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- JГ¶rg Liesen, Zdenek Strakos, Shun Wang, Feliks Ruvimovich, Sabine Van Huffel, Joos Vandewalle
- 2009

The mathematical theory of Krylov subspace methods with a focus on solving systems of linear algebraic equations is given a detailed treatment in this principles-based book. Starting from the idea of projections, Krylov subspace methods are characterised by their orthogonality and minimisation properties. Projections onto highly nonlinear Krylov subspaces… (More)

- Anne Greenbaum, Vlastimil Pták, Zdenek Strakos
- SIAM J. Matrix Analysis Applications
- 1996

Given a nonincreasing positive sequence, f(0) f(1) : : : f(n ? 1) > 0, it is shown that there exists an n by n matrix A and a vector r 0 with kr 0 k = f(0) such that f(k) = kr k k, k = 1; : : :; n ? 1, where r k is the residual at step k of the GMRES algorithm applied to the linear system Ax = b, with initial residual r 0 = b?Ax 0. Moreover, the matrix A… (More)

- Christopher C. Paige, Zdenek Strakos
- SIAM J. Matrix Analysis Applications
- 2005

- Christopher C. Paige, Zdenek Strakos
- Numerische Mathematik
- 2002

The standard approaches to solving overdetermined linear systems Bx ≈ c construct minimal corrections to the vector c and/or the matrix B such that the corrected system is compatible. In ordinary least squares (LS) the correction is restricted to c, while in data least squares (DLS) it is restricted to B. In scaled total least squares (STLS) [22],… (More)

- Christopher C. Paige, Miroslav Rozlozník, Zdenek Strakos
- SIAM J. Matrix Analysis Applications
- 2006

The generalized minimum residual method (GMRES) for solving linear systems Ax = b is implemented as a sequence of least squares problems involving Krylov subspaces of increasing dimensions. The most usual implementation is Modified Gram-Schmidt GMRES (MGS-GMRES). Here we show that MGS-GMRES is backward stable. The result depends on a more general result on… (More)

- Pavel Jiránek, Zdenek Strakos, Martin Vohralík
- SIAM J. Scientific Computing
- 2010

For the finite volume discretization of a second-order elliptic model problem, we derive a posteriori error estimates which take into account an inexact solution of the associated linear algebraic system. We show that the algebraic error can be bounded by constructing an equilibrated Raviart–Thomas–Nédélec discrete vector field whose divergence is given by… (More)

- Christopher C. Paige, Zdenek Strakos
- SIAM J. Scientific Computing
- 2002

Minimum residual norm iterative methods for solving linear systems Ax = b can be viewed as, and are often implemented as, sequences of least squares problems involving Krylov sub-spaces of increasing dimensions. The minimum residual method (MINRES) [C. Bounds for the least squares distance using scaled total least squares, Numer. Math., to appear] revealing… (More)

- Christopher C. Paige, Zdenek Strakos
- Numerische Mathematik
- 2002

The standard approaches to solving overdetermined linear systems Bx ≈ c construct minimal corrections to the data to make the corrected system compatible. In ordinary least squares (LS) the correction is restricted to the right hand side c, while in scaled total least squares (STLS) [14, 12] corrections to both c and B are allowed, and their relative sizes… (More)

- Jörg Liesen, Zdenek Strakos
- SIAM Review
- 2008

In this talk I will discuss necessary and sufficient conditions on a nonsingu-lar matrix A, such that for any initial vector r 0 , an orthogonal basis of the Krylov subspaces K n (A, r 0) is generated by a short recurrence. Orthogonality here is meant with respect to some unspecified positive definite inner product. This question is closely related to the… (More)

The aim of the paper is to compile and compare basic theoretical facts on Krylov subspaces and block Krylov subspaces. Many Krylov (sub)space methods for solving a linear system Ax = b have the property that in exact computer arithmetic the true solution is found after ν iterations, where ν is the dimension of the largest Krylov subspace generated by A from… (More)