## Fully Symmetric Interpolatory Rules for Multiple Integrals over Infinite Regions with Gaussian Weight (1996)

Citations: | 16 - 3 self |

### BibTeX

@MISC{Genz96fullysymmetric,

author = {Alan Genz and B. D. Keister},

title = {Fully Symmetric Interpolatory Rules for Multiple Integrals over Infinite Regions with Gaussian Weight},

year = {1996}

}

### Years of Citing Articles

### OpenURL

### Abstract

Fully symmetric interpolatory integration rules are constructed for multidimensional integrals over infinite integration regions with a Gaussian weight function. The points for these rules are determined by successive extensions of the one dimensional three point Gauss-Hermite rule. The new rules are shown to be efficient and only moderately unstable.

### Citations

332 |
Methods of Numerical Integration
- Davis, Rabinowitz
- 1997
(Show Context)
Citation Context ...iformly from U n , then r m (f; z) is an unbiased degree 2m+1 stochastic rule for I(f ). This method for randomizing a polynomial rule is a type of \control variates" method (see Davis and Rabino=-=witz [-=-1], p. 389) for reducing variance. In many cases the sum EN (f) + Q (m;n) (f) will be a better estimate for I(f) than Q (m;n) (f ). A simple heuristic for these cases is to compare of SN (f) with EN... |

197 |
Approximate Calculation of Multiple Integrals
- Stroud
- 1973
(Show Context)
Citation Context ... 2 1 + z 2 2 + z 2 n = 1g, and where is an element of surface on U n . This is an important problem in pure and applied science, which has been studied by various authors. The books by Stroud [7] and Mysovskikh [5] both contain a number of formulas, and the paper by Keast and Diaz [4] provides a general method for constructing fully symmetric rules. Recent work by Xu [9] provides some fully s... |

93 |
The Efficient Generation of Random Orthogonal Matrices with an Application to Conditional Estimators
- Stewart
- 1980
(Show Context)
Citation Context ...;n) (f; Z) by Q (m;n) N (f) = 1 N N X i=1 Q (m;n) (f; Z i ): If random orthogonal matrices Z i are generated with Haar distribution from the set of all matrices in the orthogonal group (see Stewart [=-=6], 1-=-980), then the \stochastic" rule Q (m;n) N (f) is an unbiased degree 2m+ 1 estimate for I(f ). A robust unbiased degree 2m+ 1 error estimate Q (m;n) N (f) is provided by the Monte Carlo standar... |

35 | Numerical Quadrature and Cubature - Engels - 1980 |

29 | The optimum addition of points to quadrature formulae - Patterson - 1968 |

28 | Orthogonal polynomials and cubature formulae on spheres and on simplices, Methods Anal
- Xu
- 1998
(Show Context)
Citation Context .... The books by Stroud [7] and Mysovskikh [5] both contain a number of formulas, and the paper by Keast and Diaz [4] provides a general method for constructing fully symmetric rules. Recent work by Xu =-=[9]-=- provides some fully symmetric rules with explicit formulas for the rule weights. The purpose of this paper is to show how to modify a method for construction of the numerical integration rules descri... |

12 | Nodes and Weights of Quadrature Formulas - KRONROD - 1965 |

11 | Multidimensional quadrature algorithms at higher degree and/or dimension - Capstick, Keister - 1996 |

9 | The Nuclear Shell Model - Heyde - 1990 |

7 |
Stochastic quadrature formulas
- HABER
- 1969
(Show Context)
Citation Context ... EN (f) = 1 N(N 1) N X i=1 (Q (m;n) (f; Z i ) Q (m;n) N (f)) 2 1 2 : 4.2 Polynomial Model Randomizations for Q (m;n) Rules The method described in this section was initially developed by Haber ([3], 1969) for interpolatory rules and generalized by Genz ([2], 1998) for fully symmetric interpolatory rules. Dene e m (f; z) = ffzg M (m;n) (f; z); for a point z 2 U n . If f(z) = z k and jkj 2m + ... |

7 | Gauss-Kronrod quadrature – a survey, in Numerical Methods and Approximation Theory - Gautschi - 1988 |

6 | Nonexistence of extended Gauss-Laguerre and Gauss-Hermite quadrature rules with positive weights - Kahaner, Monegato - 1978 |

5 |
Fully symmetric integration formulas for the surface of the sphere in s dimensions
- Keast, Diaz
- 1983
(Show Context)
Citation Context ...important problem in pure and applied science, which has been studied by various authors. The books by Stroud [7] and Mysovskikh [5] both contain a number of formulas, and the paper by Keast and Diaz =-=[4]-=- provides a general method for constructing fully symmetric rules. Recent work by Xu [9] provides some fully symmetric rules with explicit formulas for the rule weights. The purpose of this paper is t... |

4 | Symmetric quadrature formulae for simplexes - Sylvester - 1970 |

4 | An Imbedded Family of Cubature Formulas for n-Dimensional Product - Cools, Haegemans - 1994 |

3 | Stochastic methods for multiple integrals over unbounded regions
- Genz
- 1998
(Show Context)
Citation Context ...omial degree of precision is greater than 7. The rule construction method also allows the construction of two types of randomized rules, using methods previously described by the present author (Genz [2]) for randomized rules over R n with Gaussian weight. 2 Interpolatory Rules for U n Let T n 1 be the n 1-simplex dened by T n 1 = fx j x 2 R n 1 ; 0 x 1 +x 2 +x n 1 1g; and, for any x 2 T n... |

2 | Some Integration Strategies for - Evans, Swartz - 1992 |

2 | Atomic and Molecular Orbital Theory - Offenhartz |

1 | Construction of Fully Symmetric - McNamee, Stenger - 1967 |