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Backward Error and Condition of Polynomial Eigenvalue Problems
 Linear Algebra Appl
, 1999
"... We develop normwise backward errors and condition numbers for the polynomial eigenvalue problem. The standard way of dealing with this problem is to reformulate it as a generalized eigenvalue problem (GEP). For the special case of the quadratic eigenvalue problem (QEP), we show that solving the QEP ..."
Abstract

Cited by 49 (12 self)
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We develop normwise backward errors and condition numbers for the polynomial eigenvalue problem. The standard way of dealing with this problem is to reformulate it as a generalized eigenvalue problem (GEP). For the special case of the quadratic eigenvalue problem (QEP), we show that solving the QEP by applying the QZ algorithm to a corresponding GEP can be backward unstable. The QEP can be reformulated as a GEP in many ways. We investigate the sensitivity of a given eigenvalue to perturbations in each of the GEP formulations and identify which formulations are to be preferred for large and small eigenvalues, respectively. Key words. Polynomial eigenvalue problem, quadratic eigenvalue problem, generalized eigenvalue problem, backward error, condition number. 1 Introduction We are concerned with backward error analysis and conditioning for the nonlinear eigenvalue problem P ()x = 0; (1.1) where P () is a matrix whose elements are polynomials in a scalar . We write P in the form P () =...
ScaleInvariant Image Recognition Based On Higher Order Autocorrelation Features
 Pattern Recognition
, 1996
"... We propose a framework and a complete implementation of a translation and scale invariant image recognition system for natural indoor scenes. The system employs higher order autocorrelation features of scale space data which permit linear classification. An optimal linear classification method is pr ..."
Abstract

Cited by 12 (1 self)
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We propose a framework and a complete implementation of a translation and scale invariant image recognition system for natural indoor scenes. The system employs higher order autocorrelation features of scale space data which permit linear classification. An optimal linear classification method is presented, which is able to cope with a large number of classes represented by many, as well as very few samples. In the course of the analysis of our system, we examine which numerical methods for feature transformation and classification show sufficient stability to fulfill these demands. The implementation has been extensively tested. We present the results of our own application and several classification benchmarks. Image recognition Face recognition Scale invariancy Scale space Higher order autocorrelation Optimal linear classification 1. INTRODUCTION The task of visual recognition which was defined by Marr (1) with the question: "What objects are where in the environment?" is still ...