o o h;@sddlmZddlmZddlmZddlmZddlm Z m Z ddl m Z ddl mZddlmZdd lmZdd lmZdd lmZdd lmZdd lmZddlmZddlmZddlm Z m!Z!m"Z"m#Z#ddl$m%Z%ddl&m'Z'ddl(m)Z)gdZ*e#+e"ddZ,GdddeZ-Gddde-Z.Gddde.Z/Gddde.Z0Gdd d e.Z1d!d"Z2d#d$Z3d%d&Z4d'd(Z5Gd)d*d*e-Z6Gd+d,d,e6Z7Gd-d.d.e6Z8Gd/d0d0e6Z9d1d2Z:d3d4Z;d5d6Zd;d<Z?d=d>Z@d?S)@)Product)Sum)Basic)Lambda)Ipi)S)Dummy)Abs)exp)gamma)Integral) MatrixSymbol)Trace) IndexedBase)_sympify)_symbol_converterDensityRandomMatrixSymbol is_random)JointDistributionHandmade)RandomMatrixPSpace)ArrayComprehension) CircularEnsembleCircularUnitaryEnsembleCircularOrthogonalEnsembleCircularSymplecticEnsembleGaussianEnsembleGaussianUnitaryEnsembleGaussianOrthogonalEnsembleGaussianSymplecticEnsemblejoint_eigen_distributionJointEigenDistributionlevel_spacing_distributioncCsdS)NT)xr$r$t/var/www/html/construction_image-detection-poc/venv/lib/python3.10/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|}}|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&6 z"RandomMatrixEnsembleModel.cCr/)Nr0r2r$r$r&r47r5cCst|SN)rr3exprr$r$r&density9sz!RandomMatrixEnsembleModel.densitycCs ||Sr7)r:r8r$r$r&__call__<s 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&r4Ws.zGGaussianEnsembleModel._compute_normalization_constant..rGTintegerpositiver6rE)rr rdoitr)r3rIn prod_termrGterm1term2term3r$rHr&_compute_normalization_constantKs ( z5GaussianEnsembleModel._compute_normalization_constantc Cs|j}|||}td}tdddd}tdddd}tdddd}tt| dt||d|d|f}t|t t |||||||d|f} t | ||d|df} t |||d|f} tt | || |S) z Helper function for computing the joint probability distribution of eigen values of the random matrix. liTrJrGkrEr6) rBrTrr r rrrNrrr rtuple) r3rIrOZbnrUrVrGrWrQsub_termrRsymsr$r$r&!_compute_joint_eigen_distribution^s .. z7GaussianEnsembleModel._compute_joint_eigen_distributionN)r<r=r>r?rTr\r$r$r$r&rD?s rDc@0eZdZeddZddZddZddZd S) GaussianUnitaryEnsembleModelcCs*|j}dt|dtt|ddSNrE)rBrr)r3rOr$r$r&normalization_constantqs$z3GaussianUnitaryEnsembleModel.normalization_constantcCsV|j|j}}td|d}td|||d}t|tt| dt|d||SNPmodelHpspacerErBr`rrrr rr)r3r9rOZGUEh_pspacerer$r$r&r:v ,z$GaussianUnitaryEnsembleModel.densitycC|tdSr_r\rr2r$r$r&r!|z5GaussianUnitaryEnsembleModel.joint_eigen_distributioncCs:td}dtd|dtdt|d}t||S)Ns rEr rr rr3rofr$r$r&r#s( z7GaussianUnitaryEnsembleModel.level_spacing_distributionNr<r=r>r@r`r:r!r#r$r$r$r&r^ps   r^c@r]) GaussianOrthogonalEnsembleModelcCs4|j}td||}ttt| dt|dS)N_HrMrErBrr r rrr3rOrwr$r$r&r`s "z6GaussianOrthogonalEnsembleModel.normalization_constantcCsV|j|j}}td|d}td|||d}t|tt| dt|d||S)NrbrcrerfrMrErh)r3r9rOZGOErjrer$r$r&r:rkz'GaussianOrthogonalEnsembleModel.densitycC |tjSr7r\rrFr2r$r$r&r! z8GaussianOrthogonalEnsembleModel.joint_eigen_distributioncCs4td}td|tt d|d}t||S)NrorErMrrrsr$r$r&r#s" z:GaussianOrthogonalEnsembleModel.level_spacing_distributionNrur$r$r$r&rv   rvc@r]) GaussianSymplecticEnsembleModelcCs0|j}td||}ttt| t|dS)NrwrErxryr$r$r&r`s z6GaussianSymplecticEnsembleModel.normalization_constantcCsR|j|j}}td|d}td|||d}t|tt| t|d||Srarh)r3r9rOZGSErjrer$r$r&r:s (z'GaussianSymplecticEnsembleModel.densitycCrlNrMrmr2r$r$r&r!rnz8GaussianSymplecticEnsembleModel.joint_eigen_distributioncCsRtd}tddtddtd|dtddt|d}t||S) NrorErMi )r rrr rrsr$r$r&r#s@ z:GaussianSymplecticEnsembleModel.level_spacing_distributionNrur$r$r$r&rr~rcC8t|t|}}t||}t||d}t||||dSNrcrf)rrrDrrr-r.rdrmpr$r$r&r  rcCr)a- Represents Gaussian Unitary Ensembles. Examples ======== >>> from sympy.stats import GaussianUnitaryEnsemble as GUE, density >>> from sympy import MatrixSymbol >>> G = GUE('U', 2) >>> X = MatrixSymbol('X', 2, 2) >>> density(G)(X) exp(-Trace(X**2))/(2*pi**2) rcrf)rrr^rrrr$r$r&r  rcCr)aN Represents Gaussian Orthogonal Ensembles. Examples ======== >>> from sympy.stats import GaussianOrthogonalEnsemble as GOE, density >>> from sympy import MatrixSymbol >>> G = GOE('U', 2) >>> X = MatrixSymbol('X', 2, 2) >>> density(G)(X) exp(-Trace(X**2)/2)/Integral(exp(-Trace(_H**2)/2), _H) rcrf)rrrvrrrr$r$r&rrrcCr)aN Represents Gaussian Symplectic Ensembles. Examples ======== >>> from sympy.stats import GaussianSymplecticEnsemble as GSE, density >>> from sympy import MatrixSymbol >>> G = GSE('U', 2) >>> X = MatrixSymbol('X', 2, 2) >>> density(G)(X) exp(-2*Trace(X**2))/Integral(exp(-2*Trace(_H**2)), _H) rcrf)rrrrrrr$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$r&r:szCircularEnsembleModel.densityc Cs|j}dt|t||ddtt|dd|}td}tdddtdddtddd}}}t|||d|f}ttt t t ||t t |||||d|f|d|df} t t || |S) z Helper function to compute the joint distribution of phases of the complex eigen values of matrices belonging to any circular ensembles. rEr6trVT)rKrGrW)rBrr rrr rrNrr r rrrX) r3rIrOrYrrVrGrWr[rtr$r$r&r\s8  < z7CircularEnsembleModel._compute_joint_eigen_distributionN)r<r=r>r?r:r\r$r$r$r&rs rc@eZdZddZdS)CircularUnitaryEnsembleModelcCrlr_rmr2r$r$r&r!rnz5CircularUnitaryEnsembleModel.joint_eigen_distributionNr<r=r>r!r$r$r$r&r rc@r)CircularOrthogonalEnsembleModelcCr{r7r|r2r$r$r&r!r}z8CircularOrthogonalEnsembleModel.joint_eigen_distributionNrr$r$r$r&rrrc@r)CircularSymplecticEnsembleModelcCrlrrmr2r$r$r&r!rnz8CircularSymplecticEnsembleModel.joint_eigen_distributionNrr$r$r$r&rrrcCrr)rrrrrrr$r$r&rrrcCr)a8 Represents Circular Unitary Ensembles. Examples ======== >>> from sympy.stats import CircularUnitaryEnsemble as CUE >>> from sympy.stats import joint_eigen_distribution >>> C = CUE('U', 1) >>> joint_eigen_distribution(C) Lambda(t[1], Product(Abs(exp(I*t[_j]) - exp(I*t[_k]))**2, (_j, _k + 1, 1), (_k, 1, 0))/(2*pi)) Note ==== As can be seen above in the example, density of CiruclarUnitaryEnsemble is not evaluated because the exact definition is based on haar measure of unitary group which is not unique. rcrf)rrrrrrr$r$r&r!  rcCr)a> Represents Circular Orthogonal Ensembles. Examples ======== >>> from sympy.stats import CircularOrthogonalEnsemble as COE >>> from sympy.stats import joint_eigen_distribution >>> C = COE('O', 1) >>> joint_eigen_distribution(C) Lambda(t[1], Product(Abs(exp(I*t[_j]) - exp(I*t[_k])), (_j, _k + 1, 1), (_k, 1, 0))/(2*pi)) Note ==== As can be seen above in the example, density of CiruclarOrthogonalEnsemble is not evaluated because the exact definition is based on haar measure of unitary group which is not unique. rcrf)rrrrrrr$r$r&r:rrcCr)aA Represents Circular Symplectic Ensembles. Examples ======== >>> from sympy.stats import CircularSymplecticEnsemble as CSE >>> from sympy.stats import joint_eigen_distribution >>> C = CSE('S', 1) >>> joint_eigen_distribution(C) Lambda(t[1], Product(Abs(exp(I*t[_j]) - exp(I*t[_k]))**4, (_j, _k + 1, 1), (_k, 1, 0))/(2*pi)) Note ==== As can be seen above in the example, density of CiruclarSymplecticEnsemble is not evaluated because the exact definition is based on haar measure of unitary group which is not unique. rcrf)rrrrrrr$r$r&rSrrcCs"t|ts td||jjS)aA For obtaining joint probability distribution of eigen values of random matrix. Parameters ========== mat: RandomMatrixSymbol The matrix symbol whose eigen values are to be considered. Returns ======= Lambda Examples ======== >>> from sympy.stats import GaussianUnitaryEnsemble as GUE >>> from sympy.stats import joint_eigen_distribution >>> U = GUE('U', 2) >>> joint_eigen_distribution(U) Lambda((l[1], l[2]), exp(-l[1]**2 - l[2]**2)*Product(Abs(l[_i] - l[_j])**2, (_j, _i + 1, 2), (_i, 1, 1))/pi) z&%s is not of type, RandomMatrixSymbol.) isinstancerr*rgrdr!matr$r$r&r!ls   r!cCs2|jdd}tddt|Dstdt|S)a Creates joint distribution of eigen values of matrices with random expressions. Parameters ========== mat: Matrix The matrix under consideration. Returns ======= JointDistributionHandmade Examples ======== >>> from sympy.stats import Normal, JointEigenDistribution >>> from sympy import Matrix >>> A = [[Normal('A00', 0, 1), Normal('A01', 0, 1)], ... [Normal('A10', 0, 1), Normal('A11', 0, 1)]] >>> JointEigenDistribution(Matrix(A)) JointDistributionHandmade(-sqrt(A00**2 - 2*A00*A11 + 4*A01*A10 + A11**2)/2 + A00/2 + A11/2, sqrt(A00**2 - 2*A00*A11 + 4*A01*A10 + A11**2)/2 + A00/2 + A11/2) T)multiplecss|]}t|VqdSr7)r).0eigenvalr$r$r& sz)JointEigenDistribution..zWEigen values do not have any random expression, joint distribution cannot be generated.) eigenvalsallsetr*r)rrr$r$r&r"s r"cCs |jjS)a For obtaining distribution of level spacings. Parameters ========== mat: RandomMatrixSymbol The random matrix symbol whose eigen values are to be considered for finding the level spacings. Returns ======= Lambda Examples ======== >>> from sympy.stats import GaussianUnitaryEnsemble as GUE >>> from sympy.stats import level_spacing_distribution >>> U = GUE('U', 2) >>> level_spacing_distribution(U) Lambda(_s, 32*_s**2*exp(-4*_s**2/pi)/pi**2) References ========== .. [1] https://en.wikipedia.org/wiki/Random_matrix#Distribution_of_level_spacings )rgrdr#rr$r$r&r#s r#N)Asympy.concrete.productsrsympy.concrete.summationsrsympy.core.basicrsympy.core.functionrsympy.core.numbersrrsympy.core.singletonrsympy.core.symbolr $sympy.functions.elementary.complexesr &sympy.functions.elementary.exponentialr 'sympy.functions.special.gamma_functionsr sympy.integrals.integralsr "sympy.matrices.expressions.matexprr sympy.matrices.expressions.tracersympy.tensor.indexedrsympy.core.sympifyrsympy.stats.rvrrrrsympy.stats.joint_rv_typesrsympy.stats.random_matrixrsympy.tensor.arrayr__all__registerr'r(rDr^rvrrrrr rrrrrrrrr!r"r#r$r$r$r&sT                 1" "