugate gradient (PCG) method is applied since the direct methods cannot be applied due to the nonzero structure of the coefficient matrix. However, it is difficult in terms of memory requirements to simply implement the shift-and-invert Lanczos and the PCG methods since the size of T grows rapidly by the sizes of A, B, and C. In this paper, we present the following two techniques: (1) efficient implementations of the shift-and-invert Lanczos method for the eigenvalue problem of TT T and the PCG method for TT T − ˜σ2 Iℓmn using three-dimensional arrays (third-order tensors) and the n-mode products, and (2) preconditioning matrices of the PCG method based on the eigenvalue and the Schur decomposition of T. Finally, we show the effectiveness of the proposed methods through numerical experiments.

DOI： 10.3390/axioms10030211

A thick-restart Lanczos type method for HermitianJ-symmetric eigenvalue problems Reviewed Open Access

Ken-Ichi Ishikawa, Tomohiro Sogabe

JAPAN JOURNAL OF INDUSTRIAL AND APPLIED MATHEMATICS Vol. 38 ( 1 ) page： 233 - 256 2021.2

Language：English Publishing type：Research paper (scientific journal) Publisher：SPRINGER JAPAN KK

A thick-restart Lanczos type algorithm is proposed for HermitianJ-symmetric matrices. Since HermitianJ-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 theJ-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 theJ-symmetry. We test our algorithm on randomly generated small dense matrices and a sparse large matrix originating from a quantum field theory.

DOI： 10.1007/s13160-020-00435-x

Other Link： http://link.springer.com/article/10.1007/s13160-020-00435-x/fulltext.html

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

Takeo Hoshi, Mitsuaki Kawamura, Kazuyoshi Yoshimi, Yuichi Motoyama, Takahiro Misawa, Youhei Yamaji, Synge Todo, Naoki Kawashima, Tomohiro Sogabe

COMPUTER PHYSICS COMMUNICATIONS Vol. 258 2021.1

Language：English Publishing type：Research paper (scientific journal) Publisher：ELSEVIER

We develop K omega, an open-source linear algebra library for the shifted Krylov subspace methods. The methods solve a set of shifted linear equations (z(k)l - 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 H Phi, K omega can realize parallel computation of excitation spectra and intermediate eigenvalues for various quantum lattice models.Program summaryProgram Title: K omega [kei-oumig@]CPC Library link to program files: http://dx.doi.org/10.17632/mt928nz5r3.1Developer's repository link: https://github.com/issp-center-dev/KomegaLicensing provisions: GNU Lesser General Public License Version 3.Programming language: Fortran 90External routines/libraries: BLAS library, LAPACK library (Used in the sample program), MPI library (Optional).Nature of problem: Efficient algorithms, called shifted Krylov subspace algorithms, designed to solve the shifted linear equations.Solution method: Shifted conjugate gradient method, Shifted conjugate orthogonal conjugate gradient method, shifted bi-conjugate gradient method.Additional comments: The present paper is accompanied by a frozen copy of K omega release 2.0.0 that is made publicly available on GitHub (repository https://github.com/issp-center-dev/Komega commit hash fd5455328b102ec4fa13432496e41c404a0f5a9d). (C) 2020 The Authors. Published by Elsevier B.V.

DOI： 10.1016/j.cpc.2020.107536

Computing the matrix fractional power with the double exponential formula Reviewed Open Access

Tatsuoka F., Sogabe T., Miyatake Y., Kemmochi T., Zhang S.L.

Electronic Transactions on Numerical Analysis Vol. 54 page： 558 - 580 2021

Language：Japanese Publishing type：Research paper (scientific journal) Publisher：Electronic Transactions on Numerical Analysis

Two quadrature-based algorithms for computing the matrix fractional power Aα 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 into an infinite interval. Therefore, it is necessary to truncate the infinite interval to an appropriate finite interval. In this paper, a truncation method, which is based on a truncation error analysis specialized to the computation of Aα, is proposed. Then, two algorithms are presented-one where Aα is computed with a fixed number of abscissa points and one with Aα computed 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 Gaussian quadrature when A is ill-conditioned and α is a non-unit fraction. Numerical results show that our algorithms achieve the required accuracy and are faster than other algorithms in several situations.

DOI： 10.1553/0X003CC757

On a transformation of the (*)-congruence Sylvester equation for the least squares optimization Reviewed

Yuki Satake, Tomohiro Sogabe, Tomoya Kemmochi, Shao-Liang Zhang

OPTIMIZATION METHODS & SOFTWARE Vol. 35 ( 5 ) page： 974 - 981 2020.9

Language：English Publishing type：Research paper (scientific journal) Publisher：TAYLOR & FRANCIS LTD

The star-congruence Sylvester equation is the matrix equation AX + (XB)-B-star = C, where A is an element of F-mxm, B is an element of F-nxm and C is an element of F-mxm are given, whereas X is an element of F-nxm is to be determined. Here, F = R or C, and star = T (transposed) or (*) (conjugate transposed). Very recently, Satake et al. showed that under some conditions, the matrix equation for the case star = T is equivalent to the generalized Sylvester equation. In this paper, we demonstrate that the result can be extended to the case star = (*). Through this extension, the least squares solution of the (*)-congruence Sylvester equation may be obtained using well-researched results on the least squares solution of the generalized Sylvester equation.

DOI： 10.1080/10556788.2020.1734004

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

Fuminori Tatsuoka, Tomohiro Sogabe, Yuto Miyatake, Shao-Liang Zhang

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS Vol. 373 2020.8

Language：English Publishing type：Research paper (scientific journal) Publisher：ELSEVIER

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. (C) 2019 Elsevier B.V. All rights reserved.

DOI： 10.1016/j.cam.2019.112396

Generalized Sherman-Morrison-Woodbury formula based algorithm for the inverses of opposite-bordered tridiagonal matrices Reviewed

Ji-Teng Jia, Tomohiro Sogabe

JOURNAL OF MATHEMATICAL CHEMISTRY Vol. 58 ( 7 ) page： 1466 - 1480 2020.8

Language：English Publishing type：Research paper (scientific journal) Publisher：SPRINGER

Matrix inverse computation is one of the fundamental mathematical problems of linear algebra and has been widely used in many fields of science and engineering. In this paper, we consider the inverse computation of an opposite-bordered tridiagonal matrix which has attracted much attention in recent years. By exploiting the low-rank structure of the matrix, first we show that an explicit formula for the inverse of the opposite-bordered tridiagonal matrix can be obtained based on the combination of a specific matrix splitting and the generalized Sherman-Morrison-Woodbury formula. Accordingly, a numerical algorithm is outlined. Second, we present a breakdown-free symbolic algorithm of O(n(2)) for computing the inverse of an n-by-n opposite-bordered tridiagonal matrix, which is based on the use of GTINV algorithm and the generalized symbolic Thomas algorithm. Finally, we have provided the results of some numerical experiments for the sake of illustration.

DOI： 10.1007/s10910-020-01138-x

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