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‎README.md

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RandomMatrices.jl
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=================
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Random matrix repository for Julia
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Random matrix package for [Julia](http://julialang.org).
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This extends the [Distributions](https://github.com/JuliaStats/Distributions.jl)
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package to provide methods for working with matrix-valued random variables,
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a.k.a. random matrices. State of the art methods for computing random matrix
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samples and their associated distributions are provided.
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## License
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Copyright (c) 2013 Jiahao Chen <[email protected]> @jiahao
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Copyright (c) 2013 [Jiahao Chen](https://github.com/jiahao) <[email protected]>
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This Julia package is distribtued under the [MIT License](http://opensource.org/licenses/MIT).
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Distributed under the [MIT License](http://opensource.org/licenses/MIT).
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# Gaussian matrix ensembles
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- `GaussianHermiteMatrix(n, beta)`, `GaussianLaguerreMatrix(n, m, beta)`,
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`GaussianJacobiMatrix(n, m1, m2, beta)`
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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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each construct a sample dense `n`x`n` matrix for the corresponding matrix
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ensemble with `beta=1,2,4`
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- `GaussianHermiteTridiagonalMatrix(n, beta)`,
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`GaussianLaguerreTridiagonalMatrix(n, m, beta)`,
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`GaussianJacobiSparseMatrix(n, m1, m2, beta)`
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each construct a sparse `n`x`n` matrix for the corresponding matrix ensemble
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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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These sampled matrices have the same eigenvalues as above but are much faster
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to diagonalize oweing to their sparsity. They also extend Dyson's threefold
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way to arbitrary `beta`.
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- `GaussianHermiteSamples(n, beta)`,
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`GaussianLaguerreSamples(n, m, beta)`,
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`GaussianJacobiSamples(n, m1, m2, beta)`
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return a set of `n` eigenvalues from the sparse random matrix samples
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(Note the parameters of the Laguerre and Jacobi ensembles are not yet defined consistently.
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For the first set they are integers but for the rest they are reals.)
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For the first set of methods they are integers but for the rest they are reals.)
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# Formal power series
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Allows for manipulations of formal power series (fps) and formal Laurent series.
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This defines the new types
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- `FormalPowerSeries`: power series with coefficients allowed only for non-negative integer powers
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- `FormalLaurentSeries`: power series with coefficients allowed for all integer powers
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Allows for manipulations of formal power series (fps) and formal Laurent series
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(fLs), which come in handy for the computation of free cumulants.
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## FormalPowerSeries
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## Types
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- `FormalPowerSeries`: power series with coefficients allowed only for
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non-negative integer powers
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- `FormalLaurentSeries`: power series with coefficients allowed for all
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integer powers
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In addition to basic arithmetic operations `==`, `+`, `-`, `^`, this also provides:
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## FormalPowerSeries methods
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- `tovector` returns the series coefficients
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- `trim` removes extraneous zeroes
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### Elementary operations
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- basic arithmetic operations `==`, `+`, `-`, `^`
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- `*` computes the Cauchy product (discrete convolution)
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- `.*` computes the Hadamard product (elementwise multiplication)
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- `isunit(P)` determines if `P` is a unit series
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- `isnonunit(P)` determines if `P` is a non-unit series
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- `MatrixForm(P)` returns a matrix representation of `P` as an upper triangular Toeplitz matrix
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- `reciprocal` computes the series reciprocal
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- `derivative` computes the series derivative
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- `isconstant(P)` determines if `P` is a constant series
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- `compose(P,Q)` computes the series composition P.Q
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- `derivative` computes the series derivative
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- `reciprocal` computes the series reciprocal
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### Utility methods
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- `trim(P)` removes extraneous zeroes in the internal representation of `P`
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- `isalmostunit(P)` determines if `P` is an almost unit series
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- `isconstant(P)` determines if `P` is a constant series
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- `isnonunit(P)` determines if `P` is a non-unit series
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- `isunit(P)` determines if `P` is a unit series
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- `MatrixForm(P)` returns a matrix representation of `P` as an upper triangular
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Toeplitz matrix
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- `tovector` returns the series coefficients
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# Densities
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Famous distributions in random matrix theory
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- `Semicircle` provides the semicircle distribution
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- `TracyWidom` computes the Tracy-Widom density distribution by brute-force integration of the Painlevé II equation
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- `TracyWidom` computes the Tracy-Widom density distribution
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by brute-force integration of the Painlevé II equation
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# Utility functions
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- `hist_eig` computes the histogram of eigenvalues of a matrix using the method of Sturm sequences.
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For `SymTridiagonal` matrices this is significantly faster than `hist(eigvals())`
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- `hist_eig` computes the histogram of eigenvalues of a matrix using the
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method of Sturm sequences.
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This is recommended for `SymTridiagonal` matrices as it is significantly
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faster than `hist(eigvals())`
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This is also implemented for dense matrices, but it is pretty slow and
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not really practical.
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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, 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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- James Albrecht, Cy Chan, and Alan Edelman,
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"Sturm Sequences and Random Eigenvalue Distributions",
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*Foundations of Computational Mathematics*,
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vol. 9 iss. 4 (2009), pp 461-483.
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[[pdf]](www-math.mit.edu/~edelman/homepage/papers/sturm.pdf)
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[[doi]](http://dx.doi.org/10.1007/s10208-008-9037-x)
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- Ioana Dumitriu and Alan Edelman,
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"Matrix Models for Beta Ensembles",
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*Journal of Mathematical Physics*,
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vol. 43 no. 11 (2002), pp. 5830-5547
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[[doi]](http://dx.doi.org/doi: 10.1063/1.1507823)
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[arXiv:math-ph/0206043](http://arxiv.org/abs/math-ph/0206043)
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- Alan Edelman, Per-Olof Persson and Brian D Sutton,
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"The fourfold way",
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*Journal of Mathematical Physics*,
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submitted (2013).
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[[pdf]](http://www-math.mit.edu/~edelman/homepage/papers/ffw.pdf)
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- Alan Edelman and Brian D. Sutton,
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"The beta-Jacobi matrix model, the CS decomposition,
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and generalized singular value problems",
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*Foundations of Computational Mathematics*,
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vol. 8 iss. 2 (2008), pp 259-285.
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[[pdf]](http://www-math.mit.edu/~edelman/homepage/papers/betajacobi.pdf)
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[[doi]](http://dx.doi.org/10.1007/s10208-006-0215-9)
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- Peter Henrici,
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*Applied and Computational Complex Analysis,
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Volume I: Power Series---Integration---Conformal Mapping---Location of Zeros*,
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Wiley-Interscience: New York, 1974
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[[worldcat]](http://www.worldcat.org/oclc/746035)
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