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Adds beta=Inf case to GaussianHermiteTridiagonalMatrix
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README.md

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@@ -29,7 +29,8 @@ Hermite, Laguerre(m) and Jacobi(m1, m2) ensembles.
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each construct a sample dense `n`x`n` matrix for the corresponding matrix ensemble with `beta=1,2,4`
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- `GaussianHermiteTridiagonalMatrix(n, beta)`, `GaussianLaguerreTridiagonalMatrix(n, m, beta)`,
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`GaussianJacobiSparseMatrix(n, m1, m2, beta)` each construct a sparse `n`x`n` matrix for the
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corresponding matrix ensemble for arbitrary positive finite `beta`
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corresponding matrix ensemble for arbitrary positive finite `beta`.
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`GaussianHermiteTridiagonalMatrix(n, Inf)` is also allowed.
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- `GaussianHermiteSamples(n, beta)`, `GaussianLaguerreSamples(n, m, beta)`,
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`GaussianJacobiSamples(n, m1, m2, beta)` return a set of `n` eigenvalues from the previous sampled
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random matrices
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# References
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- James Albrecht, Cy Chan, and Alan Edelman, "Sturm Sequences and Random Eigenvalue Distributions", *Foundations of Computational Mathematics*, vol. 9 iss. 4 (2009), pp 461-483. [[pdf]](www-math.mit.edu/~edelman/homepage/papers/sturm.pdf) [[doi]](http://dx.doi.org/10.1007/s10208-008-9037-x)
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- Alan Edelman and Brian D. Sutton, "The beta-Jacobi matrix model, the CS decomposition, and generalized singular value problems", *Foundations of Computational Mathematics*, vol. 8 iss. 2 (2008), pp 259-285. [[pdf]](http://www-math.mit.edu/~edelman/homepage/papers/betajacobi.pdf) [[doi]](http://dx.doi.org/10.1007/s10208-006-0215-9)
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- Alan Edelman, Per-Olof Persson and Brian D Sutton, "The fourfold way", *Journal of Mathematical Physics*, submitted (2013). [[pdf]](http://www-math.mit.edu/~edelman/homepage/papers/ffw.pdf)
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u- Alan Edelman and Brian D. Sutton, "The beta-Jacobi matrix model, the CS decomposition, and generalized singular value problems", *Foundations of Computational Mathematics*, vol. 8 iss. 2 (2008), pp 259-285. [[pdf]](http://www-math.mit.edu/~edelman/homepage/papers/betajacobi.pdf) [[doi]](http://dx.doi.org/10.1007/s10208-006-0215-9)
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- Peter Henrici, *Applied and Computational Complex Analysis, Volume I: Power Series---Integration---Conformal Mapping---Location of Zeros*, Wiley-Interscience: New York, 1974 [[worldcat]](http://www.worldcat.org/title/applied-and-computational-complex-analysis/oclc/746035)
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src/GaussianEnsembleSamples.jl

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# Journal of Mathematical Physics, vol. 43 no. 11 (2002), pp. 5830--5547
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# doi: 10.1063/1.1507823
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# arXiv: math-ph/0206043
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#
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# References
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#
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# Alan Edelman, Per-Olof Persson and Brian D Sutton, "The fourfold way"
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# http://www-math.mit.edu/~edelman/homepage/papers/ffw.pdf
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using Distributions
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export GaussianHermiteMatrix, GaussianLaguerreMatrix, GaussianJacobiMatrix,
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#Generates a NxN tridiagonal Wigner matrix
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#Hermite ensemble
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#The beta=infinity case is defined in Edelman, Persson and Sutton, 2012
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function GaussianHermiteTridiagonalMatrix(n :: Integer, beta :: FloatingPoint)
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@assert beta > 0
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if beta<=0 error("beta must be positive") end
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if beta==Inf return SymTridiagonal(zeros(n), [sqrt(x/2) for x=n-1:-1:1]) end
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Hdiag = randn(n)/sqrt(n)
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Hsup = [chi(beta*i)/sqrt(2*n) for i=n-1:-1:1]
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return SymTridiagonal(Hdiag, Hsup)

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