### Abstract

We define and discuss the properties of manifolds of polynomials J_{n}(t, x) and R_{n}(t, x), called Rys polynomials, which are orthonormal with respect to the weighting factor exp(-xt^{2}) on a finite interval of t. Numerical quadrature based on Rys polynomials provides an alternative approach to the computation of integrals commonly encountered in molecular quantum mechanics. This gives rise to a curve fitting problem for the roots and quadrature weights as a function of the x parameter. We have used Chebyshev approximation for small x and an asymptotic expansion for large x. A modified Christoffel-Darboux equation applicable to Rys polynomials is derived and used to obtain alternative formulas for Rys quadrature weight factors.

Original language | English |
---|---|

Pages (from-to) | 144-165 |

Number of pages | 22 |

Journal | Journal of Computational Physics |

Volume | 21 |

Issue number | 2 |

DOIs | |

Publication status | Published - 1976 |

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### ASJC Scopus subject areas

- Computer Science Applications
- Physics and Astronomy(all)

### Cite this

*Journal of Computational Physics*,

*21*(2), 144-165. https://doi.org/10.1016/0021-9991(76)90008-5

**Numerical integration using rys polynomials.** / King, Harry F.; Dupuis, Michel.

Research output: Contribution to journal › Article

*Journal of Computational Physics*, vol. 21, no. 2, pp. 144-165. https://doi.org/10.1016/0021-9991(76)90008-5

}

TY - JOUR

T1 - Numerical integration using rys polynomials

AU - King, Harry F.

AU - Dupuis, Michel

PY - 1976

Y1 - 1976

N2 - We define and discuss the properties of manifolds of polynomials Jn(t, x) and Rn(t, x), called Rys polynomials, which are orthonormal with respect to the weighting factor exp(-xt2) on a finite interval of t. Numerical quadrature based on Rys polynomials provides an alternative approach to the computation of integrals commonly encountered in molecular quantum mechanics. This gives rise to a curve fitting problem for the roots and quadrature weights as a function of the x parameter. We have used Chebyshev approximation for small x and an asymptotic expansion for large x. A modified Christoffel-Darboux equation applicable to Rys polynomials is derived and used to obtain alternative formulas for Rys quadrature weight factors.

AB - We define and discuss the properties of manifolds of polynomials Jn(t, x) and Rn(t, x), called Rys polynomials, which are orthonormal with respect to the weighting factor exp(-xt2) on a finite interval of t. Numerical quadrature based on Rys polynomials provides an alternative approach to the computation of integrals commonly encountered in molecular quantum mechanics. This gives rise to a curve fitting problem for the roots and quadrature weights as a function of the x parameter. We have used Chebyshev approximation for small x and an asymptotic expansion for large x. A modified Christoffel-Darboux equation applicable to Rys polynomials is derived and used to obtain alternative formulas for Rys quadrature weight factors.

UR - http://www.scopus.com/inward/record.url?scp=0001142721&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0001142721&partnerID=8YFLogxK

U2 - 10.1016/0021-9991(76)90008-5

DO - 10.1016/0021-9991(76)90008-5

M3 - Article

AN - SCOPUS:0001142721

VL - 21

SP - 144

EP - 165

JO - Journal of Computational Physics

JF - Journal of Computational Physics

SN - 0021-9991

IS - 2

ER -