Complex Jacobi matrices and quadrature rules

Abstract

Given any sequence of orthogonal polynomials, satisfying the three term recurrence relation xpn(x) = βn+1pn+1(x) + αnpn(x) + βnpn−1(x), p−1(x) = 0, p0(x) = 1, with βn = 0, n ∈ N, β0 = 1, an infinite Jacobi matrix can be associated in the following way J =

The Lanczos algorithm and complex Gauss quadrature

“…Given a linear functional L on the space of sufficiently smooth functions, consider the quadrature of the form (see [12,Chapter 5], [45,Section 2], and [49, Section 7])…”

“…where the elements of T n are marked with a hat. The matching moment property in Theorem 1.4 can also be interpreted as a matrix formulation of a generalized Gauss quadrature for the approximation of quasi-definite linear functionals; see [45,49]. Moreover, given the matrix A and the vectors v and w with the associated quasi-definite linear functional defined by (1.6), the matrix T n can be determined, assuming no breakdown, by the non-Hermitian Lanczos algorithm.…”

“…A linear functional (1.1) with real moments can be naturally restricted to the space of polynomials with real coefficients R ⊂ P. If L is quasi-definite, then we can construct real monic polynomials orthogonal with respect to L with the corresponding real tridiagonal matrix T n satisfying the matching moment property (1.5). In Chapter 5 of the book [12] published in 1983, Draux introduced a generalization of the Gauss quadrature formula for approximating the real-valued linear functionals.The associated results in [45,49], obtained independently of those from [12], can be considered as straightforward generalizations to the complex case. Some results in [49] do not have, however, a straightforward real setting analogue in [12].…”

“…, (A * ) n−1 w} to the Stieltjes procedure for generating orthonormal polynomials. If n is the maximal number of steps that can be performed in the Lanczos algorithm without breakdown, then there exists no complex Gauss quadrature in the sense of [45,49] for approximating the functional (1.6) with more than n weights. This is presented in Section 3.…”

The Gauss quadrature for general linear functionals, Lanczos algorithm, and minimal partial realization

“…More precisely, a straightforward extension of Draux's definition to the case of complex-valued linear functionals will be presented. The more recent Gauss quadrature definitions in [68] and in [75,76], obtained independently of [22], can be seen as a generalization to the complex quasi-definite case. Indeed, for a real quasi-definite linear functional the quadratures in [22,68,75,76] are equivalent.…”