Faculty Profiles - SOGABE, Tomohiro

Papers - SOGABE, Tomohiro

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On decomposition of k-tridiagonal ℓ-Toeplitz matrices and its applications Reviewed

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

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

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

© 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.

DOI： 10.1515/spma-2015-0019

Web of Science

Scopus

On tensor product decomposition of k-tridiagonal Toeplitz matrices Reviewed

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

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

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

© 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.

DOI： 10.12732/ijpam.v103i3.14

Scopus

AN EXTENSION OF TWO CONJUGATE DIRECTION METHODS TO MARKOV CHAIN PROBLEMS Reviewed

Chun Wen, Ting-Zhu Huang, Tomohiro Sogabe

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

Language：English Publishing type：Research paper (scientific journal) Publisher：SLOVAK ACAD SCIENCES INST INFORMATICS

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.

Web of Science

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

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

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

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.

DOI： 10.1016/j.amc.2014.10.040

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An algorithm for solving nonsymmetric penta-diagonal Toeplitz linear systems Reviewed

Lei Du, Tomohiro Sogabe, Shao-Liang Zhang

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

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

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.

DOI： 10.1016/j.amc.2014.06.104

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Scopus

BiCR-type methods for families of shifted linear systems Reviewed

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

Language：English Publishing type：Research paper (scientific journal) Publisher：PERGAMON-ELSEVIER SCIENCE LTD

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.

DOI： 10.1016/j.camwa.2014.07.029

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Scopus

A note on symmetric k-Tridiagonal matrix family and the Fibonacci numbers Reviewed

Fatih Yilmaz, Tomohiro Sogabe

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

A novel algorithm for solving quasi penta-diagonal linear systems Reviewed

Ji-Teng Jia, Tomohiro Sogabe

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

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

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.

DOI： 10.1007/s10910-012-0122-7

Web of Science

Scopus

A novel algorithm and its parallelization for solving nearly penta-diagonal linear systems Reviewed

Ji-Teng Jia, Tomohiro Sogabe

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

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

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.

DOI： 10.1080/00207160.2012.725845

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Scopus

On particular solution of ordinary differential equations with constant coefficients Reviewed

Jiteng Jia, Tomohiro Sogabe

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

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

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.

DOI： 10.1016/j.amc.2012.12.080

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Scopus

Inversion of k-tridiagonal matrices with Toeplitz structure Reviewed

Jiteng Jia, Tomohiro Sogabe, Moawwad El-Mikkawy

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

Language：English Publishing type：Research paper (scientific journal) Publisher：PERGAMON-ELSEVIER SCIENCE LTD

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.

DOI： 10.1016/j.camwa.2012.11.001

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An interior eigenvalue problem from electronic structure calculations Reviewed

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

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

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.

DOI： 10.1007/s13160-013-0118-0

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Scopus

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

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

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.

Solution of generalized shifted linear systems with complex symmetric matrices Reviewed

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

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

Language：English Publishing type：Research paper (scientific journal) Publisher：ACADEMIC PRESS INC ELSEVIER SCIENCE

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.

DOI： 10.1016/j.jcp.2012.04.046

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An order-N electronic structure theory with generalized eigenvalue equations and its application to a ten-million-atom system Reviewed

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

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

Language：English Publishing type：Research paper (scientific journal) Publisher：IOP PUBLISHING LTD

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.

DOI： 10.1088/0953-8984/24/16/165502

Web of Science

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A new algorithm for solving nearly penta-diagonal Toeplitz linear systems Reviewed

Jiteng Jia, Qiongxiang Kong, Tomohiro Sogabe

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

Language：English Publishing type：Research paper (scientific journal) Publisher：PERGAMON-ELSEVIER SCIENCE LTD

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.

DOI： 10.1016/j.camwa.2011.12.044

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Scopus

An Efficient Variant of the GMRES(m) Method Based on the Error Equations Reviewed

Akira Imakura, Tomohiro Sogabe, Shao-Liang Zhang

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

Language：English Publishing type：Research paper (scientific journal) Publisher：GLOBAL SCIENCE PRESS

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.

DOI： 10.4208/eajam.280611.030911a

Web of Science

Scopus

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

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

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

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.

DOI： 10.11540/jsiamt.22.3_117

A Look-Back GMRES(m) Method for Solving Nonsymmetric Linear Systems Reviewed

Imakura Akira, Sogabe Tomohiro, Zhang Shao-Liang

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

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

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.

DOI： 10.11540/jsiamt.22.1_1

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Papers - SOGABE, Tomohiro