Kronrod-Patterson quadrature formula

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A quadrature formula of highest algebraic accuracy of the type

, where are the nodes of a Gauss quadrature formula and the nodes of are fixed in the construction of [a2]. Nested sequences of Kronrod–Patterson formulas are used for the numerical approximation of definite integrals with practical error estimate, in particular in the non-adaptive routines of the numerical integration package QUADPACK [a4] and in the standard numerical software libraries.

The algebraic accuracy of is at least . The free nodes of are precisely the zeros of the polynomial which satisfies

where is the system of orthogonal polynomials associated with , is the Gauss–Kronrod quadrature formula, and is the Stieltjes polynomial (cf. Stieltjes polynomials). For and , , the Chebyshev polynomial of the second kind (cf. Chebyshev polynomials), and , the Chebyshev polynomial of the first kind. In this case, all Kronrod–Patterson formulas are Gauss quadrature formulas (cf. Gauss quadrature formula). Hence, the algebraic accuracy of is , the nodes of and interlace and the formulas have positive weights. Similar properties are known for the more general Bernstein–Szegö weight functions , where is a polynomial of degree which is positive on , see [a3].

Only very little is known for , which is the most important case for practical calculations. Tables of sequences of Kronrod–Patterson formulas have been given in [a2], [a4]. A numerical investigation for and Jacobi weight functions , , can be found in [a5].


[a1] P.J. Davis, P. Rabinowitz, "Methods of numerical integration" , Acad. Press (1984) (Edition: Second)
[a2] T.N.L. Patterson, "The optimum addition of points to quadrature formulae" Math. Comput. , 22 (1968) pp. 847–856
[a3] F. Peherstorfer, "Weight functions admitting repeated positive Kronrod quadrature" BIT , 30 (1990) pp. 241–251
[a4] R. Piessens, et al., "QUADPACK: a subroutine package in automatic integration" , Springer (1983)
[a5] P. Rabinowitz, S. Elhay, J. Kautsky, "Empirical mathematics: the first Patterson extension of Gauss–Kronrod rules" Internat. J. Computer Math. , 36 (1990) pp. 119–129
How to Cite This Entry:
Kronrod-Patterson quadrature formula. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by Sven Ehrich (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article