A Fast Algorithm for Hankel Matrices Represented in Orthogonal Polynomial Bases

Gemignani, Luca (1996) A Fast Algorithm for Hankel Matrices Represented in Orthogonal Polynomial Bases. Technical Report del Dipartimento di Informatica . Università di Pisa, Pisa, IT.

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Abstract

Consider a $n \times n$ lower triangular matrix $L$ whose $(i+1)$-st row is defined by the coefficients of the real polynomial $p_i(x)$ of degree $i$ such that $\{ p_i(x)\}$ is a set of orthogonal polynomials satisfying a standard three-term recurrence relation. If $H$ is a $n\times n$ real Hankel matrix with nonsingular leading principal submatrices, then $\widehat{H}=L HL^T$ will be referred as a strongly nonsingular Hankel matrix with respect to the orthogonal polynomial basis $\{p_i(x)\}$. In this paper we develop an efficient $O(n^2)$ algorithm for the solution of a system of linear equations with a real symmetric coefficient matrix $\widehat{H}$ which is a Hankel matrix with respect to a suitable orthogonal polynomial basis. We then apply our method to the solution of a weighted finite moment problem.

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