+ i;:^RIHt^RIHt^RIHt^RIHt^RIH t H t ^RI H t ^RI Ht^RIHt^R IHt^R IHt^R IHt^R IHt^R IHt^RIHt^RIHt^RIH t H!t!H"t"H#t#^RI$H%t%^RI&H't'^RI(H)t).R3Ot*]#PV!]"4R4t,!RR]4t-!RR]-4t.!RR].4t/!RR].4t0!RR].4t1Rt2R t3R!t4R"t5!R#R$]-4t6!R%R&]64t7!R'R(]64t8!R)R*]64t9R+t:R,t;R-tR0t?R1t@R2#)4)Product)Sum)Basic)Lambda)Ipi)S)Dummy)Abs)exp)gamma)Integral) MatrixSymbol)Trace) IndexedBase)_sympify)_symbol_converterDensityRandomMatrixSymbol is_random)JointDistributionHandmade)RandomMatrixPSpace)ArrayComprehensioncR#)T)xs&^/var/www/html/photoedit/myenv/lib/python3.14/site-packages/sympy/stats/random_matrix_models.py_r#s c^a]tRt^(toRtR Rlt]!R4t]!R4tRt Rt Rt Vt R#) 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. Nc\V4\V4r!VPR8Xd\RV,4h\P !WV4#)FzGDimension of the random matrices must be integers, received %s instead.)rr is_integer ValueErrorr__new__)clssymdims&&&rr%!RandomMatrixEnsembleModel.__new__/sM$S)8C=S >>U "ABEGH H}}Ss++rc(VP^,#)rargsselfs&r"RandomMatrixEnsembleModel.6s 499Q.Ws5eAQqT ! O4U1554!9;L5MMrrNTintegerpositive)r r rdoitr)r.rOn prod_termrNterm1term2term3s&f& r_compute_normalization_constant5GaussianEnsembleModel._compute_normalization_constantKs aDM #td 3 ! q!Qi0557DFtAvq1u~a/!A#5621 }u$$rc  VPpVPW4p\R4p\RRRR7p\RRRR7p\RRRR7p\ \ V4)^, \ WG,^,V^V34P4,4p\V\\WF,WE,, 4V,We^,V344p \V !V4P4V^V^, 34P4p \WG,V^V34P4p \\V 4W,V, 4#)zh Helper function for computing the joint probability distribution of eigen values of the random matrix. liTrQrNk) rDrZrr r r rrTrrr rtuple) r.rOrUZbnr]r^rNr_rWsub_termrXsymss && 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!!$Aq 2779eDkEM3#677rrN) r=r>r?r@rArZrdrErFrGs@rrJrJ?s %&88rrJcBa]tRt^pto]R4tRtRtRtRt Vt R#)GaussianUnitaryEnsembleModelcVPp^\V4^, ,\\V^,4^, ,,#)rDr r)r.rUs& rnormalization_constant3GaussianUnitaryEnsembleModel.normalization_constantqs2 NN1Q46{R!AqD'!)_,,rc VPVPr2\RVR7p\RW"VR7p\ V\ \ V4)^, \V^,4,4V, 4!V4#PmodelHpspacerDrkrrrr r r)r.r7rUZGUEh_pspacerrs&& rr8$GaussianUnitaryEnsembleModel.densityv`..$"="=4%c6 sA :aadU1WuQT{23D89$??rc6VP\^44#rirdr r-s&rjoint_eigen_distribution5GaussianUnitaryEnsembleModel.joint_eigen_distribution|55ad;;rc\R4p^ \^,, V^,,\R\, V^,,4,p\W4#)sr rr rr.rfs& rlevel_spacing_distribution7GaussianUnitaryEnsembleModel.level_spacing_distributionsA #J AX1 c2b5!Q$,/ /a|rrN r=r>r?r@rBrkr8r|rrErFrGs@rrgrgps, --@ <rrgcBa]tRt^to]R4tRtRtRtRt Vt R#)GaussianOrthogonalEnsembleModelc VPp\RW4p\\\ V4)^, \ V^,4,44#_HrDrrr r rr.rUrs& rrk6GaussianOrthogonalEnsembleModel.normalization_constants> NN $ %QqTE!GeBEl2344rc VPVPr2\RVR7p\RW"VR7p\ V\ \ V4)^, \V^,4,4V, 4!V4#rnru)r.r7rUZGOErwrrs&& rr8'GaussianOrthogonalEnsembleModel.densityryrc@VP\P4#r5rdr rMr-s&rr|8GaussianOrthogonalEnsembleModel.joint_eigen_distribution55aee<r?r@rAr8rdrErFrGs@rrrs C**rrc&a]tRtRtoRtRtVtR#)CircularUnitaryEnsembleModelic6VP\^44#rir{r-s&rr|5CircularUnitaryEnsembleModel.joint_eigen_distributionr~rrNr=r>r?r@r|rErFrGs@rrr<>> 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. rprs)rrrrrrs&& rCircularUnitaryEnsembler!s5(!%x} ( 2E S .C c 88rct\V4\V4r\W4p\WR7p\ WWR7#)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. rprs)rrrrrrs&& rCircularOrthogonalEnsembler:5(!%x} +C 5E S .C c 88rct\V4\V4r\W4p\WR7p\ WWR7#)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. rprs)rrrrrrs&& rCircularSymplecticEnsemblerSrrc\V\4'g\RV,4hVPPP 4#)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.) isinstancerr$rtrqr|mats&rr|r|ls<2 c- . .A3GHH ::   4 4 66rcVPRR7p\;QJd%R\V44F 'dK RM RM!R\V444'g \R4h\ V!#)aZ Creates joint distribution of eigen values of matrices with random expressions. Parameters ========== mat: Matrix The matrix under consideration. Returns ======= JointDistributionHandmade Examples ======== >>> from sympy.stats import Normal, JointEigenDistribution >>> from sympy import Matrix >>> A = [[Normal('A00', 0, 1), Normal('A01', 0, 1)], ... [Normal('A10', 0, 1), Normal('A11', 0, 1)]] >>> JointEigenDistribution(Matrix(A)) JointDistributionHandmade(-sqrt(A00**2 - 2*A00*A11 + 4*A01*A10 + A11**2)/2 + A00/2 + A11/2, sqrt(A00**2 - 2*A00*A11 + 4*A01*A10 + A11**2)/2 + A00/2 + A11/2) T)multiplec38"TFp\V4xK R#5ir5)r).0eigenvals& r )JointEigenDistribution..sB>xy"">sFzWEigen values do not have any random expression, joint distribution cannot be generated.) eigenvalsallsetr$r)rrs& rJointEigenDistributionrs]8 t ,I 3B3y>B333B3y>B B BCD D $i 00rcJVPPP4#)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 )rtrqrrs&rrrs< ::   6 6 88rN) rrrrrrrrr|rr)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__registerrr!rJrgrrrrrrrrrrrrrrr|rrrrrrs+)"&&"#469.;2,'TT@81  &'(""./85/8b#8(&;*&;*9 9&9&9& *5 *D<#8<=&;=<&;<9 9292927: 1D9r