scholarly journals A Non-Krylov Subspace Method for Solving Large and Sparse Linear System of Equations

2016 ◽  
Vol 9 (2) ◽  
pp. 289-314 ◽  
Author(s):  
Wujian Peng ◽  
Qun Lin
Keyword(s):  
Iterative Methods ◽  
Numerical Experiments ◽  
Krylov Subspace ◽  
Coefficient Matrix ◽  
Projection Methods ◽  
Krylov Subspace Methods ◽  
Subspace Methods ◽  
Linear System Of Equations ◽  
Matrix Vector Multiplication ◽  
The Matrix

AbstractMost current prevalent iterative methods can be classified into the socalled extended Krylov subspace methods, a class of iterative methods which do not fall into this category are also proposed in this paper. Comparing with traditional Krylov subspace methods which always depend on the matrix-vector multiplication with a fixed matrix, the newly introduced methods (the so-called (progressively) accumulated projection methods, or AP (PAP) for short) use a projection matrix which varies in every iteration to form a subspace from which an approximate solution is sought. More importantly an accelerative approach (called APAP) is introduced to improve the convergence of PAP method. Numerical experiments demonstrate some surprisingly improved convergence behavior. Comparison between benchmark extended Krylov subspace methods (Block Jacobi and GMRES) are made and one can also see remarkable advantage of APAP in some examples. APAP is also used to solve systems with extremely ill-conditioned coefficient matrix (the Hilbert matrix) and numerical experiments shows that it can bring very satisfactory results even when the size of system is up to a few thousands.

scholarly journals Krylov Subspace Solvers and Preconditioners

2018 ◽  
Vol 63 ◽  
pp. 1-43
Author(s):  
C. Vuik
Keyword(s):  
Iterative Methods ◽  
Krylov Subspace ◽  
State Of The Art ◽  
Linear Equations ◽  
Iterative Solution ◽  
Krylov Subspace Methods ◽  
Numerical Linear Algebra ◽  
Subspace Methods ◽  
Starting Point ◽  
Lecture Notes

In these lecture notes an introduction to Krylov subspace solvers and preconditioners is presented. After a discretization of partial differential equations large, sparse systems of linear equations have to be solved. Fast solution of these systems is very urgent nowadays. The size of the problems can be 1013 unknowns and 1013 equations. Iterative solution methods are the methods of choice for these large linear systems. We start with a short introduction of Basic Iterative Methods. Thereafter preconditioned Krylov subspace methods, which are state of the art, are describeed. A distinction is made between various classes of matrices. At the end of the lecture notes many references are given to state of the art Scientific Computing methods. Here, we will discuss a number of books which are nice to use for an overview of background material. First of all the books of Golub and Van Loan [19] and Horn and Johnson [26] are classical works on all aspects of numerical linear algebra. These books also contain most of the material, which is used for direct solvers. Varga [50] is a good starting point to study the theory of basic iterative methods. Krylov subspace methods and multigrid are discussed in Saad [38] and Trottenberg, Oosterlee and Schüller [42]. Other books on Krylov subspace methods are [1, 6, 21, 34, 39].

Iterative solution of linear systems

Acta Numerica ◽  
1992 ◽  
Vol 1 ◽  
pp. 57-100 ◽  
Author(s):  
Roland W. Freund ◽  
Gene H. Golub ◽  
Noël M. Nachtigal
Keyword(s):  
Linear Systems ◽  
Iterative Methods ◽  
Conjugate Gradient ◽  
Krylov Subspace ◽  
Iterative Solution ◽  
Krylov Subspace Methods ◽  
Subspace Methods ◽  
Recent Advances ◽  
Gradient Type ◽  
Large Linear Systems

Recent advances in the field of iterative methods for solving large linear systems are reviewed. The main focus is on developments in the area of conjugate gradient-type algorithms and Krylov subspace methods for nonHermitian matrices.

scholarly journals Generalized Preconditioned MHSS Method for a Class of Complex Symmetric Linear Systems

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Cui-Xia Li ◽  
Yan-Jun Liang ◽  
Shi-Liang Wu
Keyword(s):  
Theoretical Analysis ◽  
Linear Systems ◽  
Numerical Experiments ◽  
Krylov Subspace ◽  
Krylov Subspace Methods ◽  
Point Of View ◽  
Subspace Methods ◽  
Practical Point ◽  
Complex Symmetric Linear Systems ◽  
Complex Symmetric

Based on the modified Hermitian and skew-Hermitian splitting (MHSS) and preconditioned MHSS (PMHSS) methods, a generalized preconditioned MHSS (GPMHSS) method for a class of complex symmetric linear systems is presented. Theoretical analysis gives an upper bound for the spectral radius of the iteration matrix. From a practical point of view, we have analyzed and implemented inexact GPMHSS (IGPMHSS) iteration, which employs Krylov subspace methods as its inner processes. Numerical experiments are reported to confirm the efficiency of the proposed methods.

On Preconditioners Based on HSS for the Space Fractional CNLS Equations

2017 ◽  
Vol 7 (1) ◽  
pp. 70-81 ◽  
Author(s):  
Yu-Hong Ran ◽  
Jun-Gang Wang ◽  
Dong-Ling Wang
Keyword(s):  
Krylov Subspace ◽  
Coefficient Matrix ◽  
Krylov Subspace Methods ◽  
Identity Matrix ◽  
Subspace Methods ◽  
Unconditionally Stable ◽  
Numerical Examples ◽  
Difference Formula ◽  
Conservative Difference ◽  
Centered Difference

AbstractThe space fractional coupled nonlinear Schrödinger (CNLS) equations are discretized by an implicit conservative difference scheme with the fractional centered difference formula, which is unconditionally stable. The coefficient matrix of the discretized linear system is equal to the sum of a complex scaled identity matrix which can be written as the imaginary unit times the identity matrix and a symmetric Toeplitz-plusdiagonal matrix. In this paper, we present new preconditioners based on Hermitian and skew-Hermitian splitting (HSS) for such Toeplitz-like matrix. Theoretically, we show that all the eigenvalues of the resulting preconditioned matrices lie in the interior of the disk of radius 1 centered at the point (1,0). Thus Krylov subspace methods with the proposed preconditioners converge very fast. Numerical examples are given to illustrate the effectiveness of the proposed preconditioners.

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

2012 ◽  
Vol 2 (1) ◽  
pp. 19-32
Author(s):  
Akira Imakura ◽  
Tomohiro Sogabe ◽  
Shao-Liang Zhang
Keyword(s):  
Linear Systems ◽  
Numerical Experiments ◽  
Krylov Subspace ◽  
Krylov Subspace Methods ◽  
Initial Guess ◽  
Subspace Methods ◽  
Nonsymmetric Linear Systems ◽  
Mathematical Background

AbstractThe 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.

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