Faculty Profiles - SOGABE, Tomohiro

Papers - SOGABE, Tomohiro

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A fast numerical algorithm for solving nearly penta-diagonal linear systems Reviewed

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

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

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

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.

DOI： 10.1080/00207160.2012.665903

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Fast block diagonalization of k-tridiagonal matrices Reviewed

Tomohiro Sogabe, Moawwad El-Mikkawy

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

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

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.

DOI： 10.1016/j.amc.2011.08.014

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A variant of the IDR(s) method with the quasi-minimal residual strategy Reviewed

Lei Du, Tomohiro Sogabe, Shao-Liang Zhang

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

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

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.

DOI： 10.1016/j.cam.2011.07.027

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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

A block IDR(s) method for nonsymmetric linear systems with multiple right-hand sides Reviewed

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

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

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 > 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.

DOI： 10.1016/j.cam.2011.02.035

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Efficient and accurate linear algebraic methods for large-scale electronic structure calculations with nonorthogonal atomic orbitals Reviewed

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

PHYSICAL REVIEW B Vol. 83 ( 16 ) 2011.4

Language：English Publishing type：Research paper (scientific journal) Publisher：AMER PHYSICAL SOC

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.

DOI： 10.1103/PhysRevB.83.165103

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Quasi-minimal residual smoothing technique for the IDR($s$) method Reviewed

Du Lei, Sogabe Tomohiro, Zhang Shao-Liang

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

Four-point correlation function of a passive scalar field in rapidly fluctuating turbulence: Numerical analysis of an exact closure equation Reviewed

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

PHYSICAL REVIEW E Vol. 82 ( 3 ) 2010.9

Language：English Publishing type：Research paper (scientific journal) Publisher：AMER PHYSICAL SOC

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 <[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 <[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 << 1 with R and r in the similarity range, where chi is a constant depending on the angle between R and r.

DOI： 10.1103/PhysRevE.82.036316

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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

Novel algorithm of large-scale simultaneous linear equations Reviewed

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

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

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

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.

DOI： 10.1088/0953-8984/22/7/074206

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PubMed

A new family of k-Fibonacci numbers Reviewed

Moawwad El-Mikkawy, Tomohiro Sogabe

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

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

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.

DOI： 10.1016/j.amc.2009.12.069

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Notes on particular symmetric polynomials with applications Reviewed

Moawwad El-Mikkawy, Tomohiro Sogabe

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

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

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.

DOI： 10.1016/j.amc.2009.10.019

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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(Theory)

Miyata Takafumi, Sogabe Tomohiro, Zhang Shao-Liang

Transactions of the Japan Society for Industrial and Applied Mathematics Vol. 20 ( 2 ) page： 115 - 129 2010

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

The Jacobi-Davidson (JD) iteration method was recently proposed to compute a few eigenpairs of a large-scale and sparse matrix. In the JD method, the correction equation is solved approximately to generate the basis of a subspace for the iterative computation of eigenpairs. It is empirically known that the speed of the convergence of the JD method depends on the method for solving the correction equation. In this paper, we show a shift invariance property of the Krylov subspace on a projected space and propose a new method based on the property for solving the correction equation.

DOI： 10.11540/jsiamt.20.2_115

CiNii Research

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

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

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

An extension of the conjugate residual method to nonsymmetric linear systems Reviewed

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

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

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

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.

DOI： 10.1016/j.cam.2008.05.018

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An implicit wavelet sparse approximate inverse preconditioner using block finger pattern Reviewed

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

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

Language：English Publishing type：Research paper (scientific journal) Publisher：WILEY-BLACKWELL

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

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Shifted Conjugate-Orthogonal-Conjugate-Gradient Method and Its Application to Double Orbital Extended Hubbard Model Reviewed

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

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

Tomohiro Sogabe

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

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

Recently, an efficient computational algorithm for solving periodic pentadiagonal linear systems has been proposed by Karawia [ A. A. Karawia, A computational algorithm for solving periodic pentadiagonal linear systems, Appl. Math. Comput. 174 ( 2006) 613-618]. The algorithm is based on the LU factorization of the periodic pentadiagonal matrix. In this paper, new algorithms are presented for solving periodic pentadiagonal linear systems based on the use of any pentadiagonal linear solver. In addition, an efficient way of evaluating the determinant of a periodic pentadiagonal matrix is discussed. The corresponding results in this paper can be readily obtained for solving periodic tridiagonal linear systems. (C) 2008 Elsevier Inc. All rights reserved.

DOI： 10.1016/j.amc.2008.03.030

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Papers - SOGABE, Tomohiro