The spectral connection matrix for classical real orthogonal polynomials

Sets of orthogonal polynomials are bases for polynomial spaces Pn. As a result, polynomials can be expressed in coefficients relative to a particular family of orthogonal polynomials. The connection problem refers to the task of converting from coefficients in one of these bases to coefficients in another. The entries of the matrix that applies such a change of basis, known as the connection coefficients, are well-known values that can be computed via direct computation or matrix inversion; however this can be computationally expensive. Thus their accurate and efficient computation is a relevant topic of research in numerical linear algebra, and can be found in the current literature. The two manuscripts included in this thesis address the connection problem. In the first manuscript, a connection within the classical real orthogonal polynomials of a single parameter (Hermite, Laguerre,and Gegenbauer) is discussed. The spectral connection matrix related to a connection matrix is defined. It is also shown that this spectral connection matrix in each case within the singleparameter classical families is quasiseparable, with specific generators provided. Additionally this manuscript proposes an algorithm that efficiently computes the desired connection matrix given the generators of its corresponding spectral connection matrix. The second manuscript dramatically generalizes the result of the first. It addresses the structure of the spectral connection matrix associated with a much broader group of connections. The target family is allowed to be any of the classical types, including Jacobi. The source family is allowed to be any of the classical types or Bessel, which is not considered classical here. In these cases it is shown that once again the spectral connection matrix is quasiseparable, and specific generators are provided. The algorithm from the first manuscript allows for the efficient computation of the desired connection matrix given the generators of the associated spectral connection matrix. The appendix at the conclusion provides some details for the reader’s reference. It begins with a review of orthogonal polynomials, and highlights the classical types. It then provides a review of some basic linear algebra concepts that are relevant to the manuscripts, and concludes with a survey of quasiseparable matrices. The appendix also references research activity in the field.