Kronrod-Patterson quadrature formula
A quadrature formula of highest algebraic accuracy of the type
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, 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
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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].
References
[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 |
Kronrod-Patterson quadrature formula. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Kronrod-Patterson_quadrature_formula&oldid=50288