A generalized conjugate gradient, least square method

Axelsson, Owe

doi:10.1007/bf01396750

Cited by 147 publications

(131 citation statements)

References 16 publications

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“…The same result is obtained in [5] and in [1] by Axelsson (under a more generally formulation). We also note that no particular relation on p is found when A is nondiagonalizable.…”

Section: Proposition 33supporting

confidence: 81%

“…Several authors have already tried to analyse the effects of a singular matrix on a Krylov subspace method and we give below a list of their works. We remind that Brown and Walker have introduced in [3] conditions concerning the singular matrix A under which the GMRES iterates converge safely to a solution of (1) and remark that these theoretical results are still true for any mathematically equivalent method.…”

Section: Introductionmentioning

confidence: 96%

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Spectral behaviour of GMRES applied to singular systems

Smoch

2007

Adv Comput Math

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The purpose of this paper is to develop a spectral analysis of the Hessenberg matrix obtained by the GMRES algorithm used for solving a linear system with a singular matrix. We prove that the singularity of the Hessenberg matrix depends on the nature of A and some others criteria like the zero eigenvalue multiplicity and the projection of the initial residual on particular subspaces. We also introduce some new results about the distinct kinds of breakdown which may occur in the algorithm when the system is singular.

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“…The same result is obtained in [5] and in [1] by Axelsson (under a more generally formulation). We also note that no particular relation on p is found when A is nondiagonalizable.…”

Section: Proposition 33supporting

confidence: 81%

Section: Introductionmentioning

confidence: 96%

Spectral behaviour of GMRES applied to singular systems

Smoch

2007

Adv Comput Math

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“…The similar and more general estimates can be found in the papers [5], [6], [7], [8]. These authors formulate their estimates in the form r m 2 ≤ (1− (m)) r 0 2 and formulate sufficient conditions for (m) to be in the interval 0 < (m) ≤ < 1 for all m. In this sense we will generalize first all known results in the following.…”

Section: The Recapitulation Of Various Bounds For R Msupporting

confidence: 55%

How residual bounds for restarted GMRES describe the real behaviour

Zítko

2008

Proc Appl Math and Mech

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Let us consider the non-Hermitian and non-singular system Ax = b, A ∈ C n×n , x, b ∈ C n . The following notational conventions will be observed throughout this paper; The notation x 0 will be used for the initial approximation, and r 0 = b − Ax 0 = 0 will denote the corresponding residual vector. For the matrix A and a vector u ∈ C n , we define the KrylovThe letter v will be used for the quotient v = r 0 / r 0 , S n is the unit sphere in C n , · is the Euclidean norm, P m is the set of all polynomials of degree m, P 0 m is the set of all polynomials from P m which evaluate to zero at zero. The restarted GMRES algorithm (GMRES(m)), proposed by Y. Saad and M. H. Schultz [1] and described for various situations and modifications by many authors, is the most popular Krylov method for the solution of Ax = b.We start with the estimates for r m in the first restart. The GMRES algorithm constructs the new approximation x m in the affine space (A) . In the paper [2], three bounds for the p(A), and hence for r m , are derived and their quality discussed. This paper provides the motivation for the following discussion. Elman (see [3] v. Calculation of r m yields for every q ∈ P 0 m the inequalitieswhere H q and iS q denote the Hermitian and skew-Hermitian part of the matrix q(A) respectively and i 2 = −1.is the orthogonal projector for the space span{AK m (r 0 )} and therefore.• If we substitute the estimate σ 2Another upper bounds for the quotient r m 2 / r 0 2 can be found in the papers of Strakoš, Rozložník and Liesen.To obtain non-stagnation conditions for restarted GMRES, we will concentrate on estimates of the form r m 2 / r 0 2 ≤ 1 − ρ i , wherefor α ∈ C m and 1 − ρ 3 = 1 − min The restarted version GMRES(m)Let

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“…Conjugate gradient m ethods have become one of the most widespread ways of solving not only symmetric but also nonsymmetric linear algebraic systems [2,4]. An early paper dealing w ith generalized conjugate gradient m ethods for nonsymmetric systems is [3], and a survey of available m ethods can be found in [14], which includes also the popular GMRES m ethod [13].…”

Section: Introductionmentioning

confidence: 99%

Symmetric Part Preconditioning for the Conjugate Gradient Method in Hilbert Space

Axelsson

Karátson

2003

Numerical Functional Analysis and Optimization

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Dedicated to the memory of Jean-Jacques Lions: a source of inspiration for rigour in Applied M athem atics A b s tr a c t T he conjugate gradient m ethod for non-sym m etric linear operators in H ilbert space is investigated. C onditions on th e coincidence of th e full and tru n cated versions, known from the finite-dim ensional case, are extended to th e H ilbert space setting. T he focus is on preconditioning by th e sym m etric p a rt of the operator, in which case estim ates are given for th e resulting condition num ber. A n im p o rtan t m otivation for this study is given by differential operators, for which the obtained estim ates yield m esh independent conditioning properties of th e full CGM , and are in fact achieved by th e sim pler tru n cated version.

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A generalized conjugate gradient, least square method

Cited by 147 publications

References 16 publications

Spectral behaviour of GMRES applied to singular systems

Spectral behaviour of GMRES applied to singular systems

How residual bounds for restarted GMRES describe the real behaviour

Symmetric Part Preconditioning for the Conjugate Gradient Method in Hilbert Space

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