Evolving from an elementary discussion, this book develops the Euclidean algorithm to a very powerful tool to deal with general continued fractions, non-normal Pad&eacute; tables, look-ahead algorithms for Hankel and Toeplitz matrices, and for Krylov subspace methods. It introduces the basics of fast algorithms for structured problems and shows how they deal with singular situations.<P>Links are made with more applied subjects such as linear system theory and signal processing, and with more advanced topics and recent results such as general bi-orthogonal polynomials, minimal Pad&eacute; approximation, polynomial root location problems in the complex plane, very general rational interpolation problems, and the lifting scheme for wavelet transform computation. The text serves as a supplement to existing books on structured linear algebra problems, rational approximation and orthogonal polynomials.<P>Features of this book:<P>&bull; provides a unifying approach to linear algebra, rational approximation and orthogonal polynomials<P>&bull; requires an elementary knowledge of calculus and linear algebra yet introduces advanced topics.<P>The book will be of interest to applied mathematicians and engineers and to students and researchers.