Empirical mathematics: the first patterson extension of gauss-kronrod rules

The Kronrod extension (KE) of an n-point Gauss integration rule with respect to some weight w is a (2n+l)-point rule of optimal precision which consists of the n Gauss points and n+1 additional distinct real points contained in the integration interval. KE's may or may not exist depending on w and n. For a KE which exists, we define the first Patterson extension (PE) to be a similar optimal extension of the KE. It contains at least 3n+3 points, in which case it is said to be minimal. We give here some experimental results on the computation of PE's and make some conjectures about the existence of minimal and nonminimal PE's when w is a Jacobi weight.

[1]  Giovanni Monegato,et al.  Nonexistence of extended Gauss-Laguerre and Gauss-Hermite quadrature rules with positive weights , 1978 .

[2]  T. Patterson,et al.  The optimum addition of points to quadrature formulae. , 1968 .

[3]  T. J. Rivlin,et al.  A family of Gauss-Kronrod quadrature formulae , 1988 .

[4]  G. Szegö,et al.  Über gewisse orthogonale Polynome, die zu einer oszillierenden Belegungsfunktion gehören , 1935 .

[5]  Thomas N. L. Patterson,et al.  An algorithm for generating interpolatory quadrature rules of the highest degree of precision with preassigned nodes for general weight functions , 1989, TOMS.

[6]  Walter Gautschi,et al.  On the computing Gauss-Kronrod Quadrature Formulae , 1986 .

[7]  Sotirios E. Notaris.  Gauss-Kronrod quadrature formulae for weight functions of Bernstein-Szego¨ type, II , 1990 .

[8]  Giovanni Monegato.  An overview of results and questions related to Kronrod schemes , 1979 .

[9]  Thomas N. L. Patterson,et al.  Algorithm 468: algorithm for automatic numerical integration over a finite interval [D1] , 1973, CACM.

[10]  Jaroslav Kautsky,et al.  Gauss quadratures and Jacobi matrices for weight functions not of one sign , 1984 .

[11]  G. Monegato.  A note on extended Gaussian quadrature rules , 1976 .

[12]  Giovanni Monegato,et al.  Stieltjes Polynomials and Related Quadrature Rules , 1982 .

[13]  Sotirios E. Notaris,et al.  An algebraic study of Gauss-Kronrod quadrature formulae for Jacobi weight functions , 1988 .

[14]  Giovanni Monegato,et al.  Positivity of the weights of extended Gauss-Legendre quadrature rules , 1978 .

[15]  P. Rabinowitz.  On the definiteness of Gauss-Kronrod integration rules , 1986 .

[16]  Philip Rabinowitz,et al.  Gauss-Kronrod integration rules for Cauchy principal value integrals , 1983 .

[17]  Philip Rabinowitz,et al.  The exact degree of precision of generalized Gauss-Kronrod integration rules , 1980 .