\h;8SSKJr SSKJr SSKJr SSKJr SSKJ r J r SSK J r SSK Jr SSKJr SS KJr SS KJr SS KJr SS KJr SS KJr SSKJr SSKJr SSKJ r J!r!J"r"J#r# SSK$J%r% SSK&J'r' SSK(J)r) /SQr*\#RV"\"5S5r,"SS\5r-"SS\-5r."SS\.5r/"SS\.5r0"SS\.5r1S r2S!r3S"r4S#r5"S$S%\-5r6"S&S'\65r7"S(S)\65r8"S*S+\65r9S,r:S-r;S.rS1r?S2r@g3)4)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_distributioncg)NT)xs X/var/www/auris/envauris/lib/python3.13/site-packages/sympy/stats/random_matrix_models.py_r)#s cR\rSrSrSrS Sjr\"S5r\"S5rSr Sr S r g) RandomMatrixEnsembleModel(z 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. Nc[U5[U5p!URS:Xa[SU-5e[R "XU5$)NFzGDimension of the random matrices must be integers, received %s instead.)rr is_integer ValueErrorr__new__)clssymdims r(r1!RandomMatrixEnsembleModel.__new__/sM$S)8C=S >>U "ABEGH H}}Ss++r*c URS$)Nrargsselfs r("RandomMatrixEnsembleModel.6s 499Q[ST[U5-S- -5[[RT[S5- -5- $)Nr>)r r One)jbetas r(r;GGaussianEnsembleModel._compute_normalization_constant..Ws5eAQqT ! O4U1554!9;L5MMr*rXTintegerpositiver>rV)r r rdoitr)r:rYn prod_termrXterm1term2term3s ` r(_compute_normalization_constant5GaussianEnsembleModel._compute_normalization_constantKs aDM #td 3 ! q!Qi0557DFtAvq1u~a/!A#5621 }u$$r*c @URnURX5n[S5n[SSSS9n[SSSS9n[SSSS9n[ [ U5*S- [ XGS-USU45R5-5n[U[[XFXE- 5U-XeS-U455n [U "U5R5USUS- 45R5n [XGUSU45R5n [[U 5X-U- 5$) zh Helper function for computing the joint probability distribution of eigen values of the random matrix. liTr[rXkrVr>) rOrerr r r rr_rrr rtuple) r:rYr`ZbnrhrirXrjrbsub_termrcsymss r(!_compute_joint_eigen_distribution7GaussianEnsembleModel._compute_joint_eigen_distribution^s NN224;   #td 3 #td 3 #td 3adU1WAD!GaAY 7 < < >>?!WS%5t%;aQ]KL ((*Q1q5M:??A!!$Aq 2779eDkEM3#677r*r&N)rHrIrJrKrLrerorPr&r*r(rRrR?s %&8r*rRc6\rSrSr\S5rSrSrSrSr g)GaussianUnitaryEnsembleModelpcnURnS[U5S- -[[US-5S- --$NrV)rOr r)r:r`s r(normalization_constant3GaussianUnitaryEnsembleModel.normalization_constantqs2 NN1Q46{R!AqD'!)_,,r*c URURp2[SUS9n[SX"US9n[ U[ [ U5*S- [US-5-5U- 5"U5$NPmodelHpspacerVrOrvrrrr r r)r:rBr`ZGUEh_pspacer}s r(rC$GaussianUnitaryEnsembleModel.densityv`..$"="=4%c6 sA :aadU1WuQT{23D89$??r*c6UR[S55$ruror r9s r(r"5GaussianUnitaryEnsembleModel.joint_eigen_distribution|55ad;;r*c[S5nS[S-- US--[S[- US--5-n[X5$)Ns rVr rr rr:rfs r(r$7GaussianUnitaryEnsembleModel.level_spacing_distributionsA #J AX1 c2b5!Q$,/ /a|r*r&N rHrIrJrKrMrvrCr"r$rPr&r*r(rrrrps$ --@ <r*rrc6\rSrSr\S5rSrSrSrSr g)GaussianOrthogonalEnsembleModelc URn[SX5n[[[ U5*S- [ US-5-55$)N_Hr^rVrOrrr r rr:r`rs r(rv6GaussianOrthogonalEnsembleModel.normalization_constants> NN $ %QqTE!GeBEl2344r*c URURp2[SUS9n[SX"US9n[ U[ [ U5*S- [US-5-5U- 5"U5$)Nrzr{r}r~r^rVr)r:rBr`ZGOErr}s r(rC'GaussianOrthogonalEnsembleModel.densityrr*c@UR[R5$r@ror rWr9s r(r"8GaussianOrthogonalEnsembleModel.joint_eigen_distribution55aee< <r*rch[U5[U5p[X5n[XS9n[ XXS9$Nr{r~)rrrRrrr3r4r|rmps r(rr3 %x} !# +E S .C c 88r*ch[U5[U5p[X5n[XS9n[ XXS9$)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~)rrrrrrrs r(rrs5!%x} ( 2E S .C c 88r*ch[U5[U5p[X5n[XS9n[ XXS9$)a& Represents Gaussian Orthogonal Ensembles. Examples ======== >>> from sympy.stats import GaussianOrthogonalEnsemble as GOE, density >>> from sympy import MatrixSymbol >>> G = GOE('U', 2) >>> X = MatrixSymbol('X', 2, 2) >>> density(G)(X) exp(-Trace(X**2)/2)/Integral(exp(-Trace(_H**2)/2), _H) r{r~)rrrrrrs r(r r 5!%x} +C 5E S .C c 88r*ch[U5[U5p[X5n[XS9n[ XXS9$)a& 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~)rrrrrrs r(r!r!rr*c$\rSrSrSrSrSrSrg)CircularEnsembleModelz Abstract class for Circular ensembles. Contains the properties and methods common to all the circular ensembles. References ========== .. [1] https://en.wikipedia.org/wiki/Circular_ensemble c[SU-5e)NzeSupport for Haar measure hasn't been implemented yet, therefore the density of %s cannot be computed.)NotImplementedErrorrAs r(rCCircularEnsembleModel.densitys!"#;<@#BC Cr*c 2URnS[-U-[X-S- S-5[[US- S-55U-- -n[ S5n[ SSS9[ SSS9[ SSS9pvn[ XEUSU45R5n[[[[[XG-5[[XF-5- 5U-XgS-U45R5USUS- 45R5n [[U5X- 5$) z Helper function to compute the joint distribution of phases of the complex eigen values of matrices belonging to any circular ensembles. rVr>triT)r\rXrj)rOrr r rr rr_rr r rrrk) r:rYr`rlrrirXrjrnrs r(ro7CircularEnsembleModel._compute_joint_eigen_distributions NN"qy5A.qtAvz1B/CQ/FFG  d+U3-Ed+!!$Aq 2779 GCAadF c!AD&k 9:D@1!eQ-PUUW1q5M ##'46 eDk15))r*r&N)rHrIrJrKrLrCrorPr&r*r(rrs C*r*rc\rSrSrSrSrg)CircularUnitaryEnsembleModelic6UR[S55$rurr9s r(r"5CircularUnitaryEnsembleModel.joint_eigen_distributionrr*r&NrHrIrJrKr"rPr&r*r(rr>> 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~)rrrrrrs r(rr!s5(!%x} ( 2E S .C c 88r*ch[U5[U5p[X5n[XS9n[ XXS9$)a Represents Circular Orthogonal Ensembles. Examples ======== >>> from sympy.stats import CircularOrthogonalEnsemble as COE >>> from sympy.stats import joint_eigen_distribution >>> C = COE('O', 1) >>> joint_eigen_distribution(C) Lambda(t[1], Product(Abs(exp(I*t[_j]) - exp(I*t[_k])), (_j, _k + 1, 1), (_k, 1, 0))/(2*pi)) Note ==== As can be seen above in the example, density of CiruclarOrthogonalEnsemble is not evaluated because the exact definition is based on haar measure of unitary group which is not unique. r{r~)rrrrrrs r(rr:5(!%x} +C 5E S .C c 88r*ch[U5[U5p[X5n[XS9n[ XXS9$)a  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~)rrrrrrs r(rrSrr*c[U[5(d[SU-5eURRR 5$)a For obtaining joint probability distribution of eigen values of random matrix. Parameters ========== mat: RandomMatrixSymbol The matrix symbol whose eigen values are to be considered. Returns ======= Lambda Examples ======== >>> from sympy.stats import GaussianUnitaryEnsemble as GUE >>> from sympy.stats import joint_eigen_distribution >>> U = GUE('U', 2) >>> joint_eigen_distribution(U) Lambda((l[1], l[2]), exp(-l[1]**2 - l[2]**2)*Product(Abs(l[_i] - l[_j])**2, (_j, _i + 1, 2), (_i, 1, 1))/pi) z&%s is not of type, RandomMatrixSymbol.) isinstancerr0rr|r"mats r(r"r"ls<2 c- . .A3GHH ::   4 4 66r*cURSS9n[S[U555(d [S5e[ U6$)aZ 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)multiplec38# UHn[U5v M g7fr@)r).0eigenvals r( )JointEigenDistribution..sB>xy"">szWEigen values do not have any random expression, joint distribution cannot be generated.) eigenvalsallsetr0r)rrs r(r#r#sI8 t ,I B3y>B B BCD D $i 00r*cJURRR5$)a3 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 )rr|r$rs r(r$r$s< ::   6 6 88r*N)Asympy.concrete.productsrsympy.concrete.summationsrsympy.core.basicrsympy.core.functionrsympy.core.numbersrrsympy.core.singletonr sympy.core.symbolr $sympy.functions.elementary.complexesr &sympy.functions.elementary.exponentialr 'sympy.functions.special.gamma_functionsr sympy.integrals.integralsr"sympy.matrices.expressions.matexprr sympy.matrices.expressions.tracersympy.tensor.indexedrsympy.core.sympifyrsympy.stats.rvrrrrsympy.stats.joint_rv_typesrsympy.stats.random_matrixrsympy.tensor.arrayr__all__registerr)r,rRrrrrrrr r!rrrrrrrrr"r#r$r&r*r(rs+)"&&"#469.;2,'TT@81  &'(""./85/8b#8(&;*&;*9 9&9&9& *5 *D<#8<=&;=<&;<9 9292927: 1D9r*