U 9%e; @s ddlmZddlmZddlmZddlmZddlm Z m Z ddl m Z ddl mZddlmZdd lmZdd lmZdd lmZdd lmZdd lmZddlmZddlmZddlm Z m!Z!m"Z"m#Z#ddl$m%Z%ddl&m'Z'ddl(m)Z)dddddddddddg Z*e#+e"dd Z,Gd!d"d"eZ-Gd#d$d$e-Z.Gd%d&d&e.Z/Gd'd(d(e.Z0Gd)d*d*e.Z1d+dZ2d,dZ3d-dZ4d.dZ5Gd/d0d0e-Z6Gd1d2d2e6Z7Gd3d4d4e6Z8Gd5d6d6e6Z9d7dZ:d8dZ;d9dZdS)?)Product)Sum)Basic)Lambda)Ipi)S)Dummy)Abs)exp)gamma)Integral) MatrixSymbol)Trace) IndexedBase)_sympify)_symbol_converterDensityRandomMatrixSymbol is_random)JointDistributionHandmade)RandomMatrixPSpace)ArrayComprehensionCircularEnsembleCircularUnitaryEnsembleCircularOrthogonalEnsembleCircularSymplecticEnsembleGaussianEnsembleGaussianUnitaryEnsembleGaussianOrthogonalEnsembleGaussianSymplecticEnsemblejoint_eigen_distributionJointEigenDistributionlevel_spacing_distributioncCsdS)NT)xr$r$_/var/www/html/Darija-Ai-API/env/lib/python3.8/site-packages/sympy/stats/random_matrix_models.py_#sr'c@sBeZdZdZd ddZeddZeddZdd Zd d Z dS) RandomMatrixEnsembleModelz Base class for random matrix ensembles. It acts as an umbrella and contains the methods common to all the ensembles defined in sympy.stats.random_matrix_models. NcCs6t|t|}}|jdkr(td|t|||S)NFzGDimension of the random matrices must be integers, received %s instead.)rr is_integer ValueErrorr__new__)clssymdimr$r$r&r+/s  z!RandomMatrixEnsembleModel.__new__cCs |jdS)Nrargsselfr$r$r&6z"RandomMatrixEnsembleModel.cCs |jdS)Nr/r1r$r$r&r37r4cCst|SN)rr2exprr$r$r&density9sz!RandomMatrixEnsembleModel.densitycCs ||Sr6)r9r7r$r$r&__call__<sz"RandomMatrixEnsembleModel.__call__)N) __name__ __module__ __qualname____doc__r+propertysymbol dimensionr9r:r$r$r$r&r((s    r(c@s 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)Nr5)r rOne)jbetar$r&r3Wr4zGGaussianEnsembleModel._compute_normalization_constant..rETintegerZpositiver5rC)rr rdoitr)r2rGnZ prod_termrEterm1term2Zterm3r$rFr&_compute_normalization_constantKs  (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. liTrHrEkrCr5) rArOrr r rrrKrrr rtuple) r2rGrLZbnrPrQrErRrMZsub_termrNsymsr$r$r&!_compute_joint_eigen_distribution^s .. z7GaussianEnsembleModel._compute_joint_eigen_distributionN)r;r<r=r>rOrVr$r$r$r&rB?s rBc@s0eZdZeddZddZddZddZd S) GaussianUnitaryEnsembleModelcCs*|j}dt|dtt|ddSNrC)rArr)r2rLr$r$r&normalization_constantqsz3GaussianUnitaryEnsembleModel.normalization_constantcCsV|j|j}}td|d}td|||d}t|tt| dt|d||SNPmodelHpspacerCrArYrrrr rr)r2r8rLZZGUEh_pspacer^r$r$r&r9vs z$GaussianUnitaryEnsembleModel.densitycCs|tdSrXrVrr1r$r$r&r!|sz5GaussianUnitaryEnsembleModel.joint_eigen_distributioncCs:td}dtd|dtdt|d}t||S)Ns rCr rr rr2rdfr$r$r&r#s(z7GaussianUnitaryEnsembleModel.level_spacing_distributionNr;r<r=r?rYr9r!r#r$r$r$r&rWps  rWc@s0eZdZeddZddZddZddZd S) GaussianOrthogonalEnsembleModelcCs4|j}td||}ttt| dt|dS)N_HrJrCrArr r rrr2rLrlr$r$r&rYs z6GaussianOrthogonalEnsembleModel.normalization_constantcCsV|j|j}}td|d}td|||d}t|tt| dt|d||S)Nr[r\r^r_rJrCra)r2r8rLZZGOErbr^r$r$r&r9s z'GaussianOrthogonalEnsembleModel.densitycCs |tjSr6rVrrDr1r$r$r&r!sz8GaussianOrthogonalEnsembleModel.joint_eigen_distributioncCs4td}td|tt d|d}t||S)NrdrCrJrgrhr$r$r&r#s"z:GaussianOrthogonalEnsembleModel.level_spacing_distributionNrjr$r$r$r&rks  rkc@s0eZdZeddZddZddZddZd S) GaussianSymplecticEnsembleModelcCs0|j}td||}ttt| t|dS)NrlrCrmrnr$r$r&rYs z6GaussianSymplecticEnsembleModel.normalization_constantcCsR|j|j}}td|d}td|||d}t|tt| t|d||SrZra)r2r8rLZZGSErbr^r$r$r&r9s z'GaussianSymplecticEnsembleModel.densitycCs|tdSNrJrcr1r$r$r&r!sz8GaussianSymplecticEnsembleModel.joint_eigen_distributioncCsRtd}tddtddtd|dtddt|d}t||S) NrdrCrJi )r rrr rrhr$r$r&r#s@z:GaussianSymplecticEnsembleModel.level_spacing_distributionNrjr$r$r$r&rps  rpcCs8t|t|}}t||}t||d}t||||dSNr\r_)rrrBrrr-r.r]Zrmpr$r$r&rs  cCs8t|t|}}t||}t||d}t||||dS)a- Represents Gaussian Unitary Ensembles. Examples ======== >>> from sympy.stats import GaussianUnitaryEnsemble as GUE, density >>> from sympy import MatrixSymbol >>> G = GUE('U', 2) >>> X = MatrixSymbol('X', 2, 2) >>> density(G)(X) exp(-Trace(X**2))/(2*pi**2) r\r_)rrrWrrrwr$r$r&rs  cCs8t|t|}}t||}t||d}t||||dS)aN Represents Gaussian Orthogonal Ensembles. Examples ======== >>> from sympy.stats import GaussianOrthogonalEnsemble as GOE, density >>> from sympy import MatrixSymbol >>> G = GOE('U', 2) >>> X = MatrixSymbol('X', 2, 2) >>> density(G)(X) exp(-Trace(X**2)/2)/Integral(exp(-Trace(_H**2)/2), _H) r\r_)rrrkrrrwr$r$r&rs  cCs8t|t|}}t||}t||d}t||||dS)aN Represents Gaussian Symplectic Ensembles. Examples ======== >>> from sympy.stats import GaussianSymplecticEnsemble as GSE, density >>> from sympy import MatrixSymbol >>> G = GSE('U', 2) >>> X = MatrixSymbol('X', 2, 2) >>> density(G)(X) exp(-2*Trace(X**2))/Integral(exp(-2*Trace(_H**2)), _H) r\r_)rrrprrrwr$r$r&r s  c@s eZdZdZddZddZdS)CircularEnsembleModelz Abstract class for Circular ensembles. Contains the properties and methods common to all the circular ensembles. References ========== .. [1] https://en.wikipedia.org/wiki/Circular_ensemble cCstd|dS)NzeSupport for Haar measure hasn't been implemented yet, therefore the density of %s cannot be computed.)NotImplementedErrorr7r$r$r&r9szCircularEnsembleModel.densityc Cs|j}dt|t||ddtt|dd|}td}tdddtdddtddd}}}t|||d|f}ttt t t ||t t |||||d|f|d|df} t t || |S) z Helper function to compute the joint distribution of phases of the complex eigen values of matrices belonging to any circular ensembles. rCr5trQT)rIrErR)rArr rrr rrKrr r rrrS) r2rGrLrTrzrQrErRrUrir$r$r&rVs8 < z7CircularEnsembleModel._compute_joint_eigen_distributionN)r;r<r=r>r9rVr$r$r$r&rxs rxc@seZdZddZdS)CircularUnitaryEnsembleModelcCs|tdSrXrcr1r$r$r&r!sz5CircularUnitaryEnsembleModel.joint_eigen_distributionNr;r<r=r!r$r$r$r&r{sr{c@seZdZddZdS)CircularOrthogonalEnsembleModelcCs |tjSr6ror1r$r$r&r!sz8CircularOrthogonalEnsembleModel.joint_eigen_distributionNr|r$r$r$r&r}sr}c@seZdZddZdS)CircularSymplecticEnsembleModelcCs|tdSrqrcr1r$r$r&r!sz8CircularSymplecticEnsembleModel.joint_eigen_distributionNr|r$r$r$r&r~sr~cCs8t|t|}}t||}t||d}t||||dSrv)rrrxrrrwr$r$r&rs  cCs8t|t|}}t||}t||d}t||||dS)a8 Represents Circular Unitary Ensembles. Examples ======== >>> from sympy.stats import CircularUnitaryEnsemble as CUE >>> from sympy.stats import joint_eigen_distribution >>> C = CUE('U', 1) >>> joint_eigen_distribution(C) Lambda(t[1], Product(Abs(exp(I*t[_j]) - exp(I*t[_k]))**2, (_j, _k + 1, 1), (_k, 1, 0))/(2*pi)) Note ==== As can be seen above in the example, density of CiruclarUnitaryEnsemble is not evaluated because the exact definition is based on haar measure of unitary group which is not unique. r\r_)rrr{rrrwr$r$r&r!s  cCs8t|t|}}t||}t||d}t||||dS)a> Represents Circular Orthogonal Ensembles. Examples ======== >>> from sympy.stats import CircularOrthogonalEnsemble as COE >>> from sympy.stats import joint_eigen_distribution >>> C = COE('O', 1) >>> joint_eigen_distribution(C) Lambda(t[1], Product(Abs(exp(I*t[_j]) - exp(I*t[_k])), (_j, _k + 1, 1), (_k, 1, 0))/(2*pi)) Note ==== As can be seen above in the example, density of CiruclarOrthogonalEnsemble is not evaluated because the exact definition is based on haar measure of unitary group which is not unique. r\r_)rrr}rrrwr$r$r&r:s  cCs8t|t|}}t||}t||d}t||||dS)aA Represents Circular Symplectic Ensembles. Examples ======== >>> from sympy.stats import CircularSymplecticEnsemble as CSE >>> from sympy.stats import joint_eigen_distribution >>> C = CSE('S', 1) >>> joint_eigen_distribution(C) Lambda(t[1], Product(Abs(exp(I*t[_j]) - exp(I*t[_k]))**4, (_j, _k + 1, 1), (_k, 1, 0))/(2*pi)) Note ==== As can be seen above in the example, density of CiruclarSymplecticEnsemble is not evaluated because the exact definition is based on haar measure of unitary group which is not unique. r\r_)rrr~rrrwr$r$r&rSs  cCs"t|tstd||jjS)aA For obtaining joint probability distribution of eigen values of random matrix. Parameters ========== mat: RandomMatrixSymbol The matrix symbol whose eigen values are to be considered. Returns ======= Lambda Examples ======== >>> from sympy.stats import GaussianUnitaryEnsemble as GUE >>> from sympy.stats import joint_eigen_distribution >>> U = GUE('U', 2) >>> joint_eigen_distribution(U) Lambda((l[1], l[2]), exp(-l[1]**2 - l[2]**2)*Product(Abs(l[_i] - l[_j])**2, (_j, _i + 1, 2), (_i, 1, 1))/pi) z&%s is not of type, RandomMatrixSymbol.) isinstancerr*r`r]r!matr$r$r&r!ls  cCs2|jdd}tddt|Ds*tdt|S)a Creates joint distribution of eigen values of matrices with random expressions. Parameters ========== mat: Matrix The matrix under consideration. Returns ======= JointDistributionHandmade Examples ======== >>> from sympy.stats import Normal, JointEigenDistribution >>> from sympy import Matrix >>> A = [[Normal('A00', 0, 1), Normal('A01', 0, 1)], ... [Normal('A10', 0, 1), Normal('A11', 0, 1)]] >>> JointEigenDistribution(Matrix(A)) JointDistributionHandmade(-sqrt(A00**2 - 2*A00*A11 + 4*A01*A10 + A11**2)/2 + A00/2 + A11/2, sqrt(A00**2 - 2*A00*A11 + 4*A01*A10 + A11**2)/2 + A00/2 + A11/2) T)Zmultiplecss|]}t|VqdSr6)r).0Zeigenvalr$r$r& sz)JointEigenDistribution..zWEigen values do not have any random expression, joint distribution cannot be generated.) eigenvalsallsetr*r)rrr$r$r&r"s cCs |jjS)a For obtaining distribution of level spacings. Parameters ========== mat: RandomMatrixSymbol The random matrix symbol whose eigen values are to be considered for finding the level spacings. Returns ======= Lambda Examples ======== >>> from sympy.stats import GaussianUnitaryEnsemble as GUE >>> from sympy.stats import level_spacing_distribution >>> U = GUE('U', 2) >>> level_spacing_distribution(U) Lambda(_s, 32*_s**2*exp(-4*_s**2/pi)/pi**2) References ========== .. [1] https://en.wikipedia.org/wiki/Random_matrix#Distribution_of_level_spacings )r`r]r#rr$r$r&r#sN)AZsympy.concrete.productsrZsympy.concrete.summationsrZsympy.core.basicrZsympy.core.functionrZsympy.core.numbersrrZsympy.core.singletonrZsympy.core.symbolr Z$sympy.functions.elementary.complexesr Z&sympy.functions.elementary.exponentialr Z'sympy.functions.special.gamma_functionsr Zsympy.integrals.integralsr Z"sympy.matrices.expressions.matexprrZ sympy.matrices.expressions.tracerZsympy.tensor.indexedrZsympy.core.sympifyrZsympy.stats.rvrrrrZsympy.stats.joint_rv_typesrZsympy.stats.random_matrixrZsympy.tensor.arrayr__all__registerr'r(rBrWrkrprrrr rxr{r}r~rrrrr!r"r#r$r$r$r&sh                  1""