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**D. W. Lozier and F. W. J. Olver**

** 5.6. Incomplete Bessel Functions, Incomplete Beta Function
. **

This section includes F--, t-- and von Mises' distribution functions.

5.6.1. * Incomplete Bessel Functions. *

Software Packages:

[** Hil77**, Fortran]
.

5.6.2. * Incomplete Beta Function. *

Software Packages:

[** DM92**, Fortran]
,
[** Dor68**, Algol]
,
[** Gau64**, Algol]
,
[** Hil70a**, Algol]
,
[** Lev69**, Fortran]
,
[** MB73a**, Fortran]
,
[** Mor69**, Algol]
,
[** Phi90**, Basic]
.

Intermediate Libraries:

[** Bak92**]
,
[** Mos89**]
,
[** ULI90**]
.

Comprehensive Libraries:

** IMSL**,
** NAG**,
** Numerical Recipes**,
** Scientific Desk**,
** SLATEC**.

Interactive Systems:

** Mathematica**.

5.6.3. * Inverse Incomplete Beta Function. *

Algorithms:

[** MB73b**]
.

Software Packages:

[** AS93a**, Fortran]
,
[** Hil70b**, Algol]
.

Intermediate Libraries:

[** Mos89**]
,
[** ULI90**]
.

Comprehensive Libraries:

** IMSL**,
** NAG**,
** Scientific Desk**.

Interactive Systems:

** Maple**,
** Mathematica**.

5.6.4. * Articles. *

[** AS93b**]
,
[** DJ67**]
,
[** OM68**]
,
[** Tem92b**]
.

**AS93a**-
R. W. Abernathy and R. P. Smith,
*Algorithm 724. Program to calculate*, ACM Trans. Math. Software**F**--percentiles**19**(1993), 481--483. **AS93b**-
R. W. Abernathy and R. P. Smith,
*Applying series expansion to the inverse beta distribution to find percentiles of the*, ACM Trans. Math. Software**F**--distribution**19**(1993), 474--480. **Bak92**-
L. Baker,
*C mathematical function handbook*, McGraw-Hill, Inc., New York, 1992, includes diskette. **DJ67**-
A. R. DiDonato and M. P. Jarnagin,
*The efficient calculation of the incomplete beta-function ratio for half-integer values of the parameters*, Math. Comp.**a,b****21**(1967), 652--662. **DM92**-
A. R. DiDonato and A. H. Morris, Jr.,
*Algorithm 708. Significant digit computation of the incomplete beta function ratios*, ACM Trans. Math. Software**18**(1992), 360--373. **Dor68**-
E. Dorrer,
*Algorithm 322.*, Comm. ACM**F**--distribution**11**(1968), 116--117, for certification see same journal v. 12 (1969), p. 39. **Gau64**-
W. Gautschi,
*Algorithm 222. Incomplete beta function ratios*, Comm. ACM**7**(1964), 143--144, for certification see same journal v. 7 (1964), p. 244. **Hil70a**-
G. W. Hill,
*Algorithm 395. Student's*, Comm. ACM**t**--distribution**13**(1970), 617--619, for remarks see ACM Trans. Math. Software v. 5 (1979), pp. 238--239, v. 7 (1981), pp. 247--249. **Hil70b**-
G. W. Hill,
*Algorithm 396. Student's*, Comm. ACM**t**--quantiles**13**(1970), 619--620, for remarks see ACM Trans. Math. Software v. 5 (1979), pp. 238--239, v. 7 (1981), pp. 250--251. **Hil77**-
G. W. Hill,
*Algorithm 518. Incomplete Bessel function*, ACM Trans. Math. Software*:*The von Mises distribution**3**(1977), 279--284. **Lev69**-
D. A. Levine,
*Algorithm 344. Student's*, Comm. ACM**t**--distribution**12**(1969), 37--38, for remarks see same journal v. 13 (1970), pp. 124 and 449. **MB73a**-
K. L. Majumder and G. P. Bhattacharjee,
*Algorithm AS 63. The incomplete beta integral*, Appl. Statist.**22**(1973), 409--411, for remark see same journal v. 26 (1977), pp. 111--114. **MB73b**-
K. L. Majumder and G. P. Bhattacharjee,
*Algorithm AS 64. Inverse of the incomplete beta function ratio*, Appl. Statist.**22**(1973), 411--414, for remarks and corrections see same journal v. 26 (1977), pp. 111--114, v. 39 (1990), pp. 309--310, v. 40 (1991), p. 236. **Mor69**-
J. Morris,
*Algorithm 346.*, Comm. ACM**F**--test probabilities**12**(1969), 184--185, for remark see ACM Trans. Math. Software v. 14 (1988), pp. 288--289. **Mos89**-
S. L. B. Moshier,
*Methods and programs for mathematical functions*, Ellis Horwood Limited, Chichester, 1989, separate diskette. **OM68**-
D. Osborn and R. Madey,
*The incomplete beta function and its ratio to the complete beta function*, Math. Comp.**22**(1968), 159--162. **Phi90**-
H. N. Phien,
*A note on the computation of the incomplete beta function*, Adv. Engrg. Software**12**(1990), 39--44. **Tem92**-
N. M. Temme,
*Asymptotic inversion of the incomplete beta function*, J. Comput. Appl. Math.**41**(1992), 145--157. **ULI90**-
*Mathematical function library for Microsoft--C*, United Laboratories, Inc., John Wiley & Sons, 1990, includes diskettes. Edition also exists in Fortran (1989).

The original abstract follows.

*
Higher transcendental functions continue to play varied and
important roles in investigations by engineers, mathematicians,
scientists and statisticians.
The purpose of this paper is to assist in locating useful approximations
and software for the numerical generation of these functions, and to
offer some suggestions for future developments in this field.
*

Applied and Computational Mathematics Division, National Institute of Standards and Technology, Gaithersburg, Md 20899

* E-mail address: dlozier@nist.gov*

Institute for Physical Science and Technology, University of Maryland, College Park, MD 20742

* E-mail address: olver@bessel.umd.edu*

The research of the second author has been supported by NSF Grant CCR 89-14933.

1991 * Mathematics Subject Classification*. Primary 65D20;
Secondary 33-00.

Fri Apr 7 14:14:39 EDT 1995