Numerical evaluation of special functions

Lozier, Daniel W.; Gautschi, Walter

doi:10.1090/psapm/048/1314844

Cited by 39 publications

(28 citation statements)

References 443 publications

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“…Besides being unfamiliar to most marketing researchers, multiple evaluations of the Gaussian hypergeometric are very demanding from a computational standpoint. Furthermore, the precision of some numerical procedures used to evaluate this function can vary substantially over the parameter space (Lozier and Olver 1995); this can cause major problems for numerical optimization routines as they search for the maximum of the likelihood function.…”

Section: The Pareto/nbd Modelmentioning

confidence: 99%

“Counting Your Customers” the Easy Way: An Alternative to the Pareto/NBD Model

Fader

Hardie

Lee³

2005

Marketing Science

353

237

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"Counting Your Customers" the Easy Way: An Alternative to the Pareto/NBD ModelToday's managers are very interested in predicting the future purchasing patterns of their customers, which can then serve as an input into "lifetime value" calculations. Among the models that provide such capabilities, the Pareto/NBD "Counting Your Customers" framework proposed by Schmittlein, Morrison, and Colombo (1987) is highly regarded. But despite the respect it has earned, it has proven to be a difficult model to implement, particularly because of computational challenges associated with parameter estimation.We develop a new model, the beta-geometric/NBD (BG/NBD), which represents a slight variation in the behavioral "story" associated with the Pareto/NBD, but it is vastly easier to implement. We show, for instance, how its parameters can be obtained quite easily in Microsoft Excel. The two models yield very similar results in a wide variety of purchasing environments, leading us to suggest that the BG/NBD could be viewed as an attractive alternative to the Pareto/NBD in most applications.

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Section: The Pareto/nbd Modelmentioning

confidence: 99%

“Counting Your Customers” the Easy Way: An Alternative to the Pareto/NBD Model

Fader

Hardie

Lee³

2005

Marketing Science

353

237

View full text Add to dashboard Cite

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“…• For web links to software packages for evaluating special functions we refer to the repository GAMS: Guide to Available Mathematical Software, http://gams.nist.gov/ • In 1994 a complete survey of the available software was published: Lozier and Olver (1994 (Baker 1992, Moshier 1989, Press, Teukolsky, Vetterling and Flannery 2002, Thompson 1997, Wang and Guo 1989, Zhang and Jin 1996.…”

Section: Software For Computing Special Functionsmentioning

confidence: 99%

Numerical Aspects of Special Functions

Temme

1996

Special Functions

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C e n t r u m v o o r W i s k u n d e e n I n f o r m a t i c a Modelling, Analysis and Simulation Modelling, Analysis and SimulationNumerical aspects of special functions N.M. Temme Numerical aspects of special functions ABSTRACT This paper describes methods that are important for the numerical evaluation of certain functions that frequently occur in applied mathematics, physics and mathematical statistics. This includes what we consider to be the basic methods, such as recurrence relations, series expansions (both convergent and asymptotic), and numerical quadrature. Several other methods are available and some of these will be discussed in less detail. Examples will be given on the use of special functions in certain problems from mathematical physics and mathematical statistics (integrals and series with special functions). REPORT MAS-E0711 JULY 2007 Mathematics Subject Classification: 65D20

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“…A survey on computational aspects of special functions, including information on Airy functions, can be found in [8]; [10] has a public web site that includes an extensive treatment of Scorer functions. For complex values of z the Airy functions are available in the Bessel function algorithms of [2]; see also [3] and [14].…”

Section: Introductionmentioning

confidence: 99%

On nonoscillating integrals for computing inhomogeneous Airy functions

2000

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Abstract. Integral representations are considered of solutions of the inhomogeneous Airy differential equation w − z w = ±1/π. The solutions of these equations are also known as Scorer functions. Certain functional relations for these functions are used to confine the discussion to one function and to a certain sector in the complex plane. By using steepest descent methods from asymptotics, the standard integral representations of the Scorer functions are modified in order to obtain nonoscillating integrals for complex values of z. In this way stable representations for numerical evaluations of the functions are obtained. The methods are illustrated with numerical results.

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Numerical evaluation of special functions

Cited by 39 publications

References 443 publications

“Counting Your Customers” the Easy Way: An Alternative to the Pareto/NBD Model

“Counting Your Customers” the Easy Way: An Alternative to the Pareto/NBD Model

Numerical Aspects of Special Functions

On nonoscillating integrals for computing inhomogeneous Airy functions

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