o 8Va9@sddlmZmZmZmZmZmZmZmZm Z m Z m Z m Z m Z mZmZddlmZddlmZmZmZmZddlmZddlmZddlmZgdZeedd ZGd d d eZ Gd d d e Z!Gddde!Z"Gddde!Z#Gddde!Z$ddZ%ddZ&ddZ'ddZ(Gddde Z)Gddde)Z*Gd d!d!e)Z+Gd"d#d#e)Z,d$d%Z-d&d'Z.d(d)Z/d*d+Z0d,d-Z1d.d/Z2d0d1Z3d2S)3)BasicexppiLambdaTraceS MatrixSymbolIntegralgammaProductDummySumAbs IndexedBaseI)_sympify)_symbol_converterDensityRandomMatrixSymbol is_random)JointDistributionHandmade)RandomMatrixPSpace)ArrayComprehension) CircularEnsembleCircularUnitaryEnsembleCircularOrthogonalEnsembleCircularSymplecticEnsembleGaussianEnsembleGaussianUnitaryEnsembleGaussianOrthogonalEnsembleGaussianSymplecticEnsemblejoint_eigen_distributionJointEigenDistributionlevel_spacing_distributioncCsdS)NT)xr$r$B/usr/lib/python3/dist-packages/sympy/stats/random_matrix_models.py_sr'c@sBeZdZdZd ddZeddZeddZdd Zd d Z dS) 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. NcCs6t|t|}}|jdkrtd|t|||S)NFzGDimension of the random matrices must be integers, received %s instead.)rr is_integer ValueErrorr__new__)clssymdimr$r$r&r+#s  z!RandomMatrixEnsembleModel.__new__cC |jdS)Nrargsselfr$r$r&* z"RandomMatrixEnsembleModel.cCr/)Nr0r2r$r$r&r4+r5cCst|SN)rr3exprr$r$r&density-sz!RandomMatrixEnsembleModel.densitycCs ||Sr7)r:r8r$r$r&__call__0s z"RandomMatrixEnsembleModel.__call__r7) __name__ __module__ __qualname____doc__r+propertysymbol dimensionr:r;r$r$r$r&r(s    r(c@ eZdZdZddZddZdS)GaussianEnsembleModela Abstract class for Gaussian ensembles. Contains the properties common to all the gaussian ensembles. References ========== .. [1] https://en.wikipedia.org/wiki/Random_matrix#Gaussian_ensembles .. [2] https://arxiv.org/pdf/1712.07903.pdf cs~t|}fdd}tdddd}t|||d|f}d|||dd|d}dt|d}|||S) a Helper function for computing normalization constant for joint probability density of eigen values of Gaussian ensembles. References ========== .. [1] https://en.wikipedia.org/wiki/Selberg_integral#Mehta's_integral cs.tdt|dttjtdS)Nr6)r rOne)jbetar$r&r4Ks.zGGaussianEnsembleModel._compute_normalization_constant..rGTintegerZpositiver6rE)rr r doitr)r3rInZ prod_termrGterm1term2Zterm3r$rHr&_compute_normalization_constant?s ( z5GaussianEnsembleModel._compute_normalization_constantc Cs|j}|||}td}tdddd}tdddd}tdddd}tt| dt||d|d|f}t|t t |||||||d|f} t | ||d|df} t |||d|f} tt | || |S) z Helper function for computing the joint probability distribution of eigen values of the random matrix. liTrJrGkrEr6) rBrQrr rrr rMrr rrtuple) r3rIrNZbnrRrSrGrTrOZsub_termrPsymsr$r$r&!_compute_joint_eigen_distributionRs .. z7GaussianEnsembleModel._compute_joint_eigen_distributionN)r<r=r>r?rQrXr$r$r$r&rD3s rDc@0eZdZeddZddZddZddZd S) GaussianUnitaryEnsembleModelcCs*|j}dt|dtt|ddSNrE)rBrr)r3rNr$r$r&normalization_constantes$z3GaussianUnitaryEnsembleModel.normalization_constantcCsV|j|j}}td|d}td|||d}t|tt| dt|d||SNPmodelHpspacerErBr\rrrrrr)r3r9rNZZGUEh_pspacerar$r$r&r:j ,z$GaussianUnitaryEnsembleModel.densitycC|tdSr[rXrr2r$r$r&r!pz5GaussianUnitaryEnsembleModel.joint_eigen_distributioncCs:td}dtd|dtdt|d}t||S)Ns rEr rrrr3rjfr$r$r&r#ss( z7GaussianUnitaryEnsembleModel.level_spacing_distributionNr<r=r>r@r\r:r!r#r$r$r$r&rZds   rZc@rY) GaussianOrthogonalEnsembleModelcCs4|j}td||}ttt| dt|dS)N_HrLrErBrr rrrr3rNrrr$r$r&r\ys "z6GaussianOrthogonalEnsembleModel.normalization_constantcCsV|j|j}}td|d}td|||d}t|tt| dt|d||S)Nr^r_rarbrLrErd)r3r9rNZZGOErerar$r$r&r:rfz'GaussianOrthogonalEnsembleModel.densitycC |tjSr7rXrrFr2r$r$r&r! z8GaussianOrthogonalEnsembleModel.joint_eigen_distributioncCs4td}td|tt d|d}t||S)NrjrErLrmrnr$r$r&r#s" z:GaussianOrthogonalEnsembleModel.level_spacing_distributionNrpr$r$r$r&rqx   rqc@rY) GaussianSymplecticEnsembleModelcCs0|j}td||}ttt| t|dS)NrrrErsrtr$r$r&r\s z6GaussianSymplecticEnsembleModel.normalization_constantcCsR|j|j}}td|d}td|||d}t|tt| t|d||Sr]rd)r3r9rNZZGSErerar$r$r&r:s (z'GaussianSymplecticEnsembleModel.densitycCrgNrLrhr2r$r$r&r!riz8GaussianSymplecticEnsembleModel.joint_eigen_distributioncCsRtd}tddtddtd|dtddt|d}t||S) NrjrErLi )r rrrrrnr$r$r&r#s@ z:GaussianSymplecticEnsembleModel.level_spacing_distributionNrpr$r$r$r&ryrxrycC8t|t|}}t||}t||d}t||||dSNr_rb)rrrDrrr-r.r`Zrmpr$r$r&r  rcCr)a- Represents Gaussian Unitary Ensembles. Examples ======== >>> from sympy.stats import GaussianUnitaryEnsemble as GUE, density >>> from sympy import MatrixSymbol >>> G = GUE('U', 2) >>> X = MatrixSymbol('X', 2, 2) >>> density(G)(X) exp(-Trace(X**2))/(2*pi**2) r_rb)rrrZrrrr$r$r&r  rcCr)aN Represents Gaussian Orthogonal Ensembles. Examples ======== >>> from sympy.stats import GaussianOrthogonalEnsemble as GOE, density >>> from sympy import MatrixSymbol >>> G = GOE('U', 2) >>> X = MatrixSymbol('X', 2, 2) >>> density(G)(X) exp(-Trace(X**2)/2)/Integral(exp(-Trace(_H**2)/2), _H) r_rb)rrrqrrrr$r$r&rrrcCr)aN Represents Gaussian Symplectic Ensembles. Examples ======== >>> from sympy.stats import GaussianSymplecticEnsemble as GSE, density >>> from sympy import MatrixSymbol >>> G = GSE('U', 2) >>> X = MatrixSymbol('X', 2, 2) >>> density(G)(X) exp(-2*Trace(X**2))/Integral(exp(-2*Trace(_H**2)), _H) r_rb)rrryrrrr$r$r&r rr c@rC)CircularEnsembleModelz Abstract class for Circular ensembles. Contains the properties and methods common to all the circular ensembles. References ========== .. [1] https://en.wikipedia.org/wiki/Circular_ensemble cCs td|)NzeSupport for Haar measure hasn't been implemented yet, therefore the density of %s cannot be computed.)NotImplementedErrorr8r$r$r&r:szCircularEnsembleModel.densityc Cs|j}dt|t||ddtt|dd|}td}tdddtdddtddd}}}t|||d|f}ttt t t ||t t |||||d|f|d|df} t t || |S) z Helper function to compute the joint distribution of phases of the complex eigen values of matrices belonging to any circular ensembles. rEr6trST)rKrGrT)rBrr rrr rrMr rrrrrU) r3rIrNrVrrSrGrTrWror$r$r&rXs8  < z7CircularEnsembleModel._compute_joint_eigen_distributionN)r<r=r>r?r:rXr$r$r$r&rs rc@eZdZddZdS)CircularUnitaryEnsembleModelcCrgr[rhr2r$r$r&r!riz5CircularUnitaryEnsembleModel.joint_eigen_distributionNr<r=r>r!r$r$r$r&r rc@r)CircularOrthogonalEnsembleModelcCrur7rvr2r$r$r&r!rwz8CircularOrthogonalEnsembleModel.joint_eigen_distributionNrr$r$r$r&rrrc@r)CircularSymplecticEnsembleModelcCrgrzrhr2r$r$r&r! riz8CircularSymplecticEnsembleModel.joint_eigen_distributionNrr$r$r$r&r rrcCrr)rrrrrrr$r$r&rrrcCr)a7 Represents Cicular Unitary Ensembles. Examples ======== >>> from sympy.stats import CircularUnitaryEnsemble as CUE >>> from sympy.stats import joint_eigen_distribution >>> C = CUE('U', 1) >>> joint_eigen_distribution(C) Lambda(t[1], Product(Abs(exp(I*t[_j]) - exp(I*t[_k]))**2, (_j, _k + 1, 1), (_k, 1, 0))/(2*pi)) Note ==== As can be seen above in the example, density of CiruclarUnitaryEnsemble is not evaluated becuase the exact definition is based on haar measure of unitary group which is not unique. r_rb)rrrrrrr$r$r&r  rcCr)a= Represents Cicular Orthogonal Ensembles. Examples ======== >>> from sympy.stats import CircularOrthogonalEnsemble as COE >>> from sympy.stats import joint_eigen_distribution >>> C = COE('O', 1) >>> joint_eigen_distribution(C) Lambda(t[1], Product(Abs(exp(I*t[_j]) - exp(I*t[_k])), (_j, _k + 1, 1), (_k, 1, 0))/(2*pi)) Note ==== As can be seen above in the example, density of CiruclarOrthogonalEnsemble is not evaluated becuase the exact definition is based on haar measure of unitary group which is not unique. r_rb)rrrrrrr$r$r&r.rrcCr)a@ Represents Cicular Symplectic Ensembles. Examples ======== >>> from sympy.stats import CircularSymplecticEnsemble as CSE >>> from sympy.stats import joint_eigen_distribution >>> C = CSE('S', 1) >>> joint_eigen_distribution(C) Lambda(t[1], Product(Abs(exp(I*t[_j]) - exp(I*t[_k]))**4, (_j, _k + 1, 1), (_k, 1, 0))/(2*pi)) Note ==== As can be seen above in the example, density of CiruclarSymplecticEnsemble is not evaluated becuase the exact definition is based on haar measure of unitary group which is not unique. r_rb)rrrrrrr$r$r&rGrrcCs"t|ts td||jjS)aA For obtaining joint probability distribution of eigen values of random matrix. Parameters ========== mat: RandomMatrixSymbol The matrix symbol whose eigen values are to be considered. Returns ======= Lambda Examples ======== >>> from sympy.stats import GaussianUnitaryEnsemble as GUE >>> from sympy.stats import joint_eigen_distribution >>> U = GUE('U', 2) >>> joint_eigen_distribution(U) Lambda((l[1], l[2]), exp(-l[1]**2 - l[2]**2)*Product(Abs(l[_i] - l[_j])**2, (_j, _i + 1, 2), (_i, 1, 1))/pi) z&%s is not of type, RandomMatrixSymbol.) isinstancerr*rcr`r!matr$r$r&r!`s   r!cCs2|jdd}tddt|Drtdt|S)a Creates joint distribution of eigen values of matrices with random expressions. Parameters ========== mat: Matrix The matrix under consideration. Returns ======= JointDistributionHandmade Examples ======== >>> from sympy.stats import Normal, JointEigenDistribution >>> from sympy import Matrix >>> A = [[Normal('A00', 0, 1), Normal('A01', 0, 1)], ... [Normal('A10', 0, 1), Normal('A11', 0, 1)]] >>> JointEigenDistribution(Matrix(A)) JointDistributionHandmade(-sqrt(A00**2 - 2*A00*A11 + 4*A01*A10 + A11**2)/2 + A00/2 + A11/2, sqrt(A00**2 - 2*A00*A11 + 4*A01*A10 + A11**2)/2 + A00/2 + A11/2) T)Zmultiplecss|]}t| VqdSr7)r).0Zeigenvalr$r$r& sz)JointEigenDistribution..zVEigen values don't have any random expression, joint distribution cannot be generated.) eigenvalsanysetr*r)rrr$r$r&r"}s r"cCs |jjS)a For obtaining distribution of level spacings. Parameters ========== mat: RandomMatrixSymbol The random matrix symbol whose eigen values are to be considered for finding the level spacings. Returns ======= Lambda Examples ======== >>> from sympy.stats import GaussianUnitaryEnsemble as GUE >>> from sympy.stats import level_spacing_distribution >>> U = GUE('U', 2) >>> level_spacing_distribution(U) Lambda(_s, 32*_s**2*exp(-4*_s**2/pi)/pi**2) References ========== .. [1] https://en.wikipedia.org/wiki/Random_matrix#Distribution_of_level_spacings )rcr`r#rr$r$r&r#s r#N)4Zsympyrrrrrrrr r r r r rrrZsympy.core.sympifyrZsympy.stats.rvrrrrZsympy.stats.joint_rv_typesrZsympy.stats.random_matrixrZsympy.tensor.arrayr__all__registerr'r(rDrZrqryrrrr rrrrrrrrr!r"r#r$r$r$r&s:D     1" "