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For any linear system Ax ≈ b we define a set of core problems and show that the orthogonal upper bidiagonalization of [b, A] gives such a core problem. In particular we show that these core problems have desirable properties such as minimal dimensions. When a total least squares problem is solved by first finding a core problem, we show the resulting theory… (More)

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)

We analyze the residuals of GMRES [9], when the method is applied to tridiagonal Toeplitz matrices. We first derive formulas for the residuals as well as their norms when GMRES is applied to scaled Jordan blocks. This problem has been studied previously by Ipsen [5], Eiermann and Ernst [2], but we formulate and prove our results in a different way. We then… (More)

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)

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)

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)

Let A be a real symmetric positive deenite matrix. We consider three particular questions, namely estimates for the error in linear systems Ax = b, minimizing quadratic functional min x (x T Ax ? 2b T x) subject to the constraint k x k= , <k A ?1 b k, and estimates for the entries of the matrix inverse A ?1. All of these questions can be formulated as a… (More)

We study Krylov subspace methods for solving unsymmetric linear algebraic systems 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 basic identities and bounds for the LS residual. These results… (More)