We consider the Modified Gram-Schmidt orthogonalization applied to a matrix A ∈ Rm×n. This corresponds to a QR factorization : A = QR. We study this algorithm in finite precision computation when the matrix A has a numerical rank deficiency k. This subject has already been dealt with success by Björck and Paige in 1992 [1]. They give useful bounds in term of norms. We extend their results to pr...

The numerical range of a quadratic operator acting on an indefinite inner product space is shown to have a hyperbolical shape. This result is extended to different kinds of indefinite numerical ranges, namely, indefinite higher rank numerical ranges and indefinite Davis-Wielandt shells.

The numerical range of a quadratic operator acting on an indefinite inner product space is shown to have a hyperbolical shape. This result is extended to different kinds of indefinite numerical ranges, namely, indefinite higher rank numerical ranges and indefinite Davis-Wielandt shells.

Closed-form algebraic formulae are derived for the local error incurred in the numerical solution of integral equations by iterated collocation methods, the analysis being illustrated by application to Fredholm integral equations of the second kind. The novel error analysis uses an asymptotic approach, in the small parameter of the numerical mesh size, applied to a finite-rank degenerate-kernel...

Given an m × n matrix A and a positive integer k, we introduce a randomized procedure for the approximation of A with a matrix Z of rank k. The procedure relies on applying AT to a collection of l random vectors, where l is an integer equal to or slightly greater than k; the scheme is efficient wheneverA andAT can be applied rapidly to arbitrary vectors. The discrepancy between A and Z is of th...

We prove new enclosures for the spectrum of non-selfadjoint operator matrices associated with second order linear differential equations z̈(t) +Dż(t) +A0z(t) = 0 in a Hilbert space. Our main tool is the quadratic numerical range for which we establish the spectral inclusion property under weak assumptions on the operators involved; in particular, the damping operator only needs to be accretive a...

This work is concerned with the numerical solution of large-scale linear matrix equations A1XB T 1 + · · ·+ AKXB K = C. The most straightforward approach computes X ∈ Rm×n from the solution of an mn×mn linear system, typically limiting the feasible values of m,n to a few hundreds at most. Our new approach exploits the fact that X can often be well approximated by a low-rank matrix. It combines ...

The inertia of a Hermitian matrix is defined to be a triplet composed by the numbers of the positive, negative and zero eigenvalues of the matrix counted with multiplicities, respectively. In this paper, we give various closed-form formulas for the maximal and minimal values for the rank and inertia of the Hermitian expression A + X, where A is a given Hermitian matrix and X is a variable Hermi...