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"For the past 50 years the workhorse algorithm for solving eigenvalue problems has been Francis's implictly shifted QR algorithm. We present here a new formulation of Francis's algorithm that operates on the QR factors of a matrix A rather than A itself. This is not in any existing book. A popular way to compute the roots of a polynomial is to form the companion matrix and compute its eigenvalues (which are exactly the roots of the polynomial). This is what Matlab's roots command does. Our new formulation leads to a better algorithm for solving the companion eigenvalue problem (which is unitary-plus-rank-one). Our algorithm is faster than all of the competing algorithms. We can prove it is backward stable, and it is stable in a stronger sense than the algorithm that Matlab uses. Thus our algorithm is faster and more accurate than Matlab's. In the final chapter we present a generalization of Francis's algorithm that we published (in a SIAM journal) a few years ago. This applies to all classes of eigenvalue problems, structured or not. This is not in any book. Our monograph presents a unified treatment of several classes of eigenvalue problems. We listed them above, but here they are again: unitary, unitary-plus-rank-one (companion matrices and pencils), symmetric, symmetic-plus-rank-one, matrix polynomials. We also provide new insights into matrix eigenvalue problems with no special structure. These are the results of our research of the past few years. This is hot off the press. Most of this material has appeared in our publications, but none of it is in any book. In recent conference presentations we have mentioned that we are working on a book. Informal feedback suggests there is significant interest in this book among researchers in numerical linear algebra"--

Eigenvalues

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Symmetric matrices --- Eigenvalues --- Symmetric matrices. --- Eigenvalues.

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This thesis explores the asymptotic spectral distribution of random matrices, focusing on theoretical frameworks and closed-form solutions using concepts like free independence. It delves into the properties of eigenvalues, particularly in large dimensional random matrices, and applies theories such as the Stieltjes and R-transform. The work is motivated by applications in theoretical physics and wireless communications, where random matrix theory is pivotal in analyzing systems with multiple antennas. The research characterizes classes of matrices with closed formulas for asymptotic spectral distributions, contributing to the understanding of matrix stability and invertibility. The intended audience includes mathematicians and researchers in fields reliant on matrix theory and its applications.

Random matrices. --- Eigenvalues. --- Random matrices --- Eigenvalues

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Boundary value problems --- Eigenvalues

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Eigenvalues. --- Analysis of covariance.