Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



The efficient evaluation of the hypergeometric function of a matrix argument

Authors: Plamen Koev and Alan Edelman
Journal: Math. Comp. 75 (2006), 833-846
MSC (2000): Primary 33C20, 65B10; Secondary 05A99
Published electronically: January 19, 2006
MathSciNet review: 2196994
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We present new algorithms that efficiently approximate the hypergeometric function of a matrix argument through its expansion as a series of Jack functions. Our algorithms exploit the combinatorial properties of the Jack function, and have complexity that is only linear in the size of the matrix.

References [Enhancements On Off] (What's this?)

  • 1. P.-A. Absil, A. Edelman, and P. Koev, On the largest principal angle between random subspaces, Linear Algebra Appl., to appear.
  • 2. R. W. Butler and A. T. A. Wood, Laplace approximations for hypergeometric functions with matrix argument, Ann. Statist. 30 (2002), no. 4, 1155-1177. MR 1926172 (2003h:62076)
  • 3. J. Demmel and P. Koev, Accurate and efficient evaluation of Schur and Jack functions, Math. Comp., 75 (2005), no. 253, 223-239.
  • 4. I. Dumitriu, Eigenvalue statistics for the Beta-ensembles, Ph.D. thesis, Massachusetts Institute of Technology, 2003.
  • 5. I. Dumitriu and A. Edelman, Matrix models for beta ensembles, J. Math. Phys. 43 (2002), no. 11, 5830-5847. MR 1936554 (2004g:82044)
  • 6. A. Edelman and B. Sutton, Tails of condition number distributions, SIAM J. Matrix Anal. Appl., accepted for publication, 2005.
  • 7. P. Forrester, Log-gases and random matrices, matpjf.html
  • 8. H. Gao, P.J. Smith, and M.V. Clark, Theoretical reliability of MMSE linear diversity combining in Rayleigh-fading additive interference channels, IEEE Transactions on Communications 46 (1998), no. 5, 666-672.
  • 9. K. I. Gross and D. St. P. Richards, Total positivity, spherical series, and hypergeometric functions of matrix argument, J. Approx. Theory 59 (1989), no. 2, 224-246. MR 1022118 (91i:33005)
  • 10. R. Gutiérrez, J. Rodriguez, and A. J. Sáez, Approximation of hypergeometric functions with matricial argument through their development in series of zonal polynomials, Electron. Trans. Numer. Anal. 11 (2000), 121-130.MR 1799027 (2002b:33004)
  • 11. G. H. Hardy, Ramanujan: Twelve lectures on subjects suggested by his life and work, AMS Chelsea, New York, 1999.MR 0004860 (3:71d)
  • 12. M. Kang and M.-S. Alouini, Largest eigenvalue of complex Wishart matrices and performance analysis of MIMO MRC systems, IEEE Journal on Selected Areas in Communications 21 (2003), no. 3, 418-431.
  • 13. P. Koev,
  • 14. I. G. Macdonald, Symmetric functions and Hall polynomials, Second ed., Oxford University Press, New York, 1995. MR 1354144 (96h:05207)
  • 15. The MathWorks, Inc., Natick, MA, MATLAB reference guide, 1992.
  • 16. R. J. Muirhead, Latent roots and matrix variates: a review of some asymptotic results, Ann. Statist. 6 (1978), no. 1, 5-33. MR 0458719 (56:16919)
  • 17. -, Aspects of multivariate statistical theory, John Wiley & Sons Inc., New York, 1982. MR 0652932 (84c:62073)
  • 18. K. E. Muller, Computing the confluent hypergeometric function, $ M(a,b,x)$, Numer. Math. 90 (2001), no. 1, 179-196. MR 1868767 (2003a:33044)
  • 19. A. J. Sáez, Software for calculus of zonal polynomials,, 2004.
  • 20. R. Stanley, Some combinatorial properties of Jack symmetric functions, Adv. Math. 77 (1989), no. 1, 76-115. MR 1014073 (90g:05020)
  • 21. -, Enumerative combinatorics. Vol. 1, Cambridge Studies in Advanced Mathematics, vol. 49, Cambridge University Press, Cambridge, 1997. MR 1442260 (98a:05001)
  • 22. -, Enumerative combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics, vol. 62, Cambridge University Press, Cambridge, 1999. MR 1676282 (2000k:05026)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 33C20, 65B10, 05A99

Retrieve articles in all journals with MSC (2000): 33C20, 65B10, 05A99

Additional Information

Plamen Koev
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

Alan Edelman
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

Keywords: Hypergeometric function of a matrix argument, Jack function, zonal polynomial, eigenvalues of random matrices
Received by editor(s): September 16, 2004
Received by editor(s) in revised form: February 26, 2005
Published electronically: January 19, 2006
Additional Notes: This work was supported in part by NSF Grant DMS-0314286.
Article copyright: © Copyright 2006 American Mathematical Society

American Mathematical Society