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Expressions And Bounds For The GMRES Residual
 BIT
, 1999
"... . Expressions and bounds are derived for the residual norm in GMRES. It is shown that the minimal residual norm is large as long as the Krylov basis is wellconditioned.For scaled Jordan blocks the minimal residual norm is expressed in terms of eigenvalues and departure from normality.For normal mat ..."
Abstract

Cited by 25 (0 self)
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. Expressions and bounds are derived for the residual norm in GMRES. It is shown that the minimal residual norm is large as long as the Krylov basis is wellconditioned.For scaled Jordan blocks the minimal residual norm is expressed in terms of eigenvalues and departure from normality.For normal matrices the minimal residual norm is expressed in terms of products of relative eigenvalue di#erences. Key words. linear system, Krylov methods, GMRES, MINRES, Vandermonde matrix, eigenvalues, departure from normality AMS subject classi#cation. 15A03, 15A06, 15A09, 15A12, 15A18, 15A60, 65F10, 65F15, 65F20, 65F35. 1. Introduction.. The generalised minimal residual method #GMRES# #31, 36# #and MINRES for Hermitian matrices #30## is an iterative method for solving systems of linear equations Ax = b. The approximate solution in iteration i minimises the twonorm of the residual b , Az over the Krylov space spanfb;Ab;:::;A i,1 bg. The goal of this paper is to express this minimal residual norm...
Least squares residuals and minimal residual methods
 SIAM J. Sci. Comput
"... Abstract. We study Krylov subspace methods for solving unsymmetriclinear algebraicsystems that minimize the norm of the residual at each step (minimal residual (MR) methods). MR methods are often formulated in terms of a sequence of least squares (LS) problems of increasing dimension. We present sev ..."
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Cited by 17 (2 self)
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Abstract. We study Krylov subspace methods for solving unsymmetriclinear algebraicsystems that minimize the norm of the residual at each step (minimal residual (MR) methods). MR methods are often formulated in terms of a sequence of least squares (LS) problems of increasing dimension. We present several basicidentities and bounds for the LS residual. These results are interesting in the general context of solving LS problems. When applied to MR methods, they show that the size of the MR residual is strongly related to the conditioning of different bases of the same Krylov subspace. Using different bases is useful in theory because relating convergence to the characteristics of different bases offers new insight into the behavior of MR methods. Different bases also lead to different implementations which are mathematically equivalent but can differ numerically. Our theoretical results are used for a finite precision analysis of implementations of the GMRES method [Y. Saad and M. H. Schultz, SIAM J. Sci. Statist. Comput., 7 (1986), pp. 856–869]. We explain that the choice of the basis is fundamental for the numerical stability of the implementation. As demonstrated in the case of Simpler GMRES [H. F. Walker and L. Zhou, Numer. Linear Algebra Appl., 1 (1994), pp. 571–581], the best orthogonalization technique used for computing the basis does not compensate for the loss of accuracy due to an inappropriate choice of the basis. In particular, we prove that Simpler GMRES is inherently less numerically stable than
Construction and analysis of polynomial iterative methods for nonhermitian systems of linear equations
, 1998
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A Note On The Field Of Values Of NonNormal Matrices
"... . It is shown that the distance from zero of the field of values of a matrix A depends on how large the departure from normality is compared to the distance from the zero of the field of values of the normal part of A. A connection is made to the convergence of Krylov methods for the solution of lin ..."
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. It is shown that the distance from zero of the field of values of a matrix A depends on how large the departure from normality is compared to the distance from the zero of the field of values of the normal part of A. A connection is made to the convergence of Krylov methods for the solution of linear systems. Key words. 15A60, 65F10, 15A18, 65F15, 15A42 AMS subject classification. field of values, numerical range, numerical radius, departure from normality, eigenvalues, Krylov methods 1. Introduction. The field of values (or numerical range) of a square matrix A is the set of all complex numbers 1 F(A) j f x Ax x x ; x 6= 0 is a vectorg: The field of values is used in the convergence analysis of iterative methods for the solution of systems of linear equations, such as, for instance, asymptotically stationary kstep methods [2], ADI methods [15], and the Krylov methods GMRES and FOM [16], and Orthomin [5, x2.2]. Take Orthomin(1), for instance. Applied to the linear syste...