BRET (Beta Random Ensembles Toolbox)
Current version posted: November 25, 2015.
Installation instructions:
All files are in here.
You need the
mhg package for the functions here to work.
How to cite:
Plamen Koev and Alan Edelman, The efficient evaluation of the hypergeometric function of a matrix argument, Math. Comp. 75 (2006), 833-846.
Other resources
- Ioana Dumitriu's MOPs
package for computing with Multivariate Orthogonal Polynomials.
Conventions
All functions are named as xEnsembleType.m, where
- x={r,d,p,q} stands for the {empirical, density, probability distribution, quantile} functions
- Ensemble = {Laguerre, Jacobi, Wishart, MANOVA}
- Type = {Eig, SVD, Trace}
So that for example dWishartEig returns the density of a (specified) eigenvalue of the Wishart ensemble.
Function description:
mhg, mhgi, logmhg
| Description: |
Those functions compute the Hypergeometric function of one or
two matrix arguments. See here for
details.
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| Comments: |
Note that these functions take alpha=2/beta as a parameter!
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gammam
| Syntax: |
y=gammam(beta,m,a) |
| Description: |
Computes the multivariate Gamma function of parameter beta.
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rLaguerreSVD
| Syntax: |
rLaguerreSVD(N,beta,a,m) |
| Description: |
Returns N realizations of the singular values of an mxm Laguerre matrix of parameter a.
|
| Comments: |
If you need the eigenvalues, those are the squares of the singular values. This routine returns the same as rWishartSVD(beta,2*a/beta,ones(1,m),N)
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rWishartSVD
| Syntax: |
rWishartSVD(beta,n,Sigma,N) |
| Description: |
Provides N realizations of the singular values of an mxm
beta-Wishart matrix with n degrees of freedom and
covariance diag(Sigma). Sigma is assumed to be a vector
of length m. |
| Comments: |
If you need the eigenvalues, those are the squares of the singular values.
|
rWishartTrace
| Syntax: |
rWishartTrace(beta,n,Sigma,N) |
| Description: |
Provides N realizations of the trace an mxm
beta-Wishart matrix with n degrees of freedom and
covariance diag(Sigma). Sigma must be a row vector
of length m. |
rJacobiSVD
| Syntax: |
rJacobiSVD(beta,a,b,m,N) |
| Description: |
returns N realizations of the singular values of
an m-by-m Jacobi matrix with parameters a and b.
|
| Comments: |
If you need the eigenvalues, those are the squares of the singular values.
|
pWishartEig
| Syntax: |
pWishartEig(k,MAX,beta,n,Sigma,x) |
| Description: |
Returns the distribution of the kth eigenvalue of a Wishart matrix as a function of the (vector) x. k can only be 1 (largest) or m (smallest) eigenvalue. MAX is the truncation limit for the hypergeometric function.
|
pLaguerreEig
| Syntax: |
pLaguerreEig(k,MAX,beta,a,m,x) |
| Description: |
Returns the distribution of the kth eigenvalue of a Laguerre matrix of parameter a as a function of the (vector) x. k can only be 1 (largest) or m (smallest) eigenvalue. MAX is the truncation limit for the hypergeometric function. This function reverts directly to pWishartEig, which theoretically returns the same and in practice used a wizard to recognize the identity covariance of Laguerre and call mhgi instead of mhg.
|
pJacobiEig
| Syntax: |
pJacobiEig(k,MAX,beta,a,b,m,x) |
| Description: |
Returns the distribution of the kth eigenvalue of an mxm Jacobi matrix with parameters a,b as a function of the (vector) x. k can only be 1 (largest) or m (smallest) eigenvalue. MAX is the truncation limit for the hypergeometric function.
|
dLaguerreEig
| Syntax: |
dLaguerreEig(k,MAX,beta,a,m,x) |
| Description: |
Returns the density of the kth eigenvalue of an mxm Laguerre matrix with parameter a as a function of the (vector) x. k can only be 1 (largest) or m (smallest) eigenvalue. MAX is the truncation limit for the hypergeometric function. Uses a faster approach than the straightforward derivative of the distribution (so it’s faster than dWishartEig with Sigma=I even though it’s mathematically equivalent).
|
dWishartEig
| Syntax: |
dWishartEig(k,MAX,beta,n,Sigma,x) |
| Description: |
Returns the density of the kth eigenvalue of a Wishart matrix as a function of the (vector) x. k can only be 1 (largest) or m (smallest) eigenvalue. MAX is the truncation limit for the hypergeometric function. Computed as the derivative of the distribution.
|
dWishartTrace
| Syntax: |
dWishartTrace(MAX,beta,n,Sigma,x) |
| Description: |
Returns the density, as a function of (the vector) x, of the trace of an Wishart matrix with n degrees of freedom and covariance Sigma. Sigma is assumed to be a vector. MAX is the truncation limit for the hypergeometric function.
|
dJacobiEig
| Syntax: |
dJacobiEig(k,MAX,beta,a,b,m,x) |
| Description: |
Returns the density of the kth eigenvalue of an mxm Jacobi matrix with parameters a,b as a function of the (vector) x. k can only be 1 (largest) or m (smallest) eigenvalue. MAX is the truncation limit for the hypergeometric function.
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Plamen Koev
Department of Mathematics, San Jose State University
”firstname” dot “lastname” at sjsu dot edu
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