An improved Newton iteration for the generalized inverse of a matrix, with applications

The purpose here is to clarify and illustrate the potential for the use of variants of Newton's method of solving problems of practical interest on highly personal computers. The authors show how to accelerate the method substantially and how to modify it successfully to cope with ill-conditioned matrices. The authors conclude that Newton's method can be of value for some interesting computations, especially in parallel and other computing environments in which matrix products are especially easy to work with

Inverse eigenvalue problem

A RESEARCH REPORT submitted to the Faculty of Science of the University of the Witwatersrand in partial fulfilment of the degree of MASTER OF SCIENCE Johannesburg, Republic of South Africa December1998·This work is concerned with the Inverse Eigenvalue Problem for ordinary differential equations of the Sturm-Liouville type in the general form --dd ( 7' ()xdll(t\,:rI)) + {(q) x - t\p:(r )} u (A, Xl, = 0, .1' c.r (I :::: .7' S; b. The central problem considered ill this research is the approximate reC011- struction of the unknown coefficient function q(:l') in the Sturm-Liouville equation JOIl Irom a given finite spectral data set ~i(q), for i = 1 : n . A solution is sought using a finite element discretization method. The method works br solving the non-Iinear system arising out of the difference between the eigenvalues A,(q) of the Sturm-Liouville differential equation and the given spectral data ~i(q). Numerical results me presented to illustrate the effectiveness of the discretization method ill question

Structured matrix methods for a polynomial root solver using approximate greatest common divisor computations and approximate polynomial factorisations.

This thesis discusses the use of structure preserving matrix methods for the numerical approximation of all the zeros of a univariate polynomial in the presence of noise. In particular, a robust polynomial root solver is developed for the calculation of the multiple roots and their multiplicities, such that the knowledge of the noise level is not required. This designed root solver involves repeated approximate greatest common divisor computations and polynomial divisions, both of which are ill-posed computations. A detailed description of the implementation of this root solver is presented as the main work of this thesis. Moreover, the root solver, implemented in MATLAB using 32-bit floating point arithmetic, can be used to solve non-trivial polynomials with a great degree of accuracy in numerical examples

The Orthogonal QD-Algorithm

The orthogonal qd-algorithm is presented to compute the singular value decomposition of a bidiagonal matrix. This algorithm represents a modification of Rutishauser's qd-algorithm, and it is capable of determining all the singular values to high relative precision. A generalization of the Givens transformation is also introduced, which has applications besides the orthogonal qd-algorithm. The shift strategy of the orthogonal qd-algorithm is based on Laguerre's method, which is used to compute a lower bound for the smallest singular value of the bidiagonal matrix. Special attention is devoted to the numerically stable evaluation of this shift. (Also cross-referenced as UMIACS-TR-94-9.1

Structures and Algorithms for Two-Dimensional Adaptive Signal Processing

Coordinated Science Laboratory was formerly known as Control Systems LaboratoryOpe

A circuit model for diffusive breast imaging and a numerical algorithm for its inverse problem

Thesis (M. Eng.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 1996.Includes bibliographical references (leaves 67-70).by Julie L. Wonus.M.Eng

An Examination of Some Signi cant Approaches to Statistical Deconvolution

We examine statistical approaches to two significant areas of deconvolution - Blind Deconvolution (BD) and Robust Deconvolution (RD) for stochastic stationary signals. For BD, we review some major classical and new methods in a unified framework of nonGaussian signals. The first class of algorithms we look at falls into the class of Minimum Entropy Deconvolution (MED) algorithms. We discuss the similarities between them despite differences in origins and motivations. We give new theoretical results concerning the behaviour and generality of these algorithms and give evidence of scenarios where they may fail. In some cases, we present new modifications to the algorithms to overcome these shortfalls. Following our discussion on the MED algorithms, we next look at a recently proposed BD algorithm based on the correntropy function, a function defined as a combination of the autocorrelation and the entropy functiosn. We examine its BD performance when compared with MED algorithms. We find that the BD carried out via correntropy-matching cannot be straightforwardly interpreted as simultaneous moment-matching due to the breakdown of the correntropy expansion in terms of moments. Other issues such as maximum/minimum phase ambiguity and computational complexity suggest that careful attention is required before establishing the correntropy algorithm as a superior alternative to the existing BD techniques. For the problem of RD, we give a categorisation of different kinds of uncertainties encountered in estimation and discuss techniques required to solve each individual case. Primarily, we tackle the overlooked cases of robustification of deconvolution filters based on estimated blurring response or estimated signal spectrum. We do this by utilising existing methods derived from criteria such as minimax MSE with imposed uncertainty bands and penalised MSE. In particular, we revisit the Modified Wiener Filter (MWF) which offers simplicity and flexibility in giving improved RDs to the standard plug-in Wiener Filter (WF)

A circuit for diffusive breast imaging and a numerical algorithm for its inverse problem

Includes bibliographical references (p. 84-88).Supported by the National Science Foundation. MIP91-17724Julie L. Wonus and John L. Wyatt