On the rate of convergence of the preconditioned conjugate gradient method

Axelsson, Owe; Lindskog, Gunhild

doi:10.1007/bf01389448

Cited by 273 publications

(190 citation statements)

References 10 publications

Supporting

Mentioning

186

Contrasting

Order By: Relevance

“…n A n and the properties of the generating function f (x) is very important for practical application of circulant preconditioners, because the "better" the clustering rate, the "faster" the conjugate gradient method converges [1].…”

Section: Introductionmentioning

confidence: 99%

Which circulant preconditioner is better?

Strela

Тыртышников

1996

Math. Comp.

View full text Add to dashboard Cite

Abstract. The eigenvalue clustering of matrices S −1 n An and C −1 n An is experimentally studied, where An, Sn and Cn respectively are Toeplitz matrices, Strang, and optimal circulant preconditioners generated by the Fourier expansion of a function f (x). Some illustrations are given to show how the clustering depends on the smoothness of f (x) and which preconditioner is preferable. An original technique for experimental exploration of the clustering rate is presented. This technique is based on the bisection idea and on the Toeplitz decomposition of a three-matrix product CAC, where A is a Toeplitz matrix and C is a circulant. In particular, it is proved that the Toeplitz (displacement) rank of CAC is not greater than 4, provided that C and A are symmetric.

show abstract

Section: Introductionmentioning

confidence: 99%

Which circulant preconditioner is better?

Strela

Тыртышников

1996

Math. Comp.

View full text Add to dashboard Cite

show abstract

“…The described phenomenon of superlinear convergence of Krylov subspace methods has been widely observed, and some models explaining this behavior have been proposed; see [2], [38], [39], and also [9], [10], [12]. The analysis proposed in most of the cited papers relies on the polynomial representation of the approximate solution in K m ; see section 2.…”

Section: Introductionmentioning

confidence: 99%

On the Occurrence of Superlinear Convergence of Exact and Inexact Krylov Subspace Methods

Simoncini¹,

Szyld²

2005

SIAM Rev.

View full text Add to dashboard Cite

Abstract. Krylov subspace methods often exhibit superlinear convergence. We present a general analytic model which describes this superlinear convergence, when it occurs. We take an invariant subspace approach, so that our results apply also to inexact methods, and to nondiagonalizable matrices. Thus, we provide a unified treatment of the superlinear convergence of GMRES, conjugate gradients, block versions of these, and inexact subspace methods. Numerical experiments illustrate the bounds obtained.

show abstract

“…[22][23][24]). As an example, when the matrix has small isolated eigenvalues, an improved bound is [25] …”

Section: Sce With the Cg Methodmentioning

confidence: 99%

Applications of statistical condition estimation to the solution of linear systems

Laub¹,

Xia²

2008

Numerical Linear Algebra App

View full text Add to dashboard Cite

SUMMARYThis paper discusses some applications of statistical condition estimation (SCE) to the problem of solving linear systems. Specifically, triangular and bidiagonal matrices are studied in some detail as typical of structured matrices. Such a structure, when properly respected, leads to condition estimates that are much less conservative compared with traditional non-statistical methods of condition estimation. Some examples of linear systems and Sylvester equations are presented. Vandermonde and Cauchy matrices are also studied as representative of linear systems with large condition numbers that can nonetheless be solved accurately. SCE reflects this. Moreover, SCE when applied to solving very large linear systems by iterative solvers, including conjugate gradient and multigrid methods, performs equally well and various examples are given to illustrate the performance. SCE for solving large linear systems with direct methods, such as methods for semi-separable structures, are also investigated. In all cases, the advantages of using SCE are manifold: ease of use, efficiency, and reliability.

show abstract

On the rate of convergence of the preconditioned conjugate gradient method

Cited by 273 publications

References 10 publications

Which circulant preconditioner is better?

Which circulant preconditioner is better?

On the Occurrence of Superlinear Convergence of Exact and Inexact Krylov Subspace Methods

Applications of statistical condition estimation to the solution of linear systems

Contact Info

Product

Resources

About