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## The efficient evaluation of the hypergeometric function of a matrix argument (2005)

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Venue: | MATH. COMP |

Citations: | 79 - 17 self |

### Citations

750 | Symmetric Functions and - Macdonald - 1995 |

213 |
Some combinatorial properties of Jack symmetric functions
- Stanley
- 1989
(Show Context)
Citation Context ...dC (α) κ (X) is the Jack function. The Jack function C (α) κ (X)=C (α) κ (x1,x2,...,xn) is a symmetric, homogeneous polynomial of degree |κ| in the eigenvalues x1,x2,...,xn of X [17, Rem. 2, p. 228], =-=[20]-=-. For example, when α =1,C (α) κ (X) becomes the (normalized) Schur function, and for α =2,thezonal polynomial. 1 There are several normalizations of the Jack function which are scalar multiples of on... |

173 | Matrix models for beta-ensembles
- Dumitriu, Edelman
- 2002
(Show Context)
Citation Context ...ne the hypergeometric function of a matrix argument through the series (1.1) for α =1[9,(4.1)]orα = 2 [17, p. 258] only. There is no reason for us to treat the different α’s separately (see also [4], =-=[5]-=-, [7] for the uniform treatment of the different α’s in other settings).sHYPERGEOMETRIC FUNCTION OF A MATRIX ARGUMENT 835 While there is little we can do about (A) (which is also a major problem even ... |

136 | Twelve lectures on subjects suggested by his life and works - Ramanujan - 1999 |

69 |
Largest eigenvalue of complex Wishart matrices and performance analysis of
- Kang, Alouini
- 2003
(Show Context)
Citation Context ...rix. 1. Introduction The hypergeometric function of a matrix argument has a wide area of applications in multivariate statistical analysis [17], random matrix theory [7], wireless communications [8], =-=[12]-=-, etc. Except in a few special cases, it can be expressed only as a series of multivariate homogeneous polynomials, called Jack functions. This series often converges very slowly [16, p. 390], [17], a... |

47 |
M.: Theoretical reliability of MMSE linear diversity combining in Rayleighfading additive interference channels
- Gao, Smith, et al.
- 1998
(Show Context)
Citation Context ...e matrix. 1. Introduction The hypergeometric function of a matrix argument has a wide area of applications in multivariate statistical analysis [17], random matrix theory [7], wireless communications =-=[8]-=-, [12], etc. Except in a few special cases, it can be expressed only as a series of multivariate homogeneous polynomials, called Jack functions. This series often converges very slowly [16, p. 390], [... |

38 | Eigenvalue statistics for beta-ensembles,”
- Dumitriu
- 2003
(Show Context)
Citation Context ... define the hypergeometric function of a matrix argument through the series (1.1) for α =1[9,(4.1)]orα = 2 [17, p. 258] only. There is no reason for us to treat the different α’s separately (see also =-=[4]-=-, [5], [7] for the uniform treatment of the different α’s in other settings).sHYPERGEOMETRIC FUNCTION OF A MATRIX ARGUMENT 835 While there is little we can do about (A) (which is also a major problem ... |

34 |
A.T.A.: Laplace approximations for hypergeometric functions with matrix argument
- Butler, Wood
- 2002
(Show Context)
Citation Context ... of a single Jack function is exponential [3]. The hypergeometric function of a matrix argument has thus acquired a reputation of being notoriously difficult to approximate even in the simplest cases =-=[2]-=-, [10]. In this paper we present new algorithms for approximating the value of the hypergeometric function of a matrix argument. We exploit recursive combinatorial relationships between the Jack funct... |

34 |
Reference Guide
- MathWorks, Inc, et al.
- 1992
(Show Context)
Citation Context ...omplexity that is only linear in the size of the matrix argument. In the special case when the matrix argument is a multiple of the identity, the evaluation becomes even faster. We have made a MATLAB =-=[15]-=- implementation of our algorithms available [13]. This implementation is very efficient (see performance results in Section 6), and has lead to new results [1], [6]. The hypergeometric function of a m... |

32 | D.S.P.: Total positivity, spherical series, and hypergeometric functions of matrix argument - Gross, Richards - 1989 |

28 | Tails of Condition Number Distributions
- Edelman, Sutton
- 1988
(Show Context)
Citation Context ...omes even faster. We have made a MATLAB [15] implementation of our algorithms available [13]. This implementation is very efficient (see performance results in Section 6), and has lead to new results =-=[1, 6]-=-. The hypergeometric function of a matrix argument is defined as follows. Let p ≥ 0 and q ≥ 0 be integers, and let X be an n ×n complex symmetric matrix with eigenvalues x1, x2, . . . , xn. Then (1.1)... |

24 | On the largest principal angle between random subspaces.
- Absil, Edelman, et al.
- 2006
(Show Context)
Citation Context ...omes even faster. We have made a MATLAB [15] implementation of our algorithms available [13]. This implementation is very efficient (see performance results in Section 6), and has lead to new results =-=[1]-=-, [6]. The hypergeometric function of a matrix argument is defined as follows. Let p ≥ 0andq≥0 be integers, and let X be an n ×n complex symmetric matrix with eigenvalues x1,x2,...,xn. Then ∞� � (α) (... |

24 | Developments in random matrix theory,
- Forrester, Snaith, et al.
- 2003
(Show Context)
Citation Context ...only linear in the size of the matrix. 1. Introduction The hypergeometric function of a matrix argument has a wide area of applications in multivariate statistical analysis [17], random matrix theory =-=[7]-=-, wireless communications [8, 12], etc. Except in a few special cases, it can be expressed only as a series of multivariate homogeneous polynomials, called Jack functions. This series often converges ... |

16 |
Computing the Confluent Hypergeometric Function,
- Muller
- 2001
(Show Context)
Citation Context ... of the different α’s in other settings).sHYPERGEOMETRIC FUNCTION OF A MATRIX ARGUMENT 835 While there is little we can do about (A) (which is also a major problem even in the univariate (n = 1) case =-=[18]-=-) or (B), our major contribution is in improving (C), the cost of evaluating the Jack function. We exploit the combinatorial properties of the Pochhammer symbol and the Jack function to only update th... |

16 |
Latent roots and matrix variates: a review of some asymptotic results
- Muirhead
- 1978
(Show Context)
Citation Context ...ero as |κ| → ∞. In these cases (1.4) is a good approximation to (1.1) for a large enough m. The computational difficulties in evaluating (1.4) are: (A) the series (1.1) converges slowly in many cases =-=[16]-=-; thus a rather large m may be needed before (1.4) becomes a good approximation to (1.1); (B) the number of terms in (1.4) (i.e., the number of partitions |κ| ≤ m) grows, roughly, as O(e √ m ) (see Se... |

13 | P.: Accurate and efficient evaluation of Schur and Jack functions
- Demmel, Koev
- 2006
(Show Context)
Citation Context ...ate homogeneous polynomials, called Jack functions. This series often converges very slowly [16, p. 390], [17], and the cost of the straightforward evaluation of a single Jack function is exponential =-=[3]-=-. The hypergeometric function of a matrix argument has thus acquired a reputation of being notoriously difficult to approximate even in the simplest cases [2], [10]. In this paper we present new algor... |

13 |
Log-gasses and Random matrices. http://www.ms.unimelb.edu.au/∼matpjf/matpjf.html
- Forrester
(Show Context)
Citation Context ...only linear in the size of the matrix. 1. Introduction The hypergeometric function of a matrix argument has a wide area of applications in multivariate statistical analysis [17], random matrix theory =-=[7]-=-, wireless communications [8], [12], etc. Except in a few special cases, it can be expressed only as a series of multivariate homogeneous polynomials, called Jack functions. This series often converge... |

12 |
A.J.: Approximation of hypergeometric functions with matricial argument through their development in series of zonal polynomials
- Gutiérrez, Rodriguez, et al.
- 2000
(Show Context)
Citation Context ... single Jack function is exponential [3]. The hypergeometric function of a matrix argument has thus acquired a reputation of being notoriously difficult to approximate even in the simplest cases [2], =-=[10]-=-. In this paper we present new algorithms for approximating the value of the hypergeometric function of a matrix argument. We exploit recursive combinatorial relationships between the Jack functions, ... |

4 |
Softwere for calculus of zonal polynomials
- Sáez
- 2004
(Show Context)
Citation Context ... a− ×(detY ) 2 c−a− det(I − Y ) 2 (dY ), ,andℜ(c − a) > n−1 2 . This approach, however, is restricted to the cases p = 1 or 2, q =1,andα =2. Gutiérrez, Rodriguez, and Sáez presented in [10] (see also =-=[19]-=- for the implementation) an algorithm for computing the truncation (1.4) for α = 2 (then the Jack functions are called zonal polynomials). For every k =1, 2,...,m, the authors form the upper triangula... |

2 |
Tails of condition number distributions. SIAM journal on matrix analysis and applications
- Edelman, Sutton
(Show Context)
Citation Context ...even faster. We have made a MATLAB [15] implementation of our algorithms available [13]. This implementation is very efficient (see performance results in Section 6), and has lead to new results [1], =-=[6]-=-. The hypergeometric function of a matrix argument is defined as follows. Let p ≥ 0andq≥0 be integers, and let X be an n ×n complex symmetric matrix with eigenvalues x1,x2,...,xn. Then ∞� � (α) (1.1) ... |

1 |
of multivariate statistical theory
- Aspects
- 1982
(Show Context)
Citation Context ...nd have complexity that is only linear in the size of the matrix. 1. Introduction The hypergeometric function of a matrix argument has a wide area of applications in multivariate statistical analysis =-=[17]-=-, random matrix theory [7], wireless communications [8], [12], etc. Except in a few special cases, it can be expressed only as a series of multivariate homogeneous polynomials, called Jack functions. ... |

1 | 02139 E-mail address: plamen@math.mit.edu - Vol - 1999 |