## A comparison of three high-precision quadrature schemes (2004)

### Cached

### Download Links

Venue: | Experimental Mathematics |

Citations: | 23 - 11 self |

### BibTeX

@ARTICLE{Bailey04acomparison,

author = {David H. Bailey and Karthik Jeyabalan and Xiaoye S. Li},

title = {A comparison of three high-precision quadrature schemes},

journal = {Experimental Mathematics},

year = {2004},

volume = {14},

pages = {317--329}

}

### OpenURL

### Abstract

The authors have implemented three numerical quadrature schemes, using the Arbitrary Precision (ARPREC) software package. The objective here is a quadrature facility that can efficiently evaluate to very high precision a large class of integrals typical of those encountered in experimental mathematics, relying on a minimum of a priori information regarding the function to be integrated. Such a facility is useful, for example, to permit the experimental identification of definite integrals based on their numerical values. The performance and accuracy of these three quadrature schemes are compared using a suite of 15 integrals, ranging from continuous, well-behaved functions on finite intervals to functions with infinite derivatives and blow-up singularities at endpoints, as well as several integrals on an infinite interval. In results using 412-digit arithmetic, we achieve at least 400-digit accuracy, using two of the programs, for all problems except one highly oscillatory function on an infinite interval. Similar results were obtained using 1012-digit arithmetic.

### Citations

834 | Numerical Recipes - Press, Flannery, et al. - 1986 |

385 |
Methods of numerical integration
- Davis, Rabinowitz
- 1984
(Show Context)
Citation Context ...sian quadrature. In the twentieth century, numerous additional schemes were devised, including extended Simpson rules, adaptive quadrature, Romberg integration, Clenshaw-Curtis integration and others =-=[10, 13]-=-. In addition, numerous “kernels” were devised that permit these schemes to efficiently compute definite integrals of functions that include a particular expression as a factor. Virtually all of the m... |

345 | An Introduction to Numerical Analysis - Atkinson - 1989 |

117 |
Calculation of Gauss quadrature rules
- Golub, Welsch
- 1969
(Show Context)
Citation Context ...me increases quadratically with n, since each Legendre polynomial evaluation requires n steps. The abscissa-weight pairs can alternately be computed using an eigenvector scheme due to Golub and Welch =-=[11]-=-, although this scheme requires considerably more memory, and its computational cost also increases quadratically with n. We know of no scheme for generating Gaussian abscissa-weight pairs that avoids... |

69 | A Fortran-90 based multiprecision system
- Bailey
- 1995
(Show Context)
Citation Context ...es we describe below have been implemented using the Arbitrary Precision (ARPREC) computation package [7]. This software is based in part on the multiprecision package MPFUN90 (written in Fortran-90) =-=[4]-=-, which in turn is based on the earlier MPFUN-77 package (written in Fortran-77) [3]. In MPFUN90, object-oriented facilities built into the Fortran-90 language, namely custom datatypes and operator ov... |

44 | Parallel Integer Relation Detection: Techniques and Applications
- Bailey, Broadhurst
(Show Context)
Citation Context ...value of a definite integral is given by a formula of a certain type with unknown integer or rational coefficients. The most frequently used integer relation detection algorithm is the PSLQ algorithm =-=[6]-=-. It and other integer relation schemes require very high precision (often hundreds or thousands of decimal digits) in both the input data and in the operation of the algorithm to obtain meaningful re... |

40 |
Developments in the Double Exponential Formulas for Numerical Integration
- Mori
- 1990
(Show Context)
Citation Context ...ly as p 2 log 2 p, within a range of practical precision, or in other words with the same scaling formula as with the other two methods. The tanh-sinh scheme was first introduced by Takahasi and Mori =-=[16, 14]-=-. 9s4. The Euler-Maclaurin Summation Formula The error function and tanh-sinh quadrature schemes are based on the Euler-Maclaurin summation formula, which implies that for certain bell-shaped integran... |

36 | ARPREC: An Arbitrary Precision Computation Package
- Bailey, Yozo, et al.
- 2002
(Show Context)
Citation Context ...ms that commonly arise in experimental mathematics. 2. The ARPREC Software The quadrature techniques we describe below have been implemented using the Arbitrary Precision (ARPREC) computation package =-=[7]-=-. This software is based in part on the multiprecision package MPFUN90 (written in Fortran-90) [4], which in turn is based on the earlier MPFUN-77 package (written in Fortran-77) [3]. In MPFUN90, obje... |

29 | Multiprecision translation and execution of FORTRAN programs
- Bailey
- 1993
(Show Context)
Citation Context ...mputation package [7]. This software is based in part on the multiprecision package MPFUN90 (written in Fortran-90) [4], which in turn is based on the earlier MPFUN-77 package (written in Fortran-77) =-=[3]-=-. In MPFUN90, object-oriented facilities built into the Fortran-90 language, namely custom datatypes and operator overloading, were exploited to permit Fortran programmers to utilize the MPFUN90 libra... |

23 | Topics in Advanced Scientific Computation - Crandall - 1996 |

21 | Highly Parallel, High-Precision Numerical Integration - Bailey, Robins |

16 | Integer relation detection - Bailey |

13 | Computational integration - Krommer, Überhuber - 1998 |

11 |
Private communication
- Kahan, Ivory
- 1997
(Show Context)
Citation Context ...considerably more memory, and its computational cost also increases quadratically with n. We know of no scheme for generating Gaussian abscissa-weight pairs that avoids this quadratic dependence on n =-=[12]-=-. For many wellbehaved integrand functions (as we shall see), Gaussian quadrature achieves quadratic convergence, meaning that doubling n achieves roughly twice as many correct digits in the quadratur... |

7 |
Definitely an integral
- Ahmed
(Show Context)
Citation Context .... As one example, recently one of the authors, together with Jonathan Borwein and Greg Fee of Simon Fraser University in Canada, were inspired by a recent problem in the American Mathematical Monthly =-=[1]-=-. They found by using one of the quadrature routines described in this paper, together with a PSLQ integer relation detection program, that if C(a) is defined by then C(a) = � 1 0 arctan( √ x 2 + a 2 ... |

2 |
On the evaluation of integrals related to the error function
- Chiarella, Reichel
- 1968
(Show Context)
Citation Context ...)e−(hj)2. To compute the error function erf(x), we use the following formula for erfc(t) = 1 − erf(t) given by Crandall [9, pg. 85] (who in turn attributes it to a 1968 paper by Chiarella and Reichel =-=[8]-=-): erfc(t) = e−t2αt π ⎛ ⎝ 1 � + 2 t2 k≥1 e −k2 α 2 k 2 α 2 + t 2 ⎞ ⎠ + 2 + E 1 − e2πt/α where t > 0 and |E| < 3e−π2 /α2 . The parameter α is chosen small enough to ensure that the error E is sufficien... |

1 | On the Evaluation of Integrals Related to the Error Function.” Mathematics of Computation 22 - Chiarella, Reichel - 1968 |