Rapid Generation of Jacobi Matrices for Measures Modified by Rational Factors

Orthogonal polynomials are important throughout the fields of numerical analysis and numerical linear algebra. The Jacobi matrix J for a family of n orthogonal polynomials is an n x n tridiagonal symmetric matrix constructed from the recursion coefficients for the three-term recurrence satisfied by the family. Every family of polynomials orthogonal with respect to a measure on a real interval [a,b] satisfies such a recurrence. Given a measure that is modified by multiplying by a rational weight function r(t), an important problem is to compute the modified Jacobi matrix Jmod corresponding to the new measure from knowledge of J. There already exist efficient methods to accomplish this when r(t) is a polynomial, so we focus on the case where r(t) is the reciprocal of a polynomial. Working over the field of real numbers, this means considering the case where r(t) is the reciprocal of a linear or irreducible quadratic factor, or a product of such factors. Existing methods for this type of modification are computationally expensive. Our goal is to develop a faster method based on inversion of existing procedures for the case where r(t) is a polynomial. The principal challenge in this project is that this inversion requires working around missing information. This can be accomplished by treating this information as unknown parameters and making guesses that can be corrected iteratively