Newton’s Method and Gauss-Kronrod Quadrature

Weight functions admitting repeated positive Kronrod quadrature

“…Following [2] we call Kronrod extension of Qu(f;w), in notation 9u(f;w) = Q2u+I if;w), it is known that the Kronrod nodes ~*N must be the zeros of the unique monic polynomial 7r}+l(t ) = 7r}+l(t;w) of degree N + 1 (if it exists) satisfying the orthogonality property (see [2]) (1.3) S t--1 q(t)~*+ l(t)zru(t)w(t) dt = 0 for all q ~ PN, where Pu denotes the set of polynomials of degree at most N. Since 7rN+* 1 is orthogonal with respect to a sign-changing function, one can not expect in general that the zeros of ~z* + 1 are real; even the existence of 7r* + ~ is in doubt.…”

Calculation of Gauss-Kronrod quadrature rules

“…Conversely, if φ 2n+1 is to have φ n as a factor, then φ n−1 ψ n must be divisible by φ n since b n = b n , which is nonzero by (6). But φ n−1 and φ n are consecutive terms in a sequence of orthogonal polynomials and therefore mutually prime.…”

“…In this paper we are concerned with the efficient calculation of the nodes x i and weights w i of Gauss-Kronrod rules. Several methods for computing these formulas have been suggested [2,3,4,6,14,15,21,22] but most of them, as Gautschi puts it, compute the Gauss-Kronrod formula "piecemeal." That is to say, the new points and their weights are found by one method, and the new weights for the old points by another.…”

“…One of them [2,6] is based on solving by Newton's method the 3n + 2 equations that express the exactness of the quadrature rule. In [6] the authors remark about this method: " . .…”

Stieltjes polynomials and Lagrange interpolation

“…Among them, we mention Gautschi and Notaris [16], Gautschi and Rivlin [17], and the recent papers of Peherstorfer [34] and of the first author of this paper. For a more complete history of the problem under consideration, the interested reader may consult the exhaustive surveys of Gautschi [15] and Monegato [28].…”