Numerical Algorithms, ISSN 1017-1398, 11/2019, Volume 82, Issue 3, pp. 843 - 867

For solving a class of augmented linear systems, we propose a new efficient iteration method, which is called preconditioned Richardson iteration (PR). Under...

Augmented linear system | Spectral radius | Numeric Computing | Positive definite | Theory of Computation | Convergence analysis | 65F50 | Algorithms | Algebra | SOR-like iteration | 65F10 | Numerical Analysis | Computer Science | MATHEMATICS, APPLIED | POSITIVE-DEFINITE | INEXACT | CONJUGATE-GRADIENT METHODS | UZAWA METHOD | AX PLUS XB | HERMITIAN SPLITTING METHODS | SYMMETRIC SOR METHOD | SUCCESSIVE OVERRELAXATION METHOD | POINT | OPTIMAL PARAMETERS

Augmented linear system | Spectral radius | Numeric Computing | Positive definite | Theory of Computation | Convergence analysis | 65F50 | Algorithms | Algebra | SOR-like iteration | 65F10 | Numerical Analysis | Computer Science | MATHEMATICS, APPLIED | POSITIVE-DEFINITE | INEXACT | CONJUGATE-GRADIENT METHODS | UZAWA METHOD | AX PLUS XB | HERMITIAN SPLITTING METHODS | SYMMETRIC SOR METHOD | SUCCESSIVE OVERRELAXATION METHOD | POINT | OPTIMAL PARAMETERS

Journal Article

Numerical Algorithms, ISSN 1017-1398, 2/2011, Volume 56, Issue 2, pp. 297 - 317

We propose a preconditioned variant of the modified HSS (MHSS) iteration method for solving a class of complex symmetric systems of linear equations. Under...

Convergence theory | Spectral properties | Numeric Computing | Theory of Computation | CR: G1.3 | 65F50 | Algebra | Algorithms | 65F10 | Computer Science | Mathematics, general | Complex symmetric linear system | MHSS iteration | Preconditioning | HERMITIAN SPLITTING METHODS | MATHEMATICS, APPLIED | Computer science | Analysis | Methods | Linear systems | Numerical analysis | Mathematical models | Spectra | Subspaces | Iterative methods | Convergence | Symmetry

Convergence theory | Spectral properties | Numeric Computing | Theory of Computation | CR: G1.3 | 65F50 | Algebra | Algorithms | 65F10 | Computer Science | Mathematics, general | Complex symmetric linear system | MHSS iteration | Preconditioning | HERMITIAN SPLITTING METHODS | MATHEMATICS, APPLIED | Computer science | Analysis | Methods | Linear systems | Numerical analysis | Mathematical models | Spectra | Subspaces | Iterative methods | Convergence | Symmetry

Journal Article

Applied Mathematics and Computation, ISSN 0096-3003, 10/2019, Volume 358, pp. 455 - 467

We propose a modified block splitting preconditioner for a class of complex nonsymmetric indefinite linear systems. By adopting two iteration parameters and a...

Complex nonsymmetric linear system | Preconditioning | GMRES | Relaxing parameters | HERMITIAN SPLITTING METHODS | MATHEMATICS, APPLIED | POSITIVE-DEFINITE | PSS PRECONDITIONERS | Linear systems | Analysis

Complex nonsymmetric linear system | Preconditioning | GMRES | Relaxing parameters | HERMITIAN SPLITTING METHODS | MATHEMATICS, APPLIED | POSITIVE-DEFINITE | PSS PRECONDITIONERS | Linear systems | Analysis

Journal Article

Computers and Mathematics with Applications, ISSN 0898-1221, 04/2018, Volume 75, Issue 8, pp. 2710 - 2722

In this paper, based on a convergence splitting of the matrix A, we present an inner–outer iteration method for solving the linear system Ax=b. We analyze the...

Inner-outer iterations | Accelerate | Preconditioned | Convergence | JACOBI | HERMITIAN SPLITTING METHODS | MATHEMATICS, APPLIED | SYLVESTER MATRIX EQUATIONS | ALGORITHM | INNER-OUTER ITERATION | COMPUTING PAGERANK

Inner-outer iterations | Accelerate | Preconditioned | Convergence | JACOBI | HERMITIAN SPLITTING METHODS | MATHEMATICS, APPLIED | SYLVESTER MATRIX EQUATIONS | ALGORITHM | INNER-OUTER ITERATION | COMPUTING PAGERANK

Journal Article

Journal of Engineering Mathematics, ISSN 0022-0833, 8/2015, Volume 93, Issue 1, pp. 41 - 60

A complex system of linear equations arises in many important applications. We further explore algebraic and convergence properties and present analytical and...

Analysis | Convergence theory | Spectral properties | Mechanics | Complex symmetric linear system | PMHSS iteration | Real reformulation | Applications of Mathematics | Mathematical Modeling and Industrial Mathematics | Preconditioning | Physics | SCATTERING PROBLEMS | CONJUGATE-GRADIENT METHODS | HERMITIAN SPLITTING METHODS | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | ENGINEERING, MULTIDISCIPLINARY | MATRICES | MINIMAL RESIDUAL ALGORITHM | Linear systems | Methods | Algebra | Equivalence | Mathematical analysis | Mathematical models | Iterative methods | Complex systems | Convergence

Analysis | Convergence theory | Spectral properties | Mechanics | Complex symmetric linear system | PMHSS iteration | Real reformulation | Applications of Mathematics | Mathematical Modeling and Industrial Mathematics | Preconditioning | Physics | SCATTERING PROBLEMS | CONJUGATE-GRADIENT METHODS | HERMITIAN SPLITTING METHODS | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | ENGINEERING, MULTIDISCIPLINARY | MATRICES | MINIMAL RESIDUAL ALGORITHM | Linear systems | Methods | Algebra | Equivalence | Mathematical analysis | Mathematical models | Iterative methods | Complex systems | Convergence

Journal Article

BIT Numerical Mathematics, ISSN 0006-3835, 06/2017, Volume 57, Issue 2, pp. 287 - 311

We propose a class of regularized Hermitian and skew-Hermitian splitting methods for the solution of large, sparse linear systems in saddle-point form. These...

Hermitian and skew-Hermitian splitting | Iterative methods | Preconditioning | Saddle-point linear system | Inexact implementation | Convergence | COMPUTER SCIENCE, SOFTWARE ENGINEERING | HERMITIAN SPLITTING METHODS | MATHEMATICS, APPLIED | PRECONDITIONERS | MATRICES | Linear systems | Computer science | Analysis | Methods

Hermitian and skew-Hermitian splitting | Iterative methods | Preconditioning | Saddle-point linear system | Inexact implementation | Convergence | COMPUTER SCIENCE, SOFTWARE ENGINEERING | HERMITIAN SPLITTING METHODS | MATHEMATICS, APPLIED | PRECONDITIONERS | MATRICES | Linear systems | Computer science | Analysis | Methods

Journal Article

7.
Full Text
Motivations and realizations of Krylov subspace methods for large sparse linear systems

Journal of Computational and Applied Mathematics, ISSN 0377-0427, 08/2015, Volume 283, pp. 71 - 78

We briefly introduce typical and important direct and iterative methods for solving systems of linear equations, concretely describe their fundamental...

Direct method | Iterative method | Linear system | Krylov subspace | Preconditioning | MATHEMATICS, APPLIED | RELAXATION METHODS | GMRES CONVERGENCE | HERMITIAN SPLITTING METHODS | MULTILEVEL PRECONDITIONING METHODS | BOUNDS | MATRICES | ITERATIVE METHODS | Linear systems | State of the art | Algebra | Algorithms | Searching | Mathematical models | Linear equations | Subspace methods

Direct method | Iterative method | Linear system | Krylov subspace | Preconditioning | MATHEMATICS, APPLIED | RELAXATION METHODS | GMRES CONVERGENCE | HERMITIAN SPLITTING METHODS | MULTILEVEL PRECONDITIONING METHODS | BOUNDS | MATRICES | ITERATIVE METHODS | Linear systems | State of the art | Algebra | Algorithms | Searching | Mathematical models | Linear equations | Subspace methods

Journal Article

Computers and Mathematics with Applications, ISSN 0898-1221, 11/2016, Volume 72, Issue 9, pp. 2462 - 2472

In this paper, based on accelerated overrelaxation (AOR) method and Uzawa method, we present AOR–Uzawa iterative method for solving a broad class of complex...

Complex symmetric linear systems | AOR–Uzawa iterative method | Preconditioning | Convergence | HSS METHOD | HERMITIAN SPLITTING METHODS | MATHEMATICS, APPLIED | SEMI-CONVERGENCE ANALYSIS | VARIANTS | ALGORITHMS | AOR-Uzawa iterative method | Linear systems | Equivalence | Computer simulation | Mathematical analysis | Mathematical models | Iterative methods | Symmetry

Complex symmetric linear systems | AOR–Uzawa iterative method | Preconditioning | Convergence | HSS METHOD | HERMITIAN SPLITTING METHODS | MATHEMATICS, APPLIED | SEMI-CONVERGENCE ANALYSIS | VARIANTS | ALGORITHMS | AOR-Uzawa iterative method | Linear systems | Equivalence | Computer simulation | Mathematical analysis | Mathematical models | Iterative methods | Symmetry

Journal Article

Computing, ISSN 0010-485X, 5/2010, Volume 87, Issue 3, pp. 93 - 111

In this paper, we introduce and analyze a modification of the Hermitian and skew-Hermitian splitting iteration method for solving a broad class of complex...

Computational Mathematics and Numerical Analysis | Hermitian and skew-Hermitian splitting | Complex symmetric matrix | Krylov subspace method | Iteration method | CR: G1.3 | Convergence analysis | 65F50 | 65F10 | Computer Science | Computer Science, general | Preconditioning | 65N22 | HERMITIAN SPLITTING METHODS | GMRES | COMPUTER SCIENCE, THEORY & METHODS | Computer science | Linear systems | Scientists | Analysis | Methods | Studies | Linear programming | Mathematical models

Computational Mathematics and Numerical Analysis | Hermitian and skew-Hermitian splitting | Complex symmetric matrix | Krylov subspace method | Iteration method | CR: G1.3 | Convergence analysis | 65F50 | 65F10 | Computer Science | Computer Science, general | Preconditioning | 65N22 | HERMITIAN SPLITTING METHODS | GMRES | COMPUTER SCIENCE, THEORY & METHODS | Computer science | Linear systems | Scientists | Analysis | Methods | Studies | Linear programming | Mathematical models

Journal Article

Computers and Mathematics with Applications, ISSN 0898-1221, 01/2017, Volume 73, Issue 1, pp. 87 - 95

Based on the new HSS (NHSS) iteration method proposed by Pour and Goughery (2015) and the efficient PSHSS iteration method by Zeng and Ma (2016), we introduce...

Positive definite | Complex symmetric linear system | Spectral radius | HSS iteration | Convergence analysis | HERMITIAN SPLITTING METHODS | MATHEMATICS, APPLIED | Linear systems | Analysis | Methods | High speed tool steels | Parameters | Upper bounds | Mathematical models | Iterative methods | Optimization | Symmetry

Positive definite | Complex symmetric linear system | Spectral radius | HSS iteration | Convergence analysis | HERMITIAN SPLITTING METHODS | MATHEMATICS, APPLIED | Linear systems | Analysis | Methods | High speed tool steels | Parameters | Upper bounds | Mathematical models | Iterative methods | Optimization | Symmetry

Journal Article

Numerical Algorithms, ISSN 1017-1398, 11/2019, Volume 82, Issue 3, pp. 1097 - 1115

For solving an augmented linear system, Njeru and Guo presented an accelerated SOR-like (ASOR) method in (P. N. Njeru and X.-P. Guo. Accelerated SOR-like...

65F50 | Modified accelerated SOR-like method | Augmented linear systems | Algorithms | Algebra | 65F10 | Numerical Analysis | Computer Science | Numeric Computing | Theory of Computation | 65F08 | HERMITIAN SPLITTING METHODS | MATHEMATICS, APPLIED | PRECONDITIONERS | SYMMETRIC SOR METHOD | AOR METHOD | ITERATIVE METHODS

65F50 | Modified accelerated SOR-like method | Augmented linear systems | Algorithms | Algebra | 65F10 | Numerical Analysis | Computer Science | Numeric Computing | Theory of Computation | 65F08 | HERMITIAN SPLITTING METHODS | MATHEMATICS, APPLIED | PRECONDITIONERS | SYMMETRIC SOR METHOD | AOR METHOD | ITERATIVE METHODS

Journal Article

IMA Journal of Numerical Analysis, ISSN 0272-4979, 01/2013, Volume 33, Issue 1, pp. 343 - 369

We construct a preconditioned modified Hermitian and skew-Hermitian splitting (PMHSS) iteration scheme for solving and preconditioning a class of block...

KKT systems | PMHSS iteration | spectral properties | block two-by-two matrices | PDE-constrained optimization | preconditioning | HERMITIAN SPLITTING METHODS | MATHEMATICS, APPLIED | Linear systems | Mathematical analysis | Blocking | Control systems | Spectra | Iterative methods | Galerkin methods | Convergence

KKT systems | PMHSS iteration | spectral properties | block two-by-two matrices | PDE-constrained optimization | preconditioning | HERMITIAN SPLITTING METHODS | MATHEMATICS, APPLIED | Linear systems | Mathematical analysis | Blocking | Control systems | Spectra | Iterative methods | Galerkin methods | Convergence

Journal Article

Journal of Computational and Applied Mathematics, ISSN 0377-0427, 07/2016, Volume 300, pp. 18 - 29

Block and global Krylov subspace methods have been proposed as methods adapted to the situation where one iteratively solves systems with the same matrix and...

Block methods | Krylov subspace | Multiple right-hand sides | Non-Hermitian matrices | Sparse linear systems | MATHEMATICS, APPLIED | NONSYMMETRIC SYSTEMS | ALGORITHM | QMR | BLOCK GMRES | MULTIPLE | Economics | Linear systems | Mathematical analysis | Blocking | Mathematical models | Vectors (mathematics) | Subspace methods | Arithmetic

Block methods | Krylov subspace | Multiple right-hand sides | Non-Hermitian matrices | Sparse linear systems | MATHEMATICS, APPLIED | NONSYMMETRIC SYSTEMS | ALGORITHM | QMR | BLOCK GMRES | MULTIPLE | Economics | Linear systems | Mathematical analysis | Blocking | Mathematical models | Vectors (mathematics) | Subspace methods | Arithmetic

Journal Article

Computers and Mathematics with Applications, ISSN 0898-1221, 10/2018, Volume 76, Issue 8, pp. 1906 - 1922

We propose a relaxed generalized-PSS preconditioner for generalized saddle-point linear systems with non-Hermitian positive definite leading block from steady...

Krylov subspace method | Generalized saddle-point problem | PSS | Navier–Stokes equations and Oseen problem | Preconditioner | Kiylov subspace method | HERMITIAN SPLITTING METHODS | MATHEMATICS, APPLIED | POSITIVE-DEFINITE | Navier-Stokes equations and Oseen problem | HSS PRECONDITIONER

Krylov subspace method | Generalized saddle-point problem | PSS | Navier–Stokes equations and Oseen problem | Preconditioner | Kiylov subspace method | HERMITIAN SPLITTING METHODS | MATHEMATICS, APPLIED | POSITIVE-DEFINITE | Navier-Stokes equations and Oseen problem | HSS PRECONDITIONER

Journal Article

Journal of Computational and Applied Mathematics, ISSN 0377-0427, 12/2017, Volume 325, pp. 188 - 197

In this paper, a new iteration method is proposed for solving the complex symmetric linear systems. In theory, we show that the convergence factor or the upper...

Complex symmetric linear system | PMHSS method | Numerical experiment | Iterative methods | Convergence theory | HERMITIAN SPLITTING METHODS | MATHEMATICS, APPLIED | SADDLE-POINT PROBLEMS | MATRICES | EQUATIONS | Linear systems | Analysis | Methods

Complex symmetric linear system | PMHSS method | Numerical experiment | Iterative methods | Convergence theory | HERMITIAN SPLITTING METHODS | MATHEMATICS, APPLIED | SADDLE-POINT PROBLEMS | MATRICES | EQUATIONS | Linear systems | Analysis | Methods

Journal Article

Computers and Mathematics with Applications, ISSN 0898-1221, 11/2016, Volume 72, Issue 9, pp. 2473 - 2485

In this paper, for solving a class of complex linear systems from the Helmholtz equation efficiently, a new splitting preconditioner is established and a...

Krylov subspace method | Complex linear systems | Preconditioner | PSHNS method | Convergence | HERMITIAN SPLITTING METHODS | MATHEMATICS, APPLIED | SADDLE-POINT PROBLEMS | ITERATION METHODS | MATRICES | OPTIMAL PARAMETERS | Linear systems | Helmholtz equations | Parameters | Matrices (mathematics) | Mathematical analysis | Mathematical models | Iterative methods | Matrix methods

Krylov subspace method | Complex linear systems | Preconditioner | PSHNS method | Convergence | HERMITIAN SPLITTING METHODS | MATHEMATICS, APPLIED | SADDLE-POINT PROBLEMS | ITERATION METHODS | MATRICES | OPTIMAL PARAMETERS | Linear systems | Helmholtz equations | Parameters | Matrices (mathematics) | Mathematical analysis | Mathematical models | Iterative methods | Matrix methods

Journal Article

Numerical Linear Algebra with Applications, ISSN 1070-5325, 01/2014, Volume 21, Issue 1, pp. 152 - 170

SUMMARYA generalized skew‐Hermitian triangular splitting iteration method is presented for solving non‐Hermitian linear systems with strong skew‐Hermitian...

skew‐Hermitian triangular splitting | eigenvalue estimate | convergence analysis | product‐type skew‐Hermitian triangular splitting | saddle‐point linear systems | Saddle-point linear systems | Eigenvalue estimate | Product-type skew-Hermitian triangular splitting | Skew-Hermitian triangular splitting | Convergence analysis | MATHEMATICS, APPLIED | product-type skew-Hermitian triangular splitting | saddle-point linear systems | ALGORITHM | ACCELERATION | MATHEMATICS | PRECONDITIONERS | MATRICES | AOR METHOD | CONVERGENCE | skew-Hermitian triangular splitting | Linear systems | Analysis | Methods | Splitting | Linear algebra | Constrictions | Mathematical models | Iterative methods | Convergence

skew‐Hermitian triangular splitting | eigenvalue estimate | convergence analysis | product‐type skew‐Hermitian triangular splitting | saddle‐point linear systems | Saddle-point linear systems | Eigenvalue estimate | Product-type skew-Hermitian triangular splitting | Skew-Hermitian triangular splitting | Convergence analysis | MATHEMATICS, APPLIED | product-type skew-Hermitian triangular splitting | saddle-point linear systems | ALGORITHM | ACCELERATION | MATHEMATICS | PRECONDITIONERS | MATRICES | AOR METHOD | CONVERGENCE | skew-Hermitian triangular splitting | Linear systems | Analysis | Methods | Splitting | Linear algebra | Constrictions | Mathematical models | Iterative methods | Convergence

Journal Article

Numerical Linear Algebra with Applications, ISSN 1070-5325, 08/2018, Volume 25, Issue 4, p. n/a

Summary For large sparse non‐Hermitian positive definite linear systems, we establish exact and inexact quasi‐HSS iteration methods and discuss their...

iteration method | system of linear equations | convergence | Hermitian and skew‐Hermitian splitting | preconditioning | Hermitian and skew-Hermitian splitting | MATHEMATICS, APPLIED | SOLVERS | SYMMETRIC PART | SPLITTING ITERATION | MATHEMATICS | NAVIER-STOKES EQUATIONS | MATRICES | PRECONDITIONER | FINITE-DIFFERENCE APPROXIMATIONS | Linear systems | Analysis | Methods | Robustness (mathematics) | Mathematical analysis | Solvers | Iterative methods | Matrix methods | Subspace methods

iteration method | system of linear equations | convergence | Hermitian and skew‐Hermitian splitting | preconditioning | Hermitian and skew-Hermitian splitting | MATHEMATICS, APPLIED | SOLVERS | SYMMETRIC PART | SPLITTING ITERATION | MATHEMATICS | NAVIER-STOKES EQUATIONS | MATRICES | PRECONDITIONER | FINITE-DIFFERENCE APPROXIMATIONS | Linear systems | Analysis | Methods | Robustness (mathematics) | Mathematical analysis | Solvers | Iterative methods | Matrix methods | Subspace methods

Journal Article

Computers and Mathematics with Applications, ISSN 0898-1221, 04/2019, Volume 77, Issue 7, pp. 1902 - 1916

In this paper, by adopting the preconditioned technique for the accelerated generalized successive overrelaxation method (AGSOR) proposed by Edalatpour et al....

Complex symmetric linear systems | Optimal parameters | Optimal convergence factor | Preconditioned accelerated generalized successive overrelaxation method | Convergence properties | HERMITIAN SPLITTING METHODS | MATHEMATICS, APPLIED | ITERATION METHODS | successive overrelaxation method | Preconditioned accelerated generalized | Linear systems | Analysis | Methods | Iterative methods | Convergence

Complex symmetric linear systems | Optimal parameters | Optimal convergence factor | Preconditioned accelerated generalized successive overrelaxation method | Convergence properties | HERMITIAN SPLITTING METHODS | MATHEMATICS, APPLIED | ITERATION METHODS | successive overrelaxation method | Preconditioned accelerated generalized | Linear systems | Analysis | Methods | Iterative methods | Convergence

Journal Article

Journal of Computational and Applied Mathematics, ISSN 0377-0427, 03/2012, Volume 236, Issue 9, pp. 2338 - 2353

For the singular, non-Hermitian, and positive semidefinite linear systems, we propose an alternating-direction iterative method with two parameters based on...

Iterative method | Singular linear system | Positive semidefinite matrix | Hermitian and skew-Hermitian splitting | Semi-convergence | Non-Hermitian matrix | HERMITIAN SPLITTING METHODS | ITERATION | MATHEMATICS, APPLIED | Linear systems | Analysis | Methods

Iterative method | Singular linear system | Positive semidefinite matrix | Hermitian and skew-Hermitian splitting | Semi-convergence | Non-Hermitian matrix | HERMITIAN SPLITTING METHODS | ITERATION | MATHEMATICS, APPLIED | Linear systems | Analysis | Methods

Journal Article