Generalized Kronrod Patterson type imbedded quadratures

Sylvan Elhay; Jaroslav Kautský

Applications of Mathematics (1992)

  • Volume: 37, Issue: 2, page 81-103
  • ISSN: 0862-7940

Abstract

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We present algorithms for the determination of polynomials orthogonal with respect to a positive weight function multiplied by a polynomial with simple roots inside the interval of integration. We apply these algorithms to search for and calculate all possible sequences of imbedded quadratures of maximal polynomials order of precision for the generalized Laguerre and Hermite weight functions.

How to cite

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Elhay, Sylvan, and Kautský, Jaroslav. "Generalized Kronrod Patterson type imbedded quadratures." Applications of Mathematics 37.2 (1992): 81-103. <http://eudml.org/doc/15702>.

@article{Elhay1992,
abstract = {We present algorithms for the determination of polynomials orthogonal with respect to a positive weight function multiplied by a polynomial with simple roots inside the interval of integration. We apply these algorithms to search for and calculate all possible sequences of imbedded quadratures of maximal polynomials order of precision for the generalized Laguerre and Hermite weight functions.},
author = {Elhay, Sylvan, Kautský, Jaroslav},
journal = {Applications of Mathematics},
keywords = {Gaussian quadrature; orthogonal polynomials; Kronrod extensions; Patterson sequences; imbedded quadratures; Laguerre weight function; Hermite weight function; quadrature formulae; maximal polynomial order of precision; Gaussian quadrature; orthogonal polynomials; Kronrod extensions; Patterson sequences; imbedded quadratures; Laguerre weight function; Hermite weight function; quadrature formulae; maximal polynomial order of precision},
language = {eng},
number = {2},
pages = {81-103},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Generalized Kronrod Patterson type imbedded quadratures},
url = {http://eudml.org/doc/15702},
volume = {37},
year = {1992},
}

TY - JOUR
AU - Elhay, Sylvan
AU - Kautský, Jaroslav
TI - Generalized Kronrod Patterson type imbedded quadratures
JO - Applications of Mathematics
PY - 1992
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 37
IS - 2
SP - 81
EP - 103
AB - We present algorithms for the determination of polynomials orthogonal with respect to a positive weight function multiplied by a polynomial with simple roots inside the interval of integration. We apply these algorithms to search for and calculate all possible sequences of imbedded quadratures of maximal polynomials order of precision for the generalized Laguerre and Hermite weight functions.
LA - eng
KW - Gaussian quadrature; orthogonal polynomials; Kronrod extensions; Patterson sequences; imbedded quadratures; Laguerre weight function; Hermite weight function; quadrature formulae; maximal polynomial order of precision; Gaussian quadrature; orthogonal polynomials; Kronrod extensions; Patterson sequences; imbedded quadratures; Laguerre weight function; Hermite weight function; quadrature formulae; maximal polynomial order of precision
UR - http://eudml.org/doc/15702
ER -

References

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  1. S. Elhay, J. Kautsky, A method for computing quadratures of the Kronrod-Patterson type, Australian Computer Science Communications 6 (1) (1984), 15.1-15.8. (1984) 
  2. G. H. Golub, J. Kautsky, 10.1007/BF01390210, Numer. Math. 41 (1983), 147-163. (1983) Zbl0525.65010MR0703119DOI10.1007/BF01390210
  3. D. K. Kahaner J. Waldvogel, and L. W. Fullerton, 10.1137/0905003, SIAM J. Sci. Stat. Comput. 5 (1984), 42-55. (1984) MR0731880DOI10.1137/0905003
  4. J. Kautsky, S. Elhay, Gauss quadratures and Jacobi matrices for weight functions not of one sign, Math. Соmр. 43 (168) (1984), 543-550. (1984) Zbl0556.65012MR0758201
  5. A. S. Kronrod, Nodes and weights for quadrature formulae. Sixteen place tables, Nauka, Moscow; English translation, Consultants Bureau, New York. MR0183116
  6. G. Monegato, 10.1137/1024039, SIAM Review 24(2) (1982), 137-158. (1982) Zbl0494.33010MR0652464DOI10.1137/1024039
  7. T. N. L. Patterson, The optimal addition of points to quadrature formulae, Math. Соmр. 22 (1968), 847-856. (1968) MR0242370
  8. R. Piessens, M. Branders, A note on the optimal addition of abscissae to quadrature formulae of the Gauss-Lobatto type, Math. Соmр. 28 (1974), 135-139. (1974) MR0343552
  9. P. Turan, On the theory of mechanical quadrature, Acta. Sci. Math. (Szeged) 12 (1950), 30-37. (1950) MR0036797

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