Skip to Main Content

Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

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

The 2020 MCQ for Mathematics of Computation is 1.78.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

The efficient evaluation of the hypergeometric function of a matrix argument
HTML articles powered by AMS MathViewer

by Plamen Koev and Alan Edelman PDF
Math. Comp. 75 (2006), 833-846 Request permission

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
  • P.-A. Absil, A. Edelman, and P. Koev, On the largest principal angle between random subspaces, Linear Algebra Appl., to appear.
  • Ronald W. Butler and Andrew T. A. Wood, Laplace approximations for hypergeometric functions with matrix argument, Ann. Statist. 30 (2002), no. 4, 1155–1177. MR 1926172, DOI 10.1214/aos/1031689021
  • J. Demmel and P. Koev, Accurate and efficient evaluation of Schur and Jack functions, Math. Comp., 75 (2005), no. 253, 223–239.
  • I. Dumitriu, Eigenvalue statistics for the Beta-ensembles, Ph.D. thesis, Massachusetts Institute of Technology, 2003.
  • Ioana Dumitriu and Alan Edelman, Matrix models for beta ensembles, J. Math. Phys. 43 (2002), no. 11, 5830–5847. MR 1936554, DOI 10.1063/1.1507823
  • A. Edelman and B. Sutton, Tails of condition number distributions, SIAM J. Matrix Anal. Appl., accepted for publication, 2005.
  • P. Forrester, Log-gases and random matrices, http://www.ms.unimelb.edu.au/~matpjf/matpjf.html
  • 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.
  • Kenneth I. Gross and Donald St. P. Richards, Total positivity, spherical series, and hypergeometric functions of matrix argument, J. Approx. Theory 59 (1989), no. 2, 224–246. MR 1022118, DOI 10.1016/0021-9045(89)90153-6
  • 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
  • G. H. Hardy, Ramanujan. Twelve lectures on subjects suggested by his life and work, Cambridge University Press, Cambridge, England; The Macmillan Company, New York, 1940. MR 0004860
  • 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.
  • P. Koev, http://www-math.mit.edu/~plamen.
  • I. G. Macdonald, Symmetric functions and Hall polynomials, 2nd ed., Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995. With contributions by A. Zelevinsky; Oxford Science Publications. MR 1354144
  • The MathWorks, Inc., Natick, MA, MATLAB reference guide, 1992.
  • Robb J. Muirhead, Latent roots and matrix variates: a review of some asymptotic results, Ann. Statist. 6 (1978), no. 1, 5–33. MR 458719
  • Robb J. Muirhead, Aspects of multivariate statistical theory, Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1982. MR 652932, DOI 10.1002/9780470316559
  • Keith E. Muller, Computing the confluent hypergeometric function, $M(a,b,x)$, Numer. Math. 90 (2001), no. 1, 179–196. MR 1868767, DOI 10.1007/s002110100285
  • A. J. Sáez, Software for calculus of zonal polynomials, http://estio.ujaen.es/Profesores/ajsaez/software.html, 2004.
  • Richard P. Stanley, Some combinatorial properties of Jack symmetric functions, Adv. Math. 77 (1989), no. 1, 76–115. MR 1014073, DOI 10.1016/0001-8708(89)90015-7
  • Richard P. Stanley, Enumerative combinatorics. Vol. 1, Cambridge Studies in Advanced Mathematics, vol. 49, Cambridge University Press, Cambridge, 1997. With a foreword by Gian-Carlo Rota; Corrected reprint of the 1986 original. MR 1442260, DOI 10.1017/CBO9780511805967
  • Richard P. Stanley, Enumerative combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics, vol. 62, Cambridge University Press, Cambridge, 1999. With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin. MR 1676282, DOI 10.1017/CBO9780511609589
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
  • Email: plamen@math.mit.edu
  • Alan Edelman
  • Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
  • Email: edelman@math.mit.edu
  • 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.
  • © Copyright 2006 American Mathematical Society
  • Journal: Math. Comp. 75 (2006), 833-846
  • MSC (2000): Primary 33C20, 65B10; Secondary 05A99
  • DOI: https://doi.org/10.1090/S0025-5718-06-01824-2
  • MathSciNet review: 2196994