On the condition number of certain Rayleigh-Ritz-Galerkin matrices

The Condition Number of a Class of Rayleigh-ritz-galerkin Matrices

The purpose of this note is to study the Euclidean condition number of the matrix resulting from using the well-known RayleighRitz-Galerkin method with finite dimensional subspaces of polynomial spline functions to approximate the solution of a linear, selfadjoint* two-point boundary value problem. Roughly speaking, we consider a model class of such problems of order In and determine upper boun...

The condition number of certain matrices and applications

This paper addresses the problem of estimating the eigenvalues and condition numbers of matrices of the form R = r(ti tj). We begin by mentioning some of the problems in which such matrices occur, and to which the results obtained in this paper may be applied. Examples of such problems include (i) approximation by sums of irregular translates (ii) the missing data problem and incomplete samplin...

Accuracy of Rayleigh- Ritz Approximations

New bounds on the canonical angles between an invariant subspace of A and an approximating subspace by the differences between Ritz values and the targeted eigenvalues are obtained. From this result, various bounds are readily available to estimate how accurate the Ritz vectors computed from the approximating subspace may be, based on information on approximation accuracies in the Ritz values. ...

On The Frobenius Condition Number of Positive Definite Matrices

Towards an optimal condition number of certain augmented Lagrangian-type saddle-point matrices

We present an analysis for minimizing the condition number of nonsingular parameter-dependent 2 2 block-structured saddle-point matrices with a maximally rank-deficient (1,1) block. The matrices arise from an augmented Lagrangian approach. Using quasidirect sums, we show that a decomposition akin to simultaneous diagonalization leads to an optimization based on the extremal nonzero eigenvalues ...