View Full Version : Special Functions (Hypergeometric)

physicist

09-25-2007, 08:08 AM

Are the Hypergeometric Functions ("qFp") being planned to be included in IMSL

Version 6? (p,q = 1,1;1,2;2,1;2,2; ...)

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[Of course, there are many special cases and they would generalise quite a few

of the special functions already included in and a traditional stronghold of IMSL. They are included in Mathematica, and Maple (not compiled languages) and some of it is contained in numerical recipes.]

Some references (perhaps not the newest ones) are:

(1) N.M. Temme,The numerical computation of the confluent hypergeometric function U(a, b, z), Springer-Verlag (1983).

(2) Perger, Bhalla, Nardin: A numerical evaluator for the generalized hypergeometric series, preprint, available on internet, also: ACM Trans. on Math. Software 18 (1992) 345-349.

(3) The overview made by Lozier and Oliver at NIST (1994-2000).

All of the IMSL libraries have the Hypergeometric distribution (CDF and PDF) and a Hypergeometric random distribution. In IMSL Fortran, these are HYPDF, HYPPR and RNHYP. I think what you're looking for would fall into the "Special Functions" category, but isn't currently in the library. I will file an enhancement request.

physicist

10-01-2007, 07:28 AM

Dear moderator,

Thank you for your reply. The nice thing is, if it would be programmed in some generality, it can be compared with many of the special functions already available in IMSL.

And perhaps also even cross-checked against a ('singularity-robust') numerical

differential equation solver and a numerical integration routine.

Anyhow, two further (traditional) references are:

W. Magnus, F. Oberhettinger, R.P. Soni: Formulas and Theorems for the

Special Functions of Mathematical Physics, Springer-Verlag, 1966 pp. 37-65.

Z.X. Wang and D.R. Guo (Peking University), Special Functions, World Scientific 1989, pp. 135-202 and 296-335.

With best regards,

physicist

physicist

10-01-2007, 07:31 AM

Dear moderator,

Thank you for your reply. The nice thing is, if it would be programmed in some generality, then it can be compared with many of the special functions already available in IMSL.

And perhaps also even cross-checked against a ('singularity-robust') numerical

differential equation solver and a numerical integration routine.

The singularity and/or rounding error accumulation may require double/extended precision (besides a clever program design).

Anyhow, two further (traditional) references are:

W. Magnus, F. Oberhettinger, R.P. Soni: Formulas and Theorems for the

Special Functions of Mathematical Physics, Springer-Verlag, 1966 pp. 37-65.

Z.X. Wang and D.R. Guo (Peking University), Special Functions, World Scientific 1989, pp. 135-202 and 296-335.

With best regards,

physicist

physicist

11-02-2007, 06:18 AM

Dear IMSL forum,

Some additional remarks.

(a) The suggestion is NOT to include basic hypergeometric functions as well, see [1].

(b) The log of Barnes G-function would be a nice addition to the gamma function, I think.

(c) Even modern wikipedia gives a nice overview of the

relationship between the hypergeometric functions

and many standard special functions.

(d) It is easier to suggest the project than to carry it through.

With best regards,

physicist

[1] Gasper and Rahman, Cambridge University Press, 2004 (2nd ed.).

Thanks for your detailed feedback -- this kind of information is great to have as we develop future versions of IMSL!

physicist

08-01-2008, 06:03 AM

Dear Moderator,

Thank you for welcoming the suggestion to include hypergeometric functions in

future version of IMSL.

Version 6.5?

With best regards,

physicist

P.S.: pFq (in one routine) may be too difficult to implement.

physicist

10-01-2008, 10:55 AM

withdrawn (see next e-mail)

physicist

10-01-2008, 10:57 AM

Dear Moderator,.

A brute force approach would be to file a number of special cases,

expressed in special functions that already exist in IMSL (such as

BesselI, BesselK, Dawson).

For instance: 1F1(0,k;x) for k=1,...,p=12 and 1F1(m,0;x) for m=1,...,n=12,

and then use a continuous fraction expansion for the ratio

1F1(a,b;x)/1F1(a+1,b+1;x), or its inverse. [With or without Richardson-type

extrapolation, as speed is sometimes secondary to convergence safety.]

Finally find something to interpolate for non half-integer values, and issue

warnings if x is (too) close to a negative integer.

...

Not very elegant from a theoretical point of view, perhaps, but somewhat

similar to the (past) experience with programming the Beta function.

With best regards,

physicist

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