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

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

DOI： 10.1007/s13160-020-00435-x

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Scopus

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

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

We develop K omega, an open-source linear algebra library for the shifted Krylov subspace methods. The methods solve a set of shifted linear equations (z(k)l - H)x((k)) = b(k = 0, 1, 2, ...) for a given matrix H and a vector b, simultaneously. The leading order of the operational cost is the same as that for a single equation. The shift invariance of the Krylov subspace is the mathematical foundation of the shifted Krylov subspace methods. Applications in materials science are presented to demonstrate the advantages of the algorithm over the standard Krylov subspace methods such as the Lanczos method. We introduce benchmark calculations of (i) an excited (optical) spectrum and (ii) intermediate eigenvalues by the contour integral on the complex plane. In combination with the quantum lattice solver H Phi, K omega can realize parallel computation of excitation spectra and intermediate eigenvalues for various quantum lattice models.Program summaryProgram Title: K omega [kei-oumig@]CPC Library link to program files: http://dx.doi.org/10.17632/mt928nz5r3.1Developer's repository link: https://github.com/issp-center-dev/KomegaLicensing provisions: GNU Lesser General Public License Version 3.Programming language: Fortran 90External routines/libraries: BLAS library, LAPACK library (Used in the sample program), MPI library (Optional).Nature of problem: Efficient algorithms, called shifted Krylov subspace algorithms, designed to solve the shifted linear equations.Solution method: Shifted conjugate gradient method, Shifted conjugate orthogonal conjugate gradient method, shifted bi-conjugate gradient method.Additional comments: The present paper is accompanied by a frozen copy of K omega release 2.0.0 that is made publicly available on GitHub (repository https://github.com/issp-center-dev/Komega commit hash fd5455328b102ec4fa13432496e41c404a0f5a9d). (C) 2020 The Authors. Published by Elsevier B.V.

DOI： 10.1016/j.cpc.2020.107536

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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.

DOI： 10.1553/0X003CC757

Scopus

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

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

DOI： 10.1080/10556788.2020.1734004

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

We consider the computation of the matrix logarithm by using numerical quadrature. The efficiency of numerical quadrature depends on the integrand and the choice of quadrature formula. The Gauss-Legendre quadrature has been conventionally employed; however, the convergence could be slow for ill-conditioned matrices. This effect may stem from the rapid change of the integrand values. To avoid such situations, we focus on the double exponential formula, which has been developed to address integrands with endpoint singularity. In order to utilize the double exponential formula, we must determine a suitable finite integration interval, which provides the required accuracy and efficiency. In this paper, we present a method for selecting a suitable finite interval based on an error analysis as well as two algorithms, and one of these algorithms is an adaptive quadrature algorithm. (C) 2019 Elsevier B.V. All rights reserved.

DOI： 10.1016/j.cam.2019.112396

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

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

DOI： 10.1007/s10910-020-01138-x

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

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.

DOI： 10.1007/s11075-019-00748-0

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DOI： 10.4208/csiam-am.2020-0008

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

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.

DOI： 10.1016/j.aml.2019.04.007

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

DOI： 10.1515/spma-2019-0002

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

© 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.

DOI： 10.1515/spma-2019-0009

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

©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.

DOI： 10.3934/jcd.2019018

Scopus

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>

DOI： 10.11540/jsiamt.29.1_121

J-GLOBAL

Language：English   Publishing type：Research paper (scientific journal)   Publisher：JAPAN SOC INDUSTRIAL & APPLIED MATHEMATICS-JSIAM  

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.

DOI： 10.14495/jsiaml.11.77

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DOI： 10.1016/j.cam.2018.04.013

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DOI： 10.1016/j.jcp.2018.06.002

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DOI： 10.1007/s13160-018-0308-x

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DOI： 10.1016/j.cam.2017.05.044

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

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.

DOI： 10.1016/j.aml.2017.06.016

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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^α 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>

DOI： 10.11540/jsiamt.28.3_142

J-GLOBAL

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

DOI： 10.4208/eajam.181016.300517c

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

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.

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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>

DOI： 10.11540/jsiamt.27.3_239

J-GLOBAL

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

DOI： 10.3770/j.issn:2095-2651.2017.01.010

arXiv

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

DOI： 10.3770/j.issn:2095-2651.2017.01.009

arXiv

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

DOI： 10.14495/jsiaml.8.17

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

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.

DOI： 10.1016/j.cam.2015.05.026

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

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.

DOI： 10.1016/j.cam.2014.07.004

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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

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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

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

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

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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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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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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Publishing type：Research paper (scientific journal)   Publisher：一般社団法人 日本応用数理学会  

DOI： 10.11540/jsiamt.24.1_A1

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

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.

DOI： 10.12732/ijpam.v96i2.10

Scopus

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

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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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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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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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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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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.

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

J-GLOBAL

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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Scopus

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

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Scopus

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

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

J-GLOBAL

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

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Scopus

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

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

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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Scopus

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

Web of Science

Scopus

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

Web of Science

Scopus

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

J-GLOBAL

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 &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.

DOI： 10.1016/j.cam.2011.02.035

Web of Science

Scopus

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

Web of Science

Scopus

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

DOI： 10.14495/jsiaml.3.13

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 &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.

DOI： 10.1103/PhysRevE.82.036316

Web of Science

Scopus

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

J-GLOBAL

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

Web of Science

Scopus

PubMed

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

Web of Science

Scopus

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

Web of Science

Scopus

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

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

J-GLOBAL

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

J-GLOBAL

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

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.

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

J-GLOBAL

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

Web of Science

Scopus

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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Scopus

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

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.

DOI： 10.11540/jsiamt.19.4_537

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

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.

DOI： 10.11540/jsiamt.19.4_551

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

Shifted linear systems arise in many important applications such as lattice quantum chromodynamics, large-scale electronic structure theory, and quadratic optimization problems. Recent strong need for solving the extremely large shifted linear systems enhance the importance of designing efficient solvers. As a candidate to satisfy the need, iterative methods using Krylov subspaces and the shift-invariance property have been attracting much interest. The primary aim of this paper is to survey the successful iterative methods and to classify them in terms of Krylov subspace methods.

DOI： 10.11540/bjsiam.19.3_163

CiNii Research

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

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

Web of Science

Scopus

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

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.

DOI： 10.1016/j.amc.2007.12.047

Web of Science

Scopus

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

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.

DOI： 10.1016/j.amc.2007.08.029

Web of Science

Scopus

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

J-GLOBAL

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

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.

DOI： 10.1016/j.amc.2007.07.015

Web of Science

Scopus

Authorship：Lead author   Language：English   Publishing type：Research paper (scientific journal)  

Authorship：Lead author   Language：English   Publishing type：Research paper (scientific journal)  

Authorship：Lead author   Language：English   Publishing type：Research paper (scientific journal)  

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

Language：English   Publishing type：Research paper (scientific journal)   Publisher：KENT STATE UNIVERSITY  

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.

Web of Science

Scopus

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

It is commonly known that Generalized Conjugate Residual(GCR) method with the variable preconditioning using Successive Over-Relaxation(SOR) method is an efficient iterative method for solving large and sparse linear systems. In the present paper, we consider using of Accelerated Over-Relaxation (AOR) method instead of SOR method as a variable preconditioning. Since AOR method is a natural extension of SOR method, our approach includes the previous one. It is experimentally shown by reducing the computational cost of AOR method that the present method can be very promising.

DOI： 10.11540/jsiamt.18.1_155

CiNii Research

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

J-GLOBAL

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

J-GLOBAL

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

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.

Authorship：Lead author   Language：English   Publishing type：Research paper (scientific journal)  

Authorship：Lead author   Language：English   Publishing type：Research paper (scientific journal)  

Language：English   Publishing type：Research paper (international conference proceedings)   Publisher：SPRINGER-VERLAG BERLIN  

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.

DOI： 10.1007/978-3-540-46375-7_24

Web of Science

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

Recently, the implicit wavelet sparse approximate inverse (IW-SPAI) preconditioner has been proposed for large and sparse nonsymmetric linear systems. The preconditioning matrix has a certain sparse structure called finger pattern and it can be obtained by the solutions of some least squares problems. By reusing the result of QR factorization in some least squares problems, in order to improve the performance of the IW-SPAI, in this paper, an IW-SPAI using Blocked finger pattern (BIW-SPAI) is proposed as a variant of the IW-SPAI. Numerical experiments indicate that the BIW-SPAI is often more efficient than the IW-SPAI.

DOI： 10.11540/jsiamt.17.4_523

CiNii Research

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

The Bi-CG method, a well-known Krylov subspace method for solving non-symmetric linear systems, often shows an irregular convergence behavior in the residual norm. To avoid such a problem, Freund and Nachtigal proposed the QMR method. On the other hand, recently Sogabe et al. proposed the Bi-CR method, and showed experimentally that the Bi-CR method has good convergence. The purpose of this paper is to apply the QMR approach to the Bi-CR method to derive a new method, and to report the results of numerical experiments.

DOI： 10.11540/jsiamt.17.3_301

CiNii Research

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

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

J-GLOBAL

Authorship：Lead author   Language：Japanese   Publishing type：Research paper (scientific journal)  

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．

Authorship：Lead author   Language：Japanese   Publishing type：Research paper (scientific journal)  

Language：English  

DOI： 10.1016/j.dam.2009.04.018

Web of Science

Event date： 2004.8

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

Event date： 2003.10

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