Updated on 2022/03/21

SOGABE, Tomohiro

Organization
Graduate School of Engineering Applied Physics 1 Associate professor
School of Engineering Physical Science and Engineering

### Degree 1

1. 博士（工学） （ 2006.9   東京大学 ）

### Research Areas 2

1. Others / Others  / Fundamental Engineering

2. Natural Science / Applied mathematics and statistics

1. 数値線形代数

### Research History 1

1. Nagoya University   Graduate School of Engineering Applied Physics   Associate professor

2015.4

### Professional Memberships 3

1. （一社） 日本応用数理学会   代表会員

2013.4

2. （一社） 日本応用数理学会   理事

2021.6

3. （一社） 情報処理学会　   会員

### Committee Memberships 4

1. Japan Journal of Industrial and Applied Mathematics, Springer,   Editor

2019.1

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2. Journal of Inequalities and Applications, Springer   Guest Editor

2016.1 - 2017.1

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3. 日本応用数理学会 JSIAM Letters   幹事編集委員

2012.4 - 2016.3

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4. 日本応用数理学会論文誌   編集委員

2012.4 - 2016.3

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### Papers 93

1. Ohashi Asuka, Sogabe Tomohiro

AXIOMS   Vol. 10 ( 3 )   2021.9

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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.

2. Ken-Ichi Ishikawa, Tomohiro Sogabe

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

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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.

3. 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

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

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

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

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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.

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

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

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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.

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

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS   Vol. 373   2020.8

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

7. Ji-Teng Jia, Tomohiro Sogabe

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

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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.

8. Yuto Miyatake, Tomohiro Sogabe, Shao-Liang Zhang

NUMERICAL ALGORITHMS   Vol. 84 ( 1 ) page： 117 - 132   2020.5

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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.

9. Takahira Souichi, Ohashi Asuka, Sogabe Tomohiro, Usuda Tsuyoshi Sasaki

QUANTUM INFORMATION & COMPUTATION   Vol. 20 ( 1-2 ) page： 14 - 36   2020.2

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10. Yuki Satake, Masaya Oozawa, Tomohiro Sogabe, Yuto Miyatake, Tomoya Kemmochi, Shao-Liang Zhang

APPLIED MATHEMATICS LETTERS   Vol. 96   page： 7 - 13   2019.10

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The T-congruence Sylvester equation is the matrix equation AX + (XB)-B-T = C, where A is an element of R-mxn, B is an element of R-nxm, and C is an element of R-mxm are given, and X is an element of R-nxm 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 >= n, and the other is a novel transformation for the case m <= n. (C) 2019 Elsevier Ltd. All rights reserved.

11. Takahira S., Sogabe T., Usuda T. S.

SPECIAL MATRICES   Vol. 7 ( 1 ) page： 20-26   2019.1

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12. A. Ohashi, T. Sogabe

Special Matrices   Vol. 7 ( 1 ) page： 95 - 106   2019.1

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© 2019 A. Ohashi et al., published by De Gruyter. Recently, the Lanczos bidiagonalization method over tensor space has been proposed for computing the maximum and minimum singular values of a tensor sum T. The method over tensor space is practical in memory and has a simple implementation due to recent developments in tensor computations; however, there is still room for improvement in the convergence to the minimum singular value. This study reconstructed an invert Lanczos bidiagonalization method from vector space to tensor space. The resulting algorithm requires solving linear systems at each iteration step. Using standard direct methods, such as the LU decomposition for solving the linear systems requires a huge memory of O(n6). Therefore, this paper proposes a tensor-structure-preserving direct methods of T whose memory requirements are of O(n3), which is equivalent to the order of iterative methods. Numerical examples indicate that the number of iterations tends to be much smaller than that of the conventional method.

13. Yuto Miyatake, Tai Nakagawa, Tomohiro Sogabe, Shao Liang Zhang

Journal of Computational Dynamics   Vol. 6 ( 2 ) page： 361 - 383   2019

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©2019 American Institute of Mathematical Sciences. We propose a Fourier pseudo-spectral scheme for the space-fractional nonlinear Schrödinger equation. The proposedscheme 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.

14. Lee Dongjin, 曽我部知広, 宮武勇登, Zhang Shao-Liang

日本応用数理学会論文誌(Web)   Vol. 29 ( 1 ) page： 121‐140(J‐STAGE) - 140   2019

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Language：Japanese   Publishing type：Research paper (scientific journal)   Publisher：The Japan Society for Industrial and Applied Mathematics

<p><i>Abstract.</i> We propose a numerical algorithm for the problem of computing the k-th singular triplet of a large sparse matrix. In the proposed algorithm, the problem is divided into subproblems, and then eigenvalue problems equivalent to the singular value problem are solved in each subproblem. We discuss the accuracy of the proposed algorithm and show that it can handle large matrices in numerical experiments.</p>

15. Kosuke Nakano, Tomoya Kemmochi, Yuto Miyatake, Tomohiro Sogabe, Shao-Liang Zhang

JSIAM LETTERS   Vol. 11 ( 0 ) page： 77 - 80   2019

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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.

16. On the equivalence between SOR-type methods for linear systems and the discrete gradient methods for gradient systems Reviewed

Miyatake Yuto, Sogabe Tomohiro, Zhang Shao-Liang

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS   Vol. 342   page： 58-69   2018.11

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17. Solution of the k-th eigenvalue problem in large-scale electronic structure calculations Reviewed

Lee Dongjin, Hoshi Takeo, Sogabe Tomohiro, Miyatake Yuto, Zhang Shao-Liang

JOURNAL OF COMPUTATIONAL PHYSICS   Vol. 371   page： 618-632   2018.10

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18. A Look-Back-type restart for the restarted Krylov subspace methods for solving non-Hermitian linear systems Reviewed

Imakura Akira, Sogabe Tomohiro, Zhang Shao-Liang

JAPAN JOURNAL OF INDUSTRIAL AND APPLIED MATHEMATICS   Vol. 35 ( 2 ) page： 835-859   2018.7

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19. On a relationship between the T-congruence Sylvester equation and the Lyapunov equation Reviewed

Oozawa Masaya, Sogabe Tomohiro, Miyatake Yuto, Zhang Shao-Liang

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS   Vol. 329   page： 51-56   2018.2

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20. Lei Du, Tomohiro Sogabe, Shao-Liang Zhang

APPLIED MATHEMATICS LETTERS   Vol. 75   page： 74 - 81   2018.1

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In this paper, we consider the solution of tridiagonal quasi-Toeplitz linear systems. By exploiting the special quasi-Toeplitz structure, we give a new decomposition form of the coefficient matrix. Based on this matrix decomposition form and combined with the Sherman Morrison formula, we propose an efficient algorithm for solving the tridiagonal quasi-Toeplitz linear systems. Although our algorithm takes more floating-point operations (FLOPS) than the LU decomposition method, it needs less memory storage and data transmission and is about twice faster than the LU decomposition method. Numerical examples are given to illustrate the efficiency of our algorithm. (C) 2017 Elsevier Ltd. All rights reserved.

21. 立岡文理, 曽我部知広, 宮武勇登, Zhang Shao-Liang

日本応用数理学会論文誌(Web)   Vol. 28 ( 3 ) page： 142‐161(J‐STAGE) - 161   2018

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Language：Japanese   Publishing type：Research paper (scientific journal)   Publisher：The Japan Society for Industrial and Applied Mathematics

<p><i>Abstract.</i> The matrix fractional power A<sup>α</sup> can be represented as an improper integral for a slowly decaying function over the half infinite interval. Conventional approaches to approximating the integral are to apply Gaussian quadrature after some variable transformations; however, the convergence could be slow for an ill-conditioned matrix or a specific value of α. To avoid such situation, we consider the Double Exponential (DE) formula instead of conventional approaches, propose algorithms with settings of the integration interval, and then we show some numerical results.</p>

22. Solution of a Nonlinear Eigenvalue Problem Using Signed Singular Values Reviewed

Ooi Kouhei, Mizuno Yoshinori, Sogabe Tomohiro, Yamamoto Yusaku, Zhang Shao-Liang

EAST ASIAN JOURNAL ON APPLIED MATHEMATICS   Vol. 7 ( 4 ) page： 799-809   2017.11

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23. Fatih Yilmaz, Tomohiro Sogabe, Emrullah Kirklar

JOURNAL OF INTEGER SEQUENCES   Vol. 20 ( 4 ) page： 1 - 9   2017

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This paper shows that the Pfaffians and determinants of some skew centrosymmetric matrices can be computed by a paired two-term recurrence relation, or a general number sequence of second order. As a result, the complexities of the formulas are of order It. Furthermore, the formulas have no divisions at all, i.e., they fall into the class of breakdown-free algorithms.

24. 宮武勇登, 曽我部知広, ZHANG Shao-Liang

日本応用数理学会論文誌(Web)   Vol. 27 ( 3 ) page： 239‐249(J‐STAGE) - 249   2017

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Language：Japanese   Publishing type：Research paper (scientific journal)   Publisher：The Japan Society for Industrial and Applied Mathematics

<p><i>Abstract.</i> A novel framework of generating iterative solvers for symmetric positive definite linear systems is proposed. In the framework, iterative solvers are generated by applying the discrete gradient method to initial value problems whose equilibrium coincides with the exact solution of the linear systems. The framework can generate potentially a large number of linear solvers with unconditional convergence property. As a simple example, we show that the framework includes the SOR method.</p>

25. Energy-preserving $H^1$-Galerkin schemes for the Hunter--Saxton equation Reviewed

Yuto Miyatake, Geonsik Eom, Tomohiro Sogabe, Shao-Liang Zhang

Journal of Mathematical Research with Applications   Vol. 37   page： 107 - 118   2017

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

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

Journal of Mathematical Research with Applications   ( 37 ) page： 97 - 106   2016.11

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27. Sho Mizuno, Yosuke Moriizumi, Tsuyoshi Sasaki Usuda, Tomohiro Sogabe

JSIAM Lett.   Vol. 8 ( 0 ) page： 17 - 20   2016

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28. A generalized symbolic Thomas algorithm for the solution of opposite-bordered tridiagonal linear systems Reviewed

Jiteng Jia, Tomohiro Sogabe, Sumei Li

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS   Vol. 290   page： 423 - 432   2015.12

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In the current paper, we present a generalized symbolic Thomas algorithm, that never suffers from breakdown, for solving the opposite-bordered tridiagonal (OBT) linear systems. The algorithm uses a fill-in matrix factorization and can solve an OBT linear system in 0(n) operations. Meanwhile, an efficient method of evaluating the determinant of an opposite-bordered tridiagonal matrix is derived. The computational costs of the proposed algorithms are also discussed. Moreover, three numerical examples are provided in order to demonstrate the performance and effectiveness of our algorithms and their competitiveness with some already existing algorithms. All of the experiments are performed on a computer with the aid of programs written in Matlab. (C) 2015 Elsevier B.V. All rights reserved.

29. Lei Du, Tomohiro Sogabe, Shao-Liang Zhang

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS   Vol. 274   page： 35 - 43   2015.1

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The IDR(s) method by Sonneveld and van Gijzen (2008) has recently received tremendous attention since it is effective for solving nonsymmetric linear systems. In this paper, we generalize this method to solve shifted nonsymmetric linear systems. When solving this kind of problem by existing shifted Krylov subspace methods, we know one just needs to generate one basis of the Krylov subspaces due to the shift-invariance property of Krylov subspaces. Thus the computation cost required by the basis generation of all shifted linear systems, in terms of matrix vector products, can be reduced. For the IDR(s) method, we find that there also exists a shift-invariance property of the Sonneveld subspaces. This inspires us to develop a shifted version of the IDR(s) method for solving the shifted linear systems. (C) 2014 Elsevier B.V. All rights reserved.

30. A. Ohashi, T. Sogabe, T. S. Usuda

Special Matrices   Vol. 3 ( 1 ) page： 200 - 206   2015.1

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© 2015 A. Ohashis et al., licensee De Gruyter Open. We consider a k-tridiagonal ℓ-Toeplitz matrix as one of generalizations of a tridiagonal Toeplitz matrix. In the present paper, we provide a decomposition of the matrix under a certain condition. By the decomposition, the matrix is easily analyzed since one only needs to analyze the small matrix obtained from the decomposition. Using the decomposition, eigenpairs and arbitrary integer powers of the matrix are easily shown as applications.

31. Asuka Ohashi, Tsuyoshi Sasaki Usuda, Tomohiro Sogabe, Fatih Yilmaz

International Journal of Pure and Applied Mathematics   Vol. 103 ( 3 ) page： 537 - 545   2015

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© 2015 Academic Publications, Ltd. In the present paper, we provide a decomposition of a k-tridiagonal Toeplitz matrix via tensor product. By the decomposition, the required memory of the matrix is reduced and the matrix is easily analyzed since we can use properties of tensor product.

32. Chun Wen, Ting-Zhu Huang, Tomohiro Sogabe

COMPUTING AND INFORMATICS   Vol. 34 ( 2 ) page： 495 - 516   2015

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Motivated by the recent applications of the conjugate residual method to nonsymmetric lineal systems by Sogabe, Sugihara and Zhang [An extension of the conjugate residual method to nonsymmetric linear systems. J. Comput. Appl. Math., Vol. 266, 2009, pp. 103-113], this paper describes two conjugate direction methods, BiCR and BiCG, and attempts to extend their applications to compute the stationary probability distribution for an irreducible Markov chain with the aim of finding an alternative basic solver. Numerical experiments show the feasibility of the BiCR and BiCG to some extent, with applications to several practical Markov chain problems.

33. Quasi-minimal residual variants of the COCG and COCR methods for complex symmetric linear systems in electromagnetic simulations Reviewed

IEEE Trans. Microw. Theory Techn.   Vol. 62   page： 2859-2867   2014.12

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34. A note on a fast breakdown-free algorithm for computing the determinants and the permanents of k-tridiagonal matrices Reviewed

Tomohiro Sogabe, Fatih Yilmaz

APPLIED MATHEMATICS AND COMPUTATION   Vol. 249   page： 98 - 102   2014.12

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k-Tridiagonal matrices have attracted much attention in recent years, which are a generalization of tridiagonal matrices. In this note, a breakdown-free numerical algorithm of O(n) is presented for computing the determinants and the permanents of k-tridiagonal matrices. Even though the algorithm is not a symbolic algorithm, it never suffers from breakdown. Furthermore, it produces exact values when all entries of the k-tridiagonal matrices are given in integer. In addition, the algorithm can be simplified for a general symmetric Toeplitz case, and it generates the kth powers of Fibonacci, Pell, and Jacobsthal numbers for a certain symmetric Toeplitz case. (C) 2014 Elsevier Inc. All rights reserved.

35. Lei Du, Tomohiro Sogabe, Shao-Liang Zhang

APPLIED MATHEMATICS AND COMPUTATION   Vol. 244   page： 10 - 15   2014.10

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Fast algorithms for solving symmetric penta-diagonal Toeplitz linear systems have been proposed in McNally (2010) [7], Nemani (2010) [8] and Xu (2010) [11]. In this paper, we discuss the general nonsymmetric problem and propose an algorithm for solving nonsymmetric penta-diagonal Toeplitz linear systems. Numerical examples are given to illustrate the efficiency of our algorithm. (C) 2014 Elsevier Inc. All rights reserved.

36. Xian-Ming Gu, Ting-Zhu Huang, Jing Meng, Tomohiro Sogabe, Hou-Biao Li, Liang Li

COMPUTERS & MATHEMATICS WITH APPLICATIONS   Vol. 68 ( 7 ) page： 746 - 758   2014.10

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The shifted linear systems with non-Hermitian matrices often arise from the numerical solutions for time-dependent PDEs, computing the large-scale eigenvalue problems, control theory and so on. In the present paper, we develop two shifted variants of BiCR-type methods for solving such linear systems. These variants of BiCR-type methods take advantage of the shifted structure, so that the number of matrix-vector multiplications and the number of inner products are the same as a single linear system. Finally, extensive numerical examples are reported to illustrate the performance and effectiveness of the proposed methods. (C) 2014 Elsevier Ltd. All rights reserved.

37. 曽我部 知広

日本応用数理学会論文誌   Vol. 24 ( 1 )   2014

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38. Fatih Yilmaz, Tomohiro Sogabe

International Journal of Pure and Applied Mathematics   Vol. 96 ( 2 ) page： 289 - 298   2014

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In this note, we provide arbitrary integer powers for another type of k-tridiagonal matrix family whose integer powers are specified to the famous Fibonacci numbers.

39. Ji-Teng Jia, Tomohiro Sogabe

JOURNAL OF MATHEMATICAL CHEMISTRY   Vol. 51 ( 3 ) page： 881 - 889   2013.3

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In this paper, a novel numerical algorithm for solving quasi penta-diagonal linear systems is presented. The computational costs of the algorithm is less than those of three successful algorithms given by El-Mikkawy and Rahmo (Comput Math Appl 59:1386-1396, 2010), by Lv and Le (Appl Math Comput 204:707-712, 2008), and by Jia et al. (Int J Comput Math 89:851-860, 2012). In addition, a new recursive method for inverting the quasi penta-diagonal matrices is also discussed. The implementation of the algorithm using Computer Algebra Systems (CASs) such as MATLAB and MAPLE is straightforward. Two numerical examples are given in order to demonstrate the performance and efficiency of our algorithm.

40. Ji-Teng Jia, Tomohiro Sogabe

INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS   Vol. 90 ( 2 ) page： 435 - 444   2013.2

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In the current paper, a new serial algorithm for solving nearly penta-diagonal linear systems is presented. The computational cost of the algorithm is less than or almost equal to those of recent successful algorithms [J. Jia, Q. Kong, and T. Sogabe, A fast numerical algorithm for solving nearly penta-diagonal linear systems, Int. J. Comput. Math. 89 (2012), pp. 851860; X.G. Lv and J. Le, A note on solving nearly penta-diagonal linear systems, Appl. Math. Comput. 204 (2008), pp. 707712; S.N. Neossi Nguetchue and S. Abelman, A computational algorithm for solving nearly penta-diagonal linear systems, Appl. Math. Comput. 203 (2008), pp. 629634]. Moreover, it is suitable for developing its parallel algorithms. One of the parallel algorithms is given and is shown to be promising. The implementation of the algorithms using Computer Algebra Systems such as MATLAB and MAPLE is straightforward. Two numerical examples are given in order to illustrate the validity and efficiency of our algorithms.

41. Jiteng Jia, Tomohiro Sogabe

APPLIED MATHEMATICS AND COMPUTATION   Vol. 219 ( 12 ) page： 6761 - 6767   2013.2

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In this paper, a new method is presented for obtaining the particular solution of ordinary differential equations with constant coefficients. An explicit formula of the particular solution is derived from the use of an upper triangular Toeplitz matrix. The validity of the approach is illustrated by some examples. (C) 2013 Elsevier Inc. All rights reserved.

42. Jiteng Jia, Tomohiro Sogabe, Moawwad El-Mikkawy

COMPUTERS & MATHEMATICS WITH APPLICATIONS   Vol. 65 ( 1 ) page： 116 - 125   2013.1

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In this paper, we consider an inverse problem with the k-tridiagonal Toeplitz matrices. A theoretical result is obtained that under certain assumptions the explicit inverse of a k-tridiagonal Toeplitz matrix can be derived immediately. Two numerical examples are given to demonstrate the validity of our results. (c) 2012 Elsevier Ltd. All rights reserved.

43. Dongjin Lee, Takafumi Miyata, Tomohiro Sogabe, Takeo Hoshi, Shao-Liang Zhang

JAPAN JOURNAL OF INDUSTRIAL AND APPLIED MATHEMATICS   Vol. 30 ( 3 ) page： 625 - 633   2013

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We consider the generalized eigenvalue problem Ax = lambda Bx, where A and B are real symmetric matrices and B is also positive definite. All the eigenvalues of this problem are real, and it is often necessary to compute only a few eigenvalues which are important for applications. In electronic structure calculations of materials, specific interior eigenvalues are of fundamental interest, since they play crucial roles in various industrial applications. In this paper, we propose an approach based on the inertia of the linear matrix pencil of A and B. The eigenvalue problem is restated, and a class of algorithms is presented for separating the target eigenvalues from the others.

44. An Efficient Variant of the Restarted Shifted GMRES Method for Solving Shifted Linear Systems Reviewed

Imakura,Akira, Tomohiro,Sogabe, Shao-Liang,Zhang

Journal of Mathematical Research with Applications   Vol. 33 ( 2 ) page： 127 - 141   2013

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We investigate the restart of the Restarted Shifted GMRES method for solving shifted linear systems. Recently the variant of the GMRES($m$) method with the {\it unfixed update} has been proposed to improve the convergence of the GMRES($m$) method for solving linear systems, and shown to have an efficient convergence property. In this paper, by applying the unfixed update to the Restarted Shifted GMRES method, we propose a variant of the Restarted Shifted GMRES method. We show a potentiality for efficient convergence within the variant by some numerical results.

45. On Convergence Behavior of the GMRES(m) Method with the Deflation‐Type Restart and the Look‐Back‐Type Restart Reviewed

今倉暁, YANG Ji‐Rong, 曽我部知広, ZHANG Shao‐Liang

日本応用数理学会論文誌   Vol. 22 ( 3 ) page： 117 - 141   2012.9

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46. Tomohiro Sogabe, Takeo Hoshi, Shao-Liang Zhang, Takeo Fujiwara

JOURNAL OF COMPUTATIONAL PHYSICS   Vol. 231 ( 17 ) page： 5669 - 5684   2012.7

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We develop the shifted COCG method [R. Takayama, T. Hoshi, T. Sogabe, S.-L. Zhang, T. Fujiwara, Linear algebraic calculation of Green's function for large-scale electronic structure theory, Phys. Rev. B 73 (165108) (2006) 1-9] and the shifted WQMR method [T. Sogabe, T. Hoshi, S.-L. Zhang, T. Fujiwara, On a weighted quasi-residual minimization strategy of the QMR method for solving complex symmetric shifted linear systems, Electron. Trans. Numer. Anal. 31 (2008) 126-140] for solving generalized shifted linear systems with complex symmetric matrices that arise from the electronic structure theory. The complex symmetric Lanczos process with a suitable bilinear form plays an important role in the development of the methods. The numerical examples indicate that the methods are highly attractive when the inner linear systems can efficiently be solved. (C) 2012 Elsevier Inc. All rights reserved.

47. T. Hoshi, S. Yamamoto, T. Fujiwara, T. Sogabe, S-L Zhang

JOURNAL OF PHYSICS-CONDENSED MATTER   Vol. 24 ( 16 )   2012.4

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A linear algebraic theory called the 'multiple Arnoldi method' is presented and realizes large-scale (order-N) electronic structure calculations with generalized eigenvalue equations. A set of linear equations, in the form of (zS - H)x = b, are solved simultaneously with multiple Krylov subspaces. The method is implemented in a simulation package ELSES (www.elses.jp) with tight-binding-form Hamiltonians. A finite-temperature molecular dynamics simulation is carried out for metallic and insulating materials. A calculation with 10(7) atoms was realized by a workstation. The parallel efficiency is shown up to 1024 CPU cores.

48. Jiteng Jia, Qiongxiang Kong, Tomohiro Sogabe

COMPUTERS & MATHEMATICS WITH APPLICATIONS   Vol. 63 ( 7 ) page： 1238 - 1243   2012.4

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A new numerical algorithm for solving nearly penta-diagonal Toeplitz linear systems is presented. The algorithm is suited for implementation using Computer Algebra Systems (CASs) such as MATLAB, MACSYMA and MAPLE. Numerical examples are given in order to illustrate the efficiency of our algorithm. (C) 2011 Elsevier Ltd. All rights reserved.

49. A Look‐Back GMRES(m) Method for Solving Nonsymmetric Linear Systems Reviewed

今倉暁, 曽我部知広, ZHANG Shao‐Liang

日本応用数理学会論文誌   Vol. 22 ( 1 ) page： 1 - 21   2012.3

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50. Akira Imakura, Tomohiro Sogabe, Shao-Liang Zhang

EAST ASIAN JOURNAL ON APPLIED MATHEMATICS   Vol. 2 ( 1 ) page： 19 - 32   2012.2

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The GMRES(m) method proposed by Saad and Schultz is one of the most successful Krylov subspace methods for solving nonsymmetric linear systems. In this paper, we investigate how to update the initial guess to make it converge faster, and in particular propose an efficient variant of the method that exploits an unfixed update. The mathematical background of the unfixed update variant is based on the error equations, and its potential for efficient convergence is explored in some numerical experiments.

51. Imakura Akira, Yang Ji-Rong, Sogabe Tomohiro, Zhang Shao-Liang

Transactions of the Japan Society for Industrial and Applied Mathematics   Vol. 22 ( 3 ) page： 117-141   2012

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We investigate two types of the improvement techniques for the GMRES(m) method to solve nonsymmetric linear systems: the deflation-type restart and the Look-Back-type restart. From the analysis based on the residual polynomials, we show in this paper that these restart techniques modify the convergence behavior of the GMRES(m) method by different mathematical backgrounds. Then under the knowledge from the analysis, we propose an efficient improvement of the GMRES(m) method with these restart techniques. The numerical experiments indicate that the proposed method shows the efficient convergence behavior.

52. Imakura Akira, Sogabe Tomohiro, Zhang Shao-Liang

Transactions of the Japan Society for Industrial and Applied Mathematics   Vol. 22 ( 1 ) page： 1-21   2012

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In recent years, we have focused on the restart of the GMRES(m) method, and proposed an extension of the GMRES(m) method based on the error equations. In this paper, from the analysis on the residual polynomial of the extension of the GMRES(m) method, we propose a Look-Back GMRES(m) method under a Look-Back strategy which reconstructs the past residual polynomials of the GMRES(m) method. Some numerical experiments indicate that the Look-Back GMRES(m) method has a higher performance for efficient convergence than the GMRES(m) method.

53. Ji-teng Jia, Qiong-xiang Kong, Tomohiro Sogabe

INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS   Vol. 89 ( 6 ) page： 851 - 860   2012

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In this paper, we present a fast numerical algorithm for solving nearly penta-diagonal linear systems and show that the computational cost is less than those of three algorithms in El-Mikkawy and Rahmo, [Symbolic algorithm for inverting cyclic penta-diagonal matrices recursively-Derivation and implementation, Comput. Math. Appl. 59 (2010), pp. 1386-1396], Lv and Le [A note on solving nearly penta-diagonal linear systems, Appl. Math. Comput. 204 (2008), pp. 707-712] and Neossi Nguetchue and Abelman [A computational algorithm for solving nearly penta-diagonal linear systems, Appl. Math. Comput. 203 (2008), pp. 629-634.]. In addition, an efficient way of evaluating the determinant of a nearly penta-diagonal matrix is also discussed. The algorithm is suited for implementation using computer algebra systems (CAS) such as MATLAB, MACSYMA and MAPLE. Some numerical examples are given in order to illustrate the efficiency of our algorithm.

APPLIED MATHEMATICS AND COMPUTATION   Vol. 218 ( 6 ) page： 2740 - 2743   2011.11

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In the present paper, we give a fast algorithm for block diagonalization of k-tridiagonal matrices. The block diagonalization provides us with some useful results: e. g., another derivation of a very recent result on generalized k-Fibonacci numbers in [M. E. A. El-Mikkawy, T. Sogabe, A new family of k-Fibonacci numbers, Appl. Math. Comput. 215 (2010) 4456-4461]; efficient (symbolic) algorithm for computing the matrix determinant. (C) 2011 Elsevier Inc. All rights reserved.

55. Lei Du, Tomohiro Sogabe, Shao-Liang Zhang

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS   Vol. 236 ( 5 ) page： 621 - 630   2011.10

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The IDR(s) method proposed by Sonneveld and van Gijzen is an effective method for solving nonsymmetric linear systems, but usually with irregular convergence behavior. In this paper, we reformulate the relations of residuals and their auxiliary vectors generated by the IDR(s) method in matrix form. Then, using this new formulation and motivated by other QMR-type methods, we propose a variant of the IDR(s) method, called QMRIDR(s), for overcoming the disadvantage of its irregular convergence behavior. Both fast and smooth convergence behaviors of the QMRIDR(s) method can be shown. Numerical experiments are reported to show the efficiency of our proposed method. (C) 2011 Elsevier B.V. All rights reserved.

56. An Arnoldi(M,W,G) Method for Generalized Eigenvalue Problems Reviewed

山下達也, 宮田考史, 曽我部知広, 星健夫, 藤原毅夫, ZHANG Shao‐Liang

日本応用数理学会論文誌   Vol. 21 ( 3 ) page： 241 - 254   2011.9

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57. L. Du, T. Sogabe, B. Yu, Y. Yamamoto, S. -L. Zhang

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS   Vol. 235 ( 14 ) page： 4095 - 4106   2011.5

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The IDR(s) based on the induced dimension reduction (IDR) theorem, is a new class of efficient algorithms for large nonsymmetric linear systems. IDR(1) is mathematically equivalent to BiCGStab at the even IDR(1) residuals, and IDR(s) with s &gt; 1 is competitive with most Bi-CG based methods. For these reasons, we extend the IDR(s) to solve large nonsymmetric linear systems with multiple right-hand sides. In this paper, a variant of the IDR theorem is given at first, then the block IDR(s), an extension of IDR(s) based on the variant IDR(s) theorem, is proposed. By analysis, the upper bound on the number of matrix-vector products of block IDR(s) is the same as that of the IDR(s) for a single right-hand side in generic case, i.e., the total number of matrix-vector products of IDR(s) may be m times that of of block IDR(s), where in is the number of right-hand sides. Numerical experiments are presented to show the effectiveness of our proposed method. (C) 2011 Elsevier B.V. All rights reserved.

58. H. Teng, T. Fujiwara, T. Hoshi, T. Sogabe, S. -L. Zhang, S. Yamamoto

PHYSICAL REVIEW B   Vol. 83 ( 16 )   2011.4

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The need for large-scale electronic structure calculations arises recently in the field of material physics, and efficient and accurate algebraic methods for large simultaneous linear equations become greatly important. We investigate the generalized shifted conjugate orthogonal conjugate gradient method, the generalized Lanczos method, and the generalized Arnoldi method. They are the solver methods of large simultaneous linear equations of the one-electron Schrodinger equation and map the whole Hilbert space to a small subspace called the Krylov subspace. These methods are applied to systems of fcc Au with the NRL tight-binding Hamiltonian [F. Kirchhoff et al., Phys. Rev. B 63, 195101 (2001)]. We compare results by these methods and the exact calculation and show them to be equally accurate. The system size dependence of the CPU time is also discussed. The generalized Lanczos method and the generalized Arnoldi method are the most suitable for the large-scale molecular dynamics simulations from the viewpoint of CPU time and memory size.

59. Du Lei, Sogabe Tomohiro, Zhang Shao-Liang

JSIAM Letters   Vol. 3 ( 0 ) page： 13-16   2011

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60. Y. Mizuno, K. Ohi, T. Sogabe, Y. Yamamoto, Y. Kaneda

PHYSICAL REVIEW E   Vol. 82 ( 3 )   2010.9

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A numerical analysis is made on the four-point correlation function in a similarity range of a model of two-dimensional passive scalar field psi advected by a turbulent velocity field with infinitely small correlation time. The model yields an exact closure equation for the four-point correlation Psi(4) of psi, which may be casted into the form of an eigenvalue problem in the similarity range. The analysis of the eigenvalue problem gives not only the scale dependence of Psi(4), but also the dependence on the configuration of the four points. The numerical analysis gives S-4(R) alpha R-zeta 4 in the similarity range in which S-2(R) alpha R-zeta 2, where S-N is the structure function defined by S-N(R) equivalent to &lt;[psi(x + R) - psi(x)](N) and zeta(4) not equal 2 zeta(2). The estimate of zeta(4) by the numerical analysis of the eigenvalue problem is in good agreement with numerical simulations so far reported. The agreement supports the idea of universality of the exponent zeta(4) in the sense that zeta(4) is insensitive to conditions of psi outside the similarity range. The numerical analysis also shows that the correlation C(R, r) equivalent to &lt;[psi(x + R) - psi(x)](2) X[psi(x+r) - psi(x)](2) is stronger than that given by the joint-normal approximation, and scales like C(R, r) alpha (r/R)(chi) for r/R &lt;&lt; 1 with R and r in the similarity range, where chi is a constant depending on the angle between R and r.

61. An Approach to the Correction Equation in the Jacobi‐Davidson Method―Based on a Shift Invariance Property of the Krylov Subspace on a Projected Space― Reviewed

宮田考史, 曽我部知広, ZHANG Shao‐Liang

日本応用数理学会論文誌   Vol. 20 ( 2 ) page： 115 - 129   2010.6

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62. T. Fujiwara, T. Hoshi, S. Yamamoto, T. Sogabe, S-L Zhang

JOURNAL OF PHYSICS-CONDENSED MATTER   Vol. 22 ( 7 )   2010.2

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We review our recently developed methods of solving large-scale simultaneous linear equations and applications to electronic structure calculations both in one-electron theory and many-electron theory. This is the shifted COCG (conjugate orthogonal conjugate gradient) method based on the Krylov subspace, and the most important issue for applications is the shift equation and the seed switching method, which greatly reduce the computational cost. The applications to nano-scale Si crystals and the double orbital extended Hubbard model are presented.

APPLIED MATHEMATICS AND COMPUTATION   Vol. 215 ( 12 ) page： 4456 - 4461   2010.2

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In the present paper, we give a new family of k-Fibonacci numbers and establish some properties of the relation to the ordinary Fibonacci numbers. Furthermore, we describe the recurrence relations and the generating functions of the new family for k = 2 and k = 3, and presents a few identity formulas for the family and the ordinary Fibonacci numbers. (C) 2010 Elsevier Inc. All rights reserved.

APPLIED MATHEMATICS AND COMPUTATION   Vol. 215 ( 9 ) page： 3311 - 3317   2010.1

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This paper presents some applications using several properties of three important symmetric polynomials: elementary symmetric polynomials, complete symmetric polynomials and the power sum symmetric polynomials. The applications includes a simple proof of El-Mikkawy conjecture in [M. E. A. El-Mikkawy, Appl. Math. Comput. 146 (2003) 759-769] and a very easy proof of the Newton-Girard formula. In addition, a generalization of Stirling numbers is obtained. (C) 2009 Elsevier Inc. All rights reserved.

65. On the Restart of the GMRES(m) Method Reviewed

今倉暁, 曽我部知広, ZHANG Shao‐Liang

日本応用数理学会論文誌   Vol. 19 ( 4 ) page： 551 - 564   2009.12

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66. An Extension of the Sakurai‐Sugiura Method for Eigenvalue Problems of Multiply Connected Region Reviewed

宮田考史, DU Lei, 曽我部知広, 山本有作, ZHANG Shao‐Liang

日本応用数理学会論文誌   Vol. 19 ( 4 ) page： 537 - 550   2009.12

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67. An Implicit Wavelet Sparse Approximate Inverse Preconditioner Using Block Finger Pattern Reviewed

今倉,暁, 曽我部,知広, 張,紹良

Numerical Linear Algebra with Applications   Vol. 16 ( 11-12 ) page： 915 - 928   2009.11

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Recently, an implicit wavelet sparse approximate inverse (IW-SPAI hereafter) preconditioner has been proposed by Hawkins and Chen for nonsymmetric linear systems. The preconditioning matrix of the IW-SPAI is characterized by a special sparse structure, the so-called finger pattern, which makes it possible to construct a good sparse approximate inverse. Since the IW-SPAI requires a number of QR factorizations with substantial costs to construct the preconditioner, there is a strong need to construct the IW-SPAI more efficiently. The target of this paper is to reduce the costs of the IW-SPAI using Haar basis. Under the strategy of classifying the QR factorizations into several groups and reducing the number of QR factorizations, in this paper, we introduce a block finger pattern for Haar basis in the place of the finger pattern as the nonzero pattern of the preconditioning matrix. From the block finger pattern, we propose an implicit wavelet sparse approximate inverse preconditioner using block finger pattern (BIW-SPAI hereafter). Numerical experiments indicate that the BIW-SPAI is often more efficient than the IW-SPAI.

68. 大規模シフト線形方程式の数値解法―クリロフ部分空間の性質に着目して Reviewed

曽我部知広, 張紹良

応用数理   Vol. 19 ( 3 ) page： 163 - 178   2009.9

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69. T. Sogabe, M. Sugihara, S. -L. Zhang

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS   Vol. 226 ( 1 ) page： 103 - 113   2009.4

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The Conjugate Gradient (CG) method and the Conjugate Residual (CR) method are Krylov subspace methods for solving symmetric (positive definite) linear systems. To solve nonsymmetric linear systems, the Bi-Conjugate Gradient (Bi-CG) method has been proposed as an extension of CG. Bi-CG has attractive short-term recurrences, and it is the basis for the successful variants such as Bi-CGSTAB. In this paper, we extend CR to nonsymmetric linear systems with the aim of finding an alternative basic solver. Numerical experiments show that the resulting algorithm with short-term recurrences often gives smoother convergence behavior than Bi-CG. Hence, it may take the place of Bi-CG for the successful variants. (c) 2008 Elsevier B.V. All rights reserved.

70. A. Imakura, T. Sogabe, S. -L. Zhang

NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS   Vol. 16 ( 11-12 ) page： 915 - 928   2009

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Recently, an implicit wavelet sparse approximate inverse (IW-SPAI hereafter) preconditioner has been proposed by Hawkins and Chen for nonsymmetric linear systems. The preconditioning matrix of the IW-SPAI is characterized by a special sparse structure, the so-called finger pattern, which makes it possible to construct a good sparse approximate inverse. Since the IW-SPAI requires a number of QR factorizations with substantial costs to construct the preconditioner, there is a strong need to construct the IW-SPAI more efficiently. The target of this paper is to reduce the costs of the IW-SPAI using Haar basis. Under the strategy of classifying the QR factorizations into several groups and reducing the number of QR factorizations, in this paper, we introduce a block finger pattern for Haar basis in the place of the finger pattern as the nonzero pattern of the preconditioning matrix. From the block finger pattern, we propose an implicit wavelet sparse approximate inverse preconditioner using block finger pattern (BIW-SPAI hereafter). Numerical experiments indicate that the BIW-SPAI is often more efficient than the IW-SPAI. Copyright (C) 2009 John Wiley & Sons, Ltd.

DOI： 10.1002/nla.657

71. Imakura Akira, Sogabe Tomohiro, Zhang Shao-Liang

Transactions of the Japan Society for Industrial and Applied Mathematics   Vol. 19 ( 4 ) page： 551-564   2009

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In 1986, the GMRES method and its restarted version, GMRES(m) method, were proposed by Saad and Schultz for solving nonsymmetric linear systems. In this paper, we investigate the restart of the GMRES(m) method. The main goal of this paper is to propose an extension of the GMRES(m) method based on a new framework of the restart. A mathematical background of the extension is analyzed based on the error equations. The numerical results are presented to show a potential efficiency in our framework.

72. Miyata Takafumi, Du Lei, Sogabe Tomohiro, Yamamoto Yusaku, Zhang Shao-Liang

Transactions of the Japan Society for Industrial and Applied Mathematics   Vol. 19 ( 4 ) page： 537-550   2009

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The Sakurai-Sugiura method was proposed to compute the eigenvalues which are contained in a specified region on the complex plane. In this paper, we extend the Sakurai-Sugiura method to compute the eigenvalues in a finite and multiply connected region. We also analyze the error of the computed eigenvalues.

73. Yamamoto Susumu, Sogabe Tomohiro, Hoshi Takeo, ZHANG Shao-Liang, FUJIWARA Takeo

Journal of the Physical Society of Japan   Vol. 77 ( 11 ) page： "114713-1"-"114713-8"   2008.11

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74. Tomohiro Sogabe

APPLIED MATHEMATICS AND COMPUTATION   Vol. 202 ( 2 ) page： 850 - 856   2008.8

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75. Tomohiro Sogabe

APPLIED MATHEMATICS AND COMPUTATION   Vol. 201 ( 1-2 ) page： 561 - 564   2008.7

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A fast numerical algorithm for evaluating the determinant of a pentadiagonal matrix has been recently proposed [T. Sogabe, A fast numerical algorithm for the determinant of a pentadiagonal matrix, Appl. Math. Comput. 196 (2008) 835-841]. The algorithm whose cost is 14n - 28, where n(>= 3) denotes the size of the matrix, is composed of two steps: first, transform a pentadiagonal matrix into sparse Hessenberg form; second, run a numerical algorithm for computing the determinant of the sparse Hessenberg matrix. In this note, it is shown that we have an algorithm with the cost of 13n - 24 by applying twice the idea of the sparse Hessenberg transformation to a pentadiagonal matrix. (c) 2008 Elsevier Inc. All rights reserved.

76. Tomohiro Sogabe

APPLIED MATHEMATICS AND COMPUTATION   Vol. 198 ( 1 ) page： 117 - 122   2008.4

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Recently, two efficient algorithms for solving comrade linear systems have been proposed by Karawia [A. A. Karawia, Two algorithms for solving comrade linear systems, Appl. Math. Comput. 189 (2007) 291-297]. The two algorithms are based on the LU decomposition of the comrade matrix. In this paper, two algorithms are presented for solving the comrade linear systems based on the use of conventional fast tridiagonal solvers and an efficient way of evaluating the determinant of the comrade matrix is discussed. (C) 2007 Elsevier Inc. All rights reserved.

77. An AOR‐base Variable Preconditioned GCR Method Reviewed

前田祥兵, 阿部邦美, 曽我部知広, ZHANG Shao‐Liang

日本応用数理学会論文誌   Vol. 18 ( 1 ) page： 155 - 170   2008.3

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78. Tomohiro Sogabe

APPLIED MATHEMATICS AND COMPUTATION   Vol. 196 ( 2 ) page： 835 - 841   2008.3

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Recently, a two-term recurrence for the determinant of a general matrix has been found [T. Sogabe, On a two-term recurrence for the determinant of a general matrix, Appl. Math. Comput., 187 (2007) 785-788] and it leads to a natural generalization of the DETGTRI algorithm [M. El-Mikkawy, A fast algorithm for evaluating nth order tridiagonal determinants, J. Comput. Appl. Math. 166 (2004) 581-584] for computing the determinant of a tridiagonal matrix. In this paper, we derive a fast numerical algorithm for computing the determinant of a pentadiagonal matrix from the generalization of the DETGTRI algorithm. (C) 2007 Elsevier Inc. All rights reserved.

79. New algorithms for solving periodic tridiagonal and periodic pentadiagonal linear systems Reviewed

T. Sogabe

Vol. 202   page： 850-856   2008

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80. A note on “A fast numerical algorithm for the determinant of a pentadiagonal matrix" Reviewed

T. Sogabe

Vol. 201   page： 561-564   2008

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81. AOR法を用いた可変的前処理付き一般化共役残差法 Reviewed

前田祥兵, 阿部邦美, 曽我部知広, 張紹良

日本応用数理学会論文誌   Vol. 18 ( 1 ) page： 155-170   2008

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82. Numerical algorithms for solving comrade linear systems based on tridiagonal solvers Reviewed

T. Sogabe

Vol. 198   page： 117-122   2008

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83. T. Sogabe, T. Hoshi, S. -L. Zhang, T. Fujiwara

ELECTRONIC TRANSACTIONS ON NUMERICAL ANALYSIS   Vol. 31   page： 126 - 140   2008

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We consider the solution of complex symmetric shifted linear systems. Such systems arise in large-scale electronic structure simulations, and there is a strong need of algorithms for their fast solution. With the aim of solving the systems efficiently, we consider a special case of the QMR method for non-Hermitian shifted linear systems and propose its weighted quasi-minimal residual approach. A numerical algorithm, referred to as shifted QMR_SYM(B), is obtained by the choice of a weight which is particularly cost-effective. Numerical examples are presented to show the performance of the shifted QMR_SYM(B) method.

84. An Efficient Implicit Wavelet Sparse Approximate Inverse Preconditioner Using Blocked Finger Pattern Reviewed

今倉暁, 曽我部知広, ZHANG Shao‐Liang

日本応用数理学会論文誌   Vol. 17 ( 4 ) page： 523 - 542   2007.12

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85. On an application of the QMR approach to the Bi‐CR method Reviewed

南さつき, 曽我部知広, 杉原正顯, ZHANG Shao‐Liang

日本応用数理学会論文誌   Vol. 17 ( 3 ) page： 301 - 317   2007.9

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86. ABE KUNIYOSHI, SOGABE TOMOHIRO, FUJINO SEIJI, ZHANG SHAO-LIANG

Vol. 48 ( 8 ) page： 11-21   2007.5

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We propose a product-type Krylov subspace method based on the conjugate residual (CR) method for nonsymmetric coefficient matrices. The recurrence formulas for updating an approximation and a residual vector are the same as those of the original product-type Krylov subspace method, while the recurrence coefficients α_κ and β_κ are determined so as to compute the coefficients of the residual polynomial of CR for nonsymmetric coefficient matrices. Numerical experiments show that our proposed product-type Krylov subspace method is more effective than the original.

87. On a two-term recurrence for the determinant of a general matrix Reviewed

T. Sogabe

Appl. Math. Comput.   Vol. 187   page： 785-788   2007

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88. A COCR method for solving complex symmetric linear systems Reviewed

T. Sogabe and S.-L. Zhang

Vol. 199   page： 297-303   2007

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89. A numerical method for calculating the green's function arising from electronic structure theory Reviewed

Tomohiro Sogabe, Takeo Hoshi, Shao-Liang Zhang, Takeo Fujiwara

FRONTIERS OF COMPUTATIONAL SCIENCE     page： 189 - +   2007

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We developed a fast numerical method for complex symmetric shifted linear systems, which is motivated by the quantum-mechanical (electronic-structure) theory in nanoscale materials. The method is named shifted Conjugate Orthogonal Conjugate Gradient (shifted COCG) method. The formulation is given and several numerical aspects are discussed.

90. Linear algebraic calculation of Green's function for large-scale electronic structure theory Reviewed

R. Takayama, T. Hoshi, T. Sogabe, S.-L. Zhang, and T. Fujiwara

Phys. Rev. B   Vol. 165108   page： 1-9   2006

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91. A Preconditioner of Permutation Matrix for Solving Nonsymmetric Toeplitz Linear Systems Reviewed

曽我部知広, ZHENG Bo, 橋本康, ZHANG Shao‐Liang

日本応用数理学会論文誌   Vol. 15 ( 2 ) page： 159 - 168   2005.6

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92. 共役残差法の非対称行列用への拡張 Reviewed

曽我部知広，杉原正顯，張紹良

日本応用数理学会論文誌   Vol. 15 ( 3 ) page： 445-459   2005

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本論文では，対称行列を係数に持つ連立一次方程式の効率の良い数値解法である共役残差法を短い漸化式で非対称行列用に拡張した双共役残差法を提案した．

93. CGS法の改良について Reviewed

曽我部知広，金成海，阿部邦美，張紹良

日本応用数理学会論文誌   Vol. 14 ( 1 ) page： 1-12   2004

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### Books 4

1. 20世紀のトップ10アルゴリズム

小柳義夫, 土谷隆, 曽我部知広、杉原正顯, 西山博泰, 山本有作, 平田富夫, 大浦拓哉, 張紹良, 須田礼仁（ Role： Joint author ,  第3章　線形方程式のためのクリロフ部分空間法）

共立出版  2022.1  （ ISBN:978-4-320-12267-3

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Language：Japanese Book type：Scholarly book

2. 計算科学のための基本数理アルゴリズム

曽我部知広、山本有作（ Role： Joint author）

共立出版  2019.6  （ ISBN:9784320122666

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Language：Japanese Book type：Scholarly book

3. 数値線形代数の数理とHPC

櫻井鉄也,他（ Role： Joint author ,  第４章　行列関数の数値計算法）

共立出版  2018.8  （ ISBN:978-4-320-01955-3

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4. 応用数理ハンドブック

分担執筆（ Role： Joint author ,  連立1次方程式に対する直接解法、連立1次方程式に対する反復解法）

朝倉書店  2013.7  （ ISBN:978-4-254-11141-5

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Language：Japanese

### MISC 6

1. Fast block diagonalization of k-tridiagonal matrices Reviewed

APPLIED MATHEMATICS AND COMPUTATION   Vol. 218 ( 6 ) page： 2740-2743   2011.11

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2. An Extension of the COCR Method to Solving Shifted Linear Systems with Complex Symmetric Matrices Reviewed

Sogabe Tomohiro, Zhang Shao-Liang

EAST ASIAN JOURNAL ON APPLIED MATHEMATICS   Vol. 1 ( 2 ) page： 97-107   2011.5

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3. A new family of k-Fibonacci numbers Reviewed

APPLIED MATHEMATICS AND COMPUTATION   Vol. 215 ( 12 ) page： 4456-4461   2010.2

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Language：English

4. Notes on particular symmetric polynomials with applications Reviewed

APPLIED MATHEMATICS AND COMPUTATION   Vol. 215 ( 9 ) page： 3311-3317   2010.1

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5. Lanczos-type variants of the COCR method for complex nonsymmetric linear systems Reviewed

Jing Yan-Fei, Huang Ting-Zhu, Zhang Yong, Li Liang, Cheng Guang-Hui, Ren Zhi-Gang, Duan Yong, Sogabe Tomohiro, Carpentieri Bruno

JOURNAL OF COMPUTATIONAL PHYSICS   Vol. 228 ( 17 ) page： 6376-6394   2009.9

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6. On a problem related to the Vandermonde determinant Reviewed

DISCRETE APPLIED MATHEMATICS   Vol. 157 ( 13 ) page： 2997-2999   2009.7

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### Presentations 7

1. Novel linear-algebraic algorithm and large-scale electronic structure calculation International conference

Novel linear-algebraic algorithm and large-scale electronic structure calculation

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Event date： 2008.3

Language：English   Presentation type：Oral presentation (invited, special)

2. A numerical approach to the computation of Green's function in large-scale electronic structure theory International conference

2008 International Workshop on Numerical Verification and its Applications

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Event date： 2008.3

Language：English   Presentation type：Oral presentation (invited, special)

Country：Japan

3. On an application of the QMR method to complex symmetric shifted linear systems

rnational Workshop on Large-scale Matrix Computation and Applications in Physics and Engineering Science

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Event date： 2007.12

Language：English   Presentation type：Oral presentation (invited, special)

Country：Japan

4. Recent development of Krylov subspace methods for the solution of complex symmetric shifted linear systems International conference

International Conference on Numerical Optimization and Numerical Linear Algebra

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Event date： 2007.9

Language：English   Presentation type：Oral presentation (invited, special)

5. A numerical method for calculating the Green's function arising from electronic structure theory International conference

A numerical method for calculating the Green's function arising from electronic structure theory

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Event date： 2005.12

Language：English   Presentation type：Oral presentation (invited, special)

Country：Japan

6. An extension of the conjugate residual method to nonsymmetric linear systems International conference

The 7th China-Japan Joint Seminar for Computational Mathematics and Scientific Computing

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Event date： 2004.8

Language：English   Presentation type：Oral presentation (invited, special)

7. The Bi-CR method for nonsymmetric linear systems International conference

International Conference on Numerical Linear Algebra and Optimization (ICNLAO2003),

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Event date： 2003.10

Language：English   Presentation type：Oral presentation (invited, special)

### KAKENHI (Grants-in-Aid for Scientific Research) 9

1. Grant number：21K18301  2021.7 - 2026.3

科学研究費助成事業  挑戦的研究(開拓)

宮武 勇登, 曽我部 知広

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近年，情報科学諸分野においても偏微分方程式(PDE)の数値計算の重要度が高まっている.従来，PDEの数値計算は，離散化を担う数値解析学と，行列方程式に帰着された後の数値線形代数学によって支えられてきた.学問の細分化の結果それぞれが一つの学問領域として成熟してきたが，一方で両領域間の交流は希薄になってきており，PDEの数値解法の発展の足かせになりはじめている.本研究では，数値解析学と数値線形代数学を独自の視座で再融合し，特に代数学的精神で解析学を数値的に研究する新しい学理 「数値代数解析学」を開拓する.研究期間内では，特に量子系情報・量子物理シミュレーションの限界突破を目指す.

2. 物理学・情報科学に共通する大規模行列関数の総合的数値計算法の創成

Grant number：20H00581  2020.4 - 2025.3

日本学術振興会  科学研究費助成事業 基盤研究(A)

曽我部知広, 田中健一郎, 荻田武史, 宮武勇登, 星健夫, 野中千穂, 臼田毅

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Grant amount：\45630000 （ Direct Cost: \35100000 、 Indirect Cost：\10530000 ）

高校で馴染みのある初等関数（実数乗・指数関数・対数関数・三角関数）の計算は現在では意識されないが、その行列版の計算は、超大規模行列の場合に未解決問題であるだけでなく、データ科学・気象物理学・素粒子物理学・物性物理学・量子情報科学において自然な要求として現れる。そこで超大規模行列に対する行列関数の数値計算法を確立すべく、二重指数関数型積分公式という積分の数値計算法を高度化する理論を構築し、前述した実問題に適用することで当該分野の新たな知の創出を狙う。研究期間は５年であり、理論チームと応用チームの分野横断的な知の交流に基づき解法の開発を行うことが本研究課題推進における特色の一つである。

3. 量子計算を併用した数値線形代数学の開拓

Grant number：20K20397  2018.6 - 2024.3

科学研究費助成事業 挑戦的研究(開拓)

曽我部 知広，臼田毅

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Authorship：Principal investigator

Grant amount：\25090000 （ Direct Cost: \19300000 、 Indirect Cost：\5790000 ）

数値線形代数学の主要課題である線形方程式・固有値問題・特異値問題・最小自乗問題は，計算科学・データ科学の計算効率の根幹を担う．本研究では，現在の計算機で動作する数値線形代数学上の計算アルゴリズムと量子計算機で動作する量子アルゴリズムを開発して組み合わせることで，従前にない計算アルゴリズム，つまり数値線形代数学上のアルゴリズミック・パラダイムの創成に挑戦する．研究指針は，「量子アルゴリズムを併用した線形方程式の計算アルゴリズムの創成」，および「固有値問題・特異値問題・最小自乗問題用への展開」である．
本課題は、数値線形代数（線形方程式、固有値問題、特異値問題）の計算アルゴリズムの研究と量子アルゴリズムの研究を行い、それらを融合・協調させるものである。
その一環として2019年度は、（１）複数の線形方程式の解のノルムを効率よく計算する手法の提案、（2）線形方程式の定常反復法に基づく数値解法、(3)特殊な線形方程式に対して離散フーリエ変換を活用して効率よく解く手法の提案、(4)行列方程式をシンプルかつ在来研究が多い行列方程式に変換する理論の提案、(5)テンソル構造を持つ行列の特異値計算法の提案、そして（6）行列関数の量子アルゴリズムの提案を行った。これらは全て国際誌として掲載された。特に(6)は一般の行列関数とベクトルとの積に関する量子状態を計算する研究であり汎用性が高い研究である。一方、(3)では離散フーリエ変換を用いた数値計算法が提案されており、量子フーリエ変換との関連性に基づく量子アルゴリズムの研究展開が期待される。なお、(1)は最適化と関連しているため、量子計算による最適化への展開を考慮に入れて研究を進める布石といえる。
2019年1月に量子情報ミニワークショップを研究分担者と共に滋賀県にて開催し、本課題推進のための情報共有と密な打ち合わせを行い、量子情報理論・量子計算に関する最新の情報を共有した。
なお、本プロジェクトの支援的研究として量子回路のテストを行いやすくする試み「QISKit による行列関数に対する量子アルゴリズムの実装に向けたツールについて」を実施した。
予定通り古典アルゴリズムの研究が進んでいること、および量子アルゴリズムの研究論文が採択されたことから順調に進展していると判断した。
今年度の研究推進の方策を古典計算アルゴリズムと量子アルゴリズムに分けて簡潔に説明する。まず古典計算アルゴリズムに関しては、昨年度に引き続きテンソルの視点から行列方程式（線形方程式の拡張）の数値解法の開発を行う。数値解法としては反復解法に着目する予定である。次に量子計算アルゴリズムに関しても、昨年度に引き続き線形方程式の量子アルゴリズムの開発を行う。さらに今年度は、新たに量子アニーリングの視点から固有値問題の数値解法の検討を行うことを検討している。可能であれば、今年度の１月頃に昨年度と同様ワークショップを開催（共催）し、研究分担者と研究の進捗状況を確認することで次年度の研究方針を定める予定である。

4. 電磁場解析のエクストリームスケール・コンピューティングを実現する高速数値解法開発

Grant number：17H02829  2017.4 - 2022.3

日本学術振興会  科学研究費助成事業 基盤研究(B)

荻野 正雄, 曽我部 知広, 和田 義孝, 杉本 振一郎, 武居 周

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エクサ時代に適した電磁場問題の高速な数値解析技術確立を目的とし，高性能反復法，多階層領域分割法，多倍長精度計算，曲面細分割，データ圧縮の研究開発，及び個別化医療支援の応用数値実験を研究対象としている．平成30年度の実績は以下の通りである．
(a) 複素対称線形方程式向けCOMINRES-QLP法の開発を行い，数値実験によってその有効性を示した．特に，MINRES系反復法では反復計算が進むにつれて残差が停滞してしまう問題があったが，悪条件問題では既存手法より効率的に計算できることを示した．
(b) 多階層DDMの開発として，2階層BDD前処理の開発を継続して実施した．特に，熱伝導ならびに線形弾性体の有限要素解析で得られる線形方程式を用いた数値実験を行い，開発システムの検証を完了した．
(c) DD精度演算が複素対称線形方程式の反復法に与える影響評価を継続実施した．特に，IC(p)前処理とDD演算の組合せを評価し，IC(1)前処理と組み合わせると加速パラメータ依存性が弱まり，パラメーターフリーの前処理法として利用できる可能性を示した．また，混合精度COMINRES-QLP法を開発し，特に悪条件問題においては倍精度演算に比べて反復回数と計算時間の両方を削減することに成功した．
(d) NICT人体モデルに対して4面体要素細分割で160億自由度規模モデルを用いて，電磁場解析結果の可視化実験を行った．断面，等値面，特定臓器の抽出などにより，汎用PCでも可視化できるまで情報の圧縮が可能であることを示した．
また，他の研究課題（17H03256）と連携して，計算電磁気学の専門家を集めた研究シンポジウム「第2回大規模電磁界数値解析手法に関する研究シンポジウム」（2019/03/09-10，宮崎）を主催し，本研究計画に関する情報収集および支援体制の強化に繋げることができた．

5. Comprehensive development of fast numerical methods for solving large linear systems with matrix functions

Grant number：26286088  2014.4 - 2017.3

Japan Society for the Promotion of Science  KAKENHI Grant-in-Aid for Scientific Research (B)

SOGABE Tomohiro

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Grant amount：\6370000 （ Direct Cost: \4900000 、 Indirect Cost：\1470000 ）

The purpose of the research project is to develop fast numerical algorithms for solving linear systems with matrix functions, and the research project mainly yielded the following results: (1) an efficient Krylov subspace method for solving linear systems with some matrix polynomials; (2) a method for boosting the speed of convergence of Newton's iterations to compute the matrix principal square root; (3) a cost-efficient variant of Incremental Newton method for the matrix principal pth root; (4) tensor decomposition algorithms for some special matrices. The results (2),(3) may lead to efficient Krylov solvers for the corresponding linear systems. The result (4) yields a novel direction for the case where the coefficient matrix has a tensor structure, which was not expected before the research project.

6. Fast solution for ultra-large scale systems as a basis of computational materials science

Grant number：22104004  2010.4 - 2015.3

Japan Society for the Promotion of Science  Grant-in-Aid for Scientific Research on Innovative Areas (Research in a proposed research area)

ZHANG Shao-Liang, ABE Kuniyoshi, YAMAMOTO Yusaku, SOGABE Tomohiro, IMAHORI Shinji, MIYATA Takahumi, SUGIHARA Masaaki

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Finding novel complex correlation phenomena and clarifying the non-equilibrium dynamics are of prime importance in the field of materials design. This research group tackles the challenging problems from the viewpoints of numerical linear algebra, optimization and high-performance computing. The major purpose of the researches is to develop robust and efficient numerical algorithms for solving large linear systems and eigenvalue problems in order to shed light on a breakthrough toward the challenging problems. Some of numrical algrotihms for solving linear systems we developed are as follows: the shifted COCR method for solving shifted complex symmetric linear systems; the look back GMRES(m) method for nonsymmetric linear systems; a variand of the IDR(s) method for nonsymmetric linear systems. Some of numrical algrotihms for solving eigenvalue problems we developed are as follows: the Arnoldi(M,W,G) method; an extension of the SS method;

7. Study on fast and robust iterative methods for solving large and sparse shifted linear systems arising from computational science

Grant number：21760058  2009 - 2012

Japan Society for the Promotion of Science  KAKENHI Grant-in-Aid for Young Scientists (B)

SOGABE Tomohiro

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Authorship：Principal investigator

Grant amount：\4810000 （ Direct Cost: \3700000 、 Indirect Cost：\1110000 ）

We developed some efficient iterative methods for solving large and sparse shifted linear systems that arise from computational science. As a result, the shifted COCR method was proposed for solving complex symmetric case, and a variant of the shifted GMRES method was proposed for (complex) nonsymmetric case. Remarkably, the shifted COCR method was about 26 times faster than the COCR method for a problem arising from large scale electronic structure calculation. As related work, the following results are obtained: (1) An improvement of the COCR method for solving complex symmetric linear systems; (2) Improvements of the IDR method; (3) An improvement of the GMRES(m) method for solving nonsymmetric linear systems; (4) a fast solver for linear systems with a special matrix; (5) Efficient iterative method for generalized shifted linear systems with complex symmetric matrices.

8. A study on fast solvers for large linear systems in the field of scientific computations

Grant number：19560065  2007 - 2010

Japan Society for the Promotion of Science  KAKENHI Grant-in-Aid for Scientific Research (C)

ZHANG Shao-liang

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There are many researches about Krylov subspace methods for large linear systems and some research results were utilized for solving practical problems. However, these results can be used for some special cases. We studied and developed fast solvers for large linear systems in the field of scientific computations.

9. Study on fast and stable iterative methods for solving large and sparse linear systems arising from computational science

Grant number：18760063  2006 - 2008

Japan Society for the Promotion of Science  KAKENHI Grant-in-Aid for Young Scientists (B)

SOGABE Tomohiro

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Authorship：Principal investigator

Grant amount：\3870000 （ Direct Cost: \3600000 、 Indirect Cost：\270000 ）