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�RandomMatrixEnsembleModelz�
Base class for random matrix ensembles.
It acts as an umbrella and contains
the methods common to all the ensembles
defined in sympy.stats.random_matrix_models.
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r( c @ � e Zd ZdZdd� Zdd� ZdS )�GaussianEnsembleModela
Abstract class for Gaussian ensembles.
Contains the properties common to all the
gaussian ensembles.
References
==========
.. [1] https://en.wikipedia.org/wiki/Random_matrix#Gaussian_ensembles
.. [2] https://arxiv.org/pdf/1712.07903.pdf
c s~ t |�}� fdd�}tdddd�}t||�|d|f��� }d� | � | |d d |d }dt |d }|| | S ) a
Helper function for computing normalization
constant for joint probability density of eigen
values of Gaussian ensembles.
References
==========
.. [1] https://en.wikipedia.org/wiki/Selberg_integral#Mehta's_integral
c s. t d� t| � d �t tj� td� � S )Nr6 � )r
r �One)�j��betar$ r& r4 K s . zGGaussianEnsembleModel._compute_normalization_constant.<locals>.<lambda>rG T��integer�positiver6 rE � )r r r �doitr )r3 rI �n� prod_termrG �term1�term2�term3r$ rH r& � _compute_normalization_constant? s ( z5GaussianEnsembleModel._compute_normalization_constantc C s� | j }| �||�}td�}tdddd�}tdddd�}tdddd�}tt|� d t|| d |d|f��� �}t|t t
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Helper function for computing the joint
probability distribution of eigen values
of the random matrix.
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rN r r r r �tuple) r3 rI rO �ZbnrU rV rG rW rQ �sub_termrR �symsr$ r$ r& �!_compute_joint_eigen_distributionR s .. z7GaussianEnsembleModel._compute_joint_eigen_distributionN)r< r= r> r? rT r\ r$ r$ r$ r& rD 3 s rD c @ �0 e Zd Zedd� �Zdd� Zdd� Zdd� Zd S )
� GaussianUnitaryEnsembleModelc C s* | j }dt|�d tt|d �d S �NrE )rB r r )r3 rO r$ r$ r& �normalization_constante s $z3GaussianUnitaryEnsembleModel.normalization_constantc C sV | j | j}}td| d�}td|||d�}t|tt|� d t|d � �| �|�S �N�P��model�H��pspacerE �rB r` r r r r r r )r3 r9 rO �ZGUE�h_pspacere r$ r$ r& r: j � ,z$GaussianUnitaryEnsembleModel.densityc C � | � td��S r_ �r\ r r2 r$ r$ r& r! p � z5GaussianUnitaryEnsembleModel.joint_eigen_distributionc C s: t d�}dtd |d tdt |d � }t||�S )N�s� rE ����r r r r �r3 ro �fr$ r$ r& r# s s (
z7GaussianUnitaryEnsembleModel.level_spacing_distributionN�r< r= r> r@ r` r: r! r# r$ r$ r$ r& r^ d s
r^ c @ r] )
� GaussianOrthogonalEnsembleModelc C s4 | j }td||�}ttt|� d t|d � ��S )N�_HrM rE �rB r r r r r �r3 rO rw r$ r$ r& r` y s "z6GaussianOrthogonalEnsembleModel.normalization_constantc C sV | j | j}}td| d�}td|||d�}t|tt|� d t|d � �| �|�S )Nrb rc re rf rM rE rh )r3 r9 rO �ZGOErj re r$ r$ r& r: rk z'GaussianOrthogonalEnsembleModel.densityc C � | � tj�S r7 �r\ r rF r2 r$ r$ r& r! � � z8GaussianOrthogonalEnsembleModel.joint_eigen_distributionc C s4 t d�}td | tt d |d � }t||�S )Nro rE rM rr rs r$ r$ r& r# � s "
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� GaussianSymplecticEnsembleModelc C s0 | j }td||�}ttt|� t|d � ��S )Nrw rE rx ry r$ r$ r& r` � s z6GaussianSymplecticEnsembleModel.normalization_constantc C sR | j | j}}td| d�}td|||d�}t|tt|� t|d � �| �|�S ra rh )r3 r9 rO �ZGSErj re r$ r$ r& r: � s (z'GaussianSymplecticEnsembleModel.densityc C rl �NrM rm r2 r$ r$ r& r! � rn z8GaussianSymplecticEnsembleModel.joint_eigen_distributionc C sR t d�}td�d td�d td |d tddt |d � }t||�S ) Nro rE � � � rM i����� )r r r r r rs r$ r$ r& r# � s @
z:GaussianSymplecticEnsembleModel.level_spacing_distributionNru r$ r$ r$ r& r � r~ r c C �8 t | �t|�} }t| |�}t| |d�}t| |||d�S �Nrc rf )r r rD r r �r- r. rd �rmpr$ r$ r& r � �
r c C r� )a-
Represents Gaussian Unitary Ensembles.
Examples
========
>>> from sympy.stats import GaussianUnitaryEnsemble as GUE, density
>>> from sympy import MatrixSymbol
>>> G = GUE('U', 2)
>>> X = MatrixSymbol('X', 2, 2)
>>> density(G)(X)
exp(-Trace(X**2))/(2*pi**2)
rc rf )r r r^ r r r� r$ r$ r& r � �
r c C r� )aN
Represents Gaussian Orthogonal Ensembles.
Examples
========
>>> from sympy.stats import GaussianOrthogonalEnsemble as GOE, density
>>> from sympy import MatrixSymbol
>>> G = GOE('U', 2)
>>> X = MatrixSymbol('X', 2, 2)
>>> density(G)(X)
exp(-Trace(X**2)/2)/Integral(exp(-Trace(_H**2)/2), _H)
rc rf )r r rv r r r� r$ r$ r& r � r� r c C r� )aN
Represents Gaussian Symplectic Ensembles.
Examples
========
>>> from sympy.stats import GaussianSymplecticEnsemble as GSE, density
>>> from sympy import MatrixSymbol
>>> G = GSE('U', 2)
>>> X = MatrixSymbol('X', 2, 2)
>>> density(G)(X)
exp(-2*Trace(X**2))/Integral(exp(-2*Trace(_H**2)), _H)
rc rf )r r r r r r� r$ r$ r& r � r� r c @ rC )�CircularEnsembleModelz�
Abstract class for Circular ensembles.
Contains the properties and methods
common to all the circular ensembles.
References
==========
.. [1] https://en.wikipedia.org/wiki/Circular_ensemble
c C s t d| ��)NzeSupport for Haar measure hasn't been implemented yet, therefore the density of %s cannot be computed.)�NotImplementedErrorr8 r$ r$ r& r: � s �z CircularEnsembleModel.densityc
C s� | j }dt | t|| d d �tt|d d ��| }td�}tddd�tddd�tddd�}}}t|| |d|f��� }ttt t
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|�| | �S ) z�
Helper function to compute the joint distribution of phases
of the complex eigen values of matrices belonging to any
circular ensembles.
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r� c @ � e Zd Zdd� ZdS )� CircularUnitaryEnsembleModelc C rl r_ rm r2 r$ r$ r& r! rn z5CircularUnitaryEnsembleModel.joint_eigen_distributionN�r< r= r> r! r$ r$ r$ r& r� � r� c @ r� )� CircularOrthogonalEnsembleModelc C r{ r7 r| r2 r$ r$ r& r! r} z8CircularOrthogonalEnsembleModel.joint_eigen_distributionNr� r$ r$ r$ r& r� r� r� c @ r� )� CircularSymplecticEnsembleModelc C rl r� rm r2 r$ r$ r& r! rn z8CircularSymplecticEnsembleModel.joint_eigen_distributionNr� r$ r$ r$ r& r� r� r� c C r� r� )r r r� r r r� r$ r$ r& r r� r c C r� )a7
Represents Cicular Unitary Ensembles.
Examples
========
>>> from sympy.stats import CircularUnitaryEnsemble as CUE
>>> from sympy.stats import joint_eigen_distribution
>>> C = CUE('U', 1)
>>> joint_eigen_distribution(C)
Lambda(t[1], Product(Abs(exp(I*t[_j]) - exp(I*t[_k]))**2, (_j, _k + 1, 1), (_k, 1, 0))/(2*pi))
Note
====
As can be seen above in the example, density of CiruclarUnitaryEnsemble
is not evaluated becuase the exact definition is based on haar measure of
unitary group which is not unique.
rc rf )r r r� r r r� r$ r$ r& r �
r c C r� )a=
Represents Cicular Orthogonal Ensembles.
Examples
========
>>> from sympy.stats import CircularOrthogonalEnsemble as COE
>>> from sympy.stats import joint_eigen_distribution
>>> C = COE('O', 1)
>>> joint_eigen_distribution(C)
Lambda(t[1], Product(Abs(exp(I*t[_j]) - exp(I*t[_k])), (_j, _k + 1, 1), (_k, 1, 0))/(2*pi))
Note
====
As can be seen above in the example, density of CiruclarOrthogonalEnsemble
is not evaluated becuase the exact definition is based on haar measure of
unitary group which is not unique.
rc rf )r r r� r r r� r$ r$ r& r . r� r c C r� )a@
Represents Cicular Symplectic Ensembles.
Examples
========
>>> from sympy.stats import CircularSymplecticEnsemble as CSE
>>> from sympy.stats import joint_eigen_distribution
>>> C = CSE('S', 1)
>>> joint_eigen_distribution(C)
Lambda(t[1], Product(Abs(exp(I*t[_j]) - exp(I*t[_k]))**4, (_j, _k + 1, 1), (_k, 1, 0))/(2*pi))
Note
====
As can be seen above in the example, density of CiruclarSymplecticEnsemble
is not evaluated becuase the exact definition is based on haar measure of
unitary group which is not unique.
rc rf )r r r� r r r� r$ r$ r& r G r� r c C s" t | t�s td| ��| jj�� S )aA
For obtaining joint probability distribution
of eigen values of random matrix.
Parameters
==========
mat: RandomMatrixSymbol
The matrix symbol whose eigen values are to be considered.
Returns
=======
Lambda
Examples
========
>>> from sympy.stats import GaussianUnitaryEnsemble as GUE
>>> from sympy.stats import joint_eigen_distribution
>>> U = GUE('U', 2)
>>> joint_eigen_distribution(U)
Lambda((l[1], l[2]), exp(-l[1]**2 - l[2]**2)*Product(Abs(l[_i] - l[_j])**2, (_j, _i + 1, 2), (_i, 1, 1))/pi)
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isinstancer r* rg rd r! ��matr$ r$ r& r! ` s
r! c C s2 | j dd�}tdd� t|�D ��rtd��t|� S )a�
Creates joint distribution of eigen values of matrices with random
expressions.
Parameters
==========
mat: Matrix
The matrix under consideration.
Returns
=======
JointDistributionHandmade
Examples
========
>>> from sympy.stats import Normal, JointEigenDistribution
>>> from sympy import Matrix
>>> A = [[Normal('A00', 0, 1), Normal('A01', 0, 1)],
... [Normal('A10', 0, 1), Normal('A11', 0, 1)]]
>>> JointEigenDistribution(Matrix(A))
JointDistributionHandmade(-sqrt(A00**2 - 2*A00*A11 + 4*A01*A10 + A11**2)/2
+ A00/2 + A11/2, sqrt(A00**2 - 2*A00*A11 + 4*A01*A10 + A11**2)/2 + A00/2 + A11/2)
T)�multiplec s s � | ]}t |� V qd S r7 )r )�.0�eigenvalr$ r$ r& � <genexpr>� s � z)JointEigenDistribution.<locals>.<genexpr>zVEigen values don't have any random expression, joint distribution cannot be generated.)� eigenvals�any�setr* r )r� r� r$ r$ r& r" } s r" c C s | j j�� S )a�
For obtaining distribution of level spacings.
Parameters
==========
mat: RandomMatrixSymbol
The random matrix symbol whose eigen values are
to be considered for finding the level spacings.
Returns
=======
Lambda
Examples
========
>>> from sympy.stats import GaussianUnitaryEnsemble as GUE
>>> from sympy.stats import level_spacing_distribution
>>> U = GUE('U', 2)
>>> level_spacing_distribution(U)
Lambda(_s, 32*_s**2*exp(-4*_s**2/pi)/pi**2)
References
==========
.. [1] https://en.wikipedia.org/wiki/Random_matrix#Distribution_of_level_spacings
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