+ iU^RIHt^RIHt^RIHt^RIHt^RIH t ^RI H t ^RI H t Ht^RIHtHtHtHtHtHtHtHtHt^R IHtHtHtHtHtHtH t ^R I!H"t"!R R ]4t#R t$!RR4t%!RR4t&!RR4t'R]%R]'R]'R]&/t(!RR] ]4t)!RR])4t*Rt+!RR])4t,Rt-!R R!])4t.R"t/!R#R$])4t0R%t1R&#)')prod)Basic)pi)S)exp) multigamma)sympify_sympify) ImmutableMatrixInverseTrace Determinant MatrixSymbol MatrixBase Transpose MatrixSet matrix2numpy) _value_checkRandomMatrixSymbolNamedArgsMixinPSpace_symbol_converter MatrixDomain Distribution) import_moduleca]tRt^toRtRt]!R4t]!R4t]R4t ]R4t ]R4t Rt R R lt R tVtR #) MatrixPSpacez8 Represents probability space for Matrix Distributions. c\V4p\V4\V4rCVP'dVP'g \R4h\P !WW#V4#)zDimensions should be integers)rr is_integer ValueErrorr__new__)clssym distributiondim_ndim_ms&&&&&ڀ/Users/tonyclaw/.openclaw/workspace/skills/math-calculator/venv/lib/python3.14/site-packages/sympy/stats/matrix_distributions.pyr!MatrixPSpace.__new__sQ$u   U%5%5%5<= =}}S|EBBc(VP^,#)argsselfs&r'MatrixPSpace. s 1r)c(VP^,#rr,r.s&r'r0r1!s 499Q)>??r)cv\VPVP^,VP^,V4#))rr6r-r.s&r'valueMatrixPSpace.value's'!$++tyy|TYYq\4PPr)cVP0#r5)r<r.s&r'valuesMatrixPSpace.values+s |r)cVP\4p\V4^8g\V\4'g \ R4hVP P V4#)r+ztCurrently, no algorithm has been implemented to handle general expressions containing multiple matrix distributions.)atomsrlen isinstanceNotImplementedErrorr$pdf)r/exprr-rmss&&* r'compute_densityMatrixPSpace.compute_density/sTjj+, s8a< 41C D D%'56 6  $$T**r)Nc TVPVPPWVR7/#)z] Internal sample method Returns dictionary mapping RandomMatrixSymbol to realization value. )libraryseed)r<r$sample)r/sizerLrMs&&&&r'rNMatrixPSpace.sample7s*  D--44TQU4VWWr)rQscipyN)__name__ __module__ __qualname____firstlineno____doc__r!propertyr$r6r8r<r?rIrN__static_attributes____classdictcell__ __classdict__s@r'rrswC56L / 0F @@QQ+XXr)rc\\\V44pV!V!pVP!V!VPp\ WV^,V^,4pVP #r3)listmapr check dimensionrr<)r6r"r-distdimpspaces&&& r'rvrf@sQ GT" #D :DJJ ..C &AA 7F <<r)c>a]tRt^ItoRtRRlt]R4tRtVt R#)SampleMatrixScipyz7Returns the sample from scipy of the given distributionNc&VPWV4#r5) _sample_scipyr"rcrOrMs&&&&r'r!SampleMatrixScipy.__new__K  T22r)c a ^RIHo ^RIpRV 3RlRV 3Rl/pRRRR/pVP4pVPP V9dR#Ve\ V\4'dVPPVR 7pMTpWQPP ,!V\V4V4p V PW&VPP ,!V4,4#) zSample from SciPy.)statsNWishartDistributionc<SPP\VP4\ VP \ 4VR7#))dfscalerO)wishartrvsintnr scale_matrixfloatrcrO rand_state scipy_statss&&&r'r01SampleMatrixScipy._sample_scipy..Us<+BUBUBYBYtvv;l43D3De&LSWCZCYr)MatrixNormalDistributionc<SPP\VP\4\VP \4\VP \4WR7#))meanrowcovcolcovrO random_state) matrix_normalrurlocation_matrixryscale_matrix_1scale_matrix_2rzs&&&r'r0r}WsQ{G`G`GdGd!$"6"6>#D$7$7?#D$7$7?dHeHer)c.VPP#r5rxshapercs&r'r0r}^0A0A0G0Gr)c.VPP#r5rrrs&r'r0r}_d6J6J6P6Pr)rM) rSronumpykeys __class__rTrDrvrandom default_rngrreshape) r"rcrOrMr scipy_rv_map sample_shape dist_listr{sampr|s &&&& @r'rjSampleMatrixScipy._sample_scipyNs / !$Y &)e  "#G &)P !%%' >> " ") 3 <:dC0011t1a]tRt^otoRtRRlt]R4tRtVt R#)SampleMatrixNumpyz7Returns the sample from numpy of the given distributionNc&VPWV4#r5) _sample_numpyrks&&&&r'r!SampleMatrixNumpy.__new__srmr)c /p/pVP4pVPPV9dR#^RIpVe\ V\ 4'dVP PVR7pMTpWAPP,!V\V4V4p V PW%VPP,!V4,4#)zSample from NumPy.Nr) rrrTrrDrvrrrr) r"rcrOrM numpy_rv_maprrrr{rs &&&& r'rSampleMatrixNumpy._sample_numpyvs  !%%' >> " ") 3 <:dC0011t1a]tRt^toRtRRlt]R4tRtVt R#)SampleMatrixPymcz6Returns the sample from pymc of the given distributionNc&VPWV4#r5) _sample_pymcrks&&&&r'r!SampleMatrixPymc.__new__sD11r)c  a ^RIo RV 3RlRV 3Rl/pRRRR/pVP4pVPP V9dR#^RIpVPR4PVP4S P4;_uu_4WAPP ,!V4S P\V4^R VR R R 7R ,pRRR4XPW%VPP ,!V4,4# \d ^RIo EL%i;i +'giL^;i) zSample from PyMC.Nr~c <SPR\VP\4\VP\4\VP \4VPP R7#)X)murrr) MatrixNormalrrryrrrrcpymcs&r'r0/SampleMatrixPymc._sample_pymc..sWT5F5Fs 4 4e<#D$7$7?#D$7$7?**00 6G62r)rpc<SPR\VP4\VP\ 4R7#)r)nur)WishartBartlettrvrwrrxryrs&r'r0rs30D0DStvv;,t/@/@%"H1E1Jr)c.VPP#r5rrs&r'r0rrr)c.VPP#r5rrs&r'r0rrr)rF)drawschains progressbar random_seedreturn_inferencedatacompute_convergence_checksr)r ImportErrorpymc3rrrTlogging getLoggersetLevelERRORModelrNrr) r"rcrOrM pymc_rv_maprrrsampsrs &&&& @r'rSampleMatrixPymc._sample_pymcs !  ')2 "$J   "#G &)P  $$& >> " ") 3&!**7==9 ZZ\\ // 0 6KKd4j[_v{Y^K_`cdE}}T1H1H$I$$OOPP5 !  !.\sDA D0D-,D-0 E rQr5) rTrUrVrWrXr!rrrZr[r\s@r'rrs#@2QQr)rrSrrrcJa]tRt^toRtRt]R4tRtRRlt Rt Vt R#) MatrixDistributionz) Abstract class for Matrix Distribution. cVUu.uF/p\V\4'd \V4M \V4NK1 pp\P !V.VO5!#uupir5)rDr_r r rr!)r"r-args&* r'r!MatrixDistribution.__new__sX.24.2s)33(=(=$c]#.2 4}}S(4((4s5AcR#r5rQr,s*r'raMatrixDistribution.checks r)cf\V\4'd \V4pVPV4#r5)rDr_r rF)r/rGs&&r'__call__MatrixDistribution.__call__s' dD ! !"4(Dxx~r)Nc .ROpW$9d\R\V4,4h\V4'g\RV,4h\V,!WV4pVeV#\RVP P : RV: 24h)zW Internal sample method Returns dictionary mapping RandomSymbol to realization value. z&Sampling from %s is not supported yet.zFailed to import %sz Sampling for z# is not currently implemented from )rSrrr)rEstrrr _get_sample_class_matrixrvrrT)r/rOrLrM librariesrs&&&& r'rNMatrixDistribution.samples8  #%&N*-g,'78 8W%%2W<= =*73DE  L!>>**G5 r)rQrR) rTrUrVrWrXr! staticmethodrarrNrZr[r\s@r'rrs2)    r)rcZa]tRt^toRt]R4t]R4t]R4t Rt Rt Vt R#)MatrixGammaDistributionc\V\4'g\VPR4\VPR4\VP R4\VP R4R#)+The shape matrix must be positive definite.Should be square matrix#Shape parameter should be positive.z#Scale parameter should be positive.NrDrris_positive_definite is_square is_positivealphabetarxs&&&r'raMatrixGammaDistribution.checks[, 55 ::=4 5\++. U&&(MNT%%'LMr)cpVPP^,p\W\P4#r3rxrrrRealsr/ks& r'r7MatrixGammaDistribution.set)    # #A &qww''r)c.VPP#r5rr.s&r'rb!MatrixGammaDistribution.dimension  &&&r)cfVPVPVPrCpVP^,p\ V\ 4'd \ V4p\ V\\34'g\R\V4,4h\V4)V,V, p\\V44W5V,,\W%4,, p\V4V),p\V4V\!V^,4^, , ,p Wx,V ,#r4%s should be an isinstance of Matrix or MatrixSymbol)rrrxrrDr_r rrr rr rr rrr) r/xrrrxp sigma_inv_xterm1term2term3s && r'rFMatrixGammaDistribution.pdfs$(JJ 4;L;L\   q ! a  "A!j,788&(+A/0 0 --a/$6 E+&'$5/Z=Q)QR\*uf5Q51QU8A:#56}u$$r)rQNr rTrUrVrW _argnamesrrarYr7rbrFrZr[r\s@r'rrsP1INN(('' % %r)rcj\V\4'd \V4p\V\WV34#)a Creates a random variable with Matrix Gamma Distribution. The density of the said distribution can be found at [1]. Parameters ========== alpha: Positive Real number Shape Parameter beta: Positive Real number Scale Parameter scale_matrix: Positive definite real square matrix Scale Matrix Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import density, MatrixGamma >>> from sympy import MatrixSymbol, symbols >>> a, b = symbols('a b', positive=True) >>> M = MatrixGamma('M', a, b, [[2, 1], [1, 2]]) >>> X = MatrixSymbol('X', 2, 2) >>> density(M)(X).doit() exp(Trace(Matrix([ [-2/3, 1/3], [ 1/3, -2/3]])*X)/b)*Determinant(X)**(a - 3/2)/(3**a*sqrt(pi)*b**(2*a)*gamma(a)*gamma(a - 1/2)) >>> density(M)([[1, 0], [0, 1]]).doit() exp(-4/(3*b))/(3**a*sqrt(pi)*b**(2*a)*gamma(a)*gamma(a - 1/2)) References ========== .. [1] https://en.wikipedia.org/wiki/Matrix_gamma_distribution )rDr_r rfr)r6rrrxs&&&&r' MatrixGammars1V,%%&|4 f-\/J KKr)cZa]tRtRtoRt]R4t]R4t]R4t Rt Rt Vt R#) rpiHc\V\4'g\VPR4\VPR4\VP R4R#)rrrNrrwrxs&&r'raWishartDistribution.checkLsH, 55 ::=4 5\++. Q]]$IJr)cpVPP^,p\W\P4#r3rrs& r'r7WishartDistribution.setUrr)c.VPP#r5rr.s&r'rbWishartDistribution.dimensionZrr)cVPVPr2VP^,p\V\4'd \ V4p\V\ \34'g\R\V4,4h\V4)V,\^4, p\\V44^WB,\^4, ,\V\^4, V4,, p\V4V)\^4, ,p\V4\W$, ^, 4^, ,pWg,V,#r)rwrxrrDr_r rrr rr rrr rr) r/rrwrxrrrrrs && r'rFWishartDistribution.pdf^s&&$"3"3<   q ! a  "A!j,788&(+A/0 0 --a/!A$6 E+&'!ac!A$h-:a!fa;P)PQ\*qb1g6Q1QUQY<>2}u$$r)rQNrrr\s@r'rprpHsP%IKK(('' % %r)rpch\V\4'd \V4p\V\W34#)aY Creates a random variable with Wishart Distribution. The density of the said distribution can be found at [1]. Parameters ========== n: Positive Real number Represents degrees of freedom scale_matrix: Positive definite real square matrix Scale Matrix Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import density, Wishart >>> from sympy import MatrixSymbol, symbols >>> n = symbols('n', positive=True) >>> W = Wishart('W', n, [[2, 1], [1, 2]]) >>> X = MatrixSymbol('X', 2, 2) >>> density(W)(X).doit() exp(Trace(Matrix([ [-1/3, 1/6], [ 1/6, -1/3]])*X))*Determinant(X)**(n/2 - 3/2)/(2**n*3**(n/2)*sqrt(pi)*gamma(n/2)*gamma(n/2 - 1/2)) >>> density(W)([[1, 0], [0, 1]]).doit() exp(-2/3)/(2**n*3**(n/2)*sqrt(pi)*gamma(n/2)*gamma(n/2 - 1/2)) References ========== .. [1] https://en.wikipedia.org/wiki/Wishart_distribution )rDr_r rfrp)r6rwrxs&&&r'Wishartrls/P,%%&|4 f)A+< ==r)cZa]tRtRtoRt]R4t]R4t]R4t Rt Rt Vt R#) r~ic B\V\4'g\VPR4\V\4'g\VPR4\VPR4\VPR4VP ^,pVP ^,p\VP ^,V8HR\ V4: R\ V4: 24\VP ^,V8HR\ V4: R\ V4: 24R#)r)Scale matrix 1 should be be square matrix)Scale matrix 2 should be be square matrix"Scale matrix 1 should be of shape  x "Scale matrix 2 should be of shape N)rDrrrrrr)rrrrwrs&&& r'raMatrixNormalDistribution.checks.,77 <<?4 5.,77 <<?4 5^--0 ^--0   ! !! $  ! !! $^))!,1!!fc!f4. /^))!,1!!fc!f4. /r)cfVPPwr\W\P4#r5rrrrrr/rwrs& r'r7MatrixNormalDistribution.set&##))qww''r)c.VPP#r5rr.s&r'rb"MatrixNormalDistribution.dimension##)))r)cVPVPVPrCpVPwrV\ V\ 4'd \ V4p\ V\\34'g\R\V4,4h\V4\W, 4,\V4,W, ,p\\V4)\^4, 4p^\ ,\WV,4^, ,\#V4\V4^, ,,\#V4\V4^, ,,p W, #)r)rrrrrDr_r rrr rr rrr rrr) r/rMUVrwrrnumdens && r'rFMatrixNormalDistribution.pdfs&&(;(;T=P=Paww a  "A!j,788&(+A/0 0 9QU++GAJ6>5<-!$%tqvax ;q>AaDF#;;k!nqQRtTUv>VVwr)rQN)rrrrr\s@r'r~r~sNGI//$((**  r)r~c\V\4'd \V4p\V\4'd \V4p\V\4'd \V4pWV3p\V\V4#)aC Creates a random variable with Matrix Normal Distribution. The density of the said distribution can be found at [1]. Parameters ========== location_matrix: Real ``n x p`` matrix Represents degrees of freedom scale_matrix_1: Positive definite matrix Scale Matrix of shape ``n x n`` scale_matrix_2: Positive definite matrix Scale Matrix of shape ``p x p`` Returns ======= RandomSymbol Examples ======== >>> from sympy import MatrixSymbol >>> from sympy.stats import density, MatrixNormal >>> M = MatrixNormal('M', [[1, 2]], [1], [[1, 0], [0, 1]]) >>> X = MatrixSymbol('X', 1, 2) >>> density(M)(X).doit() exp(-Trace((Matrix([ [-1], [-2]]) + X.T)*(Matrix([[-1, -2]]) + X))/2)/(2*pi) >>> density(M)([[3, 4]]).doit() exp(-4)/(2*pi) References ========== .. [1] https://en.wikipedia.org/wiki/Matrix_normal_distribution )rDr_r rfr~)r6rrrr-s&&&& r'rrsdR/4(()/:.$''(8.$''(8 ^ KA)?)NyYZY^O_)_ _`FQJM"1$&& &r)rQN)rrrrrr\s@r'r"r"sPMIUU&((** & &r)r"c\V\4'd \V4p\V\4'd \V4p\V\4'd \V4pWW43p\V\V4#)a Creates a random variable with Matrix Gamma Distribution. The density of the said distribution can be found at [1]. Parameters ========== nu: Positive Real number degrees of freedom location_matrix: Positive definite real square matrix Location Matrix of shape ``n x p`` scale_matrix_1: Positive definite real square matrix Scale Matrix of shape ``p x p`` scale_matrix_2: Positive definite real square matrix Scale Matrix of shape ``n x n`` Returns ======= RandomSymbol Examples ======== >>> from sympy import MatrixSymbol,symbols >>> from sympy.stats import density, MatrixStudentT >>> v = symbols('v',positive=True) >>> M = MatrixStudentT('M', v, [[1, 2]], [[1, 0], [0, 1]], [1]) >>> X = MatrixSymbol('X', 1, 2) >>> density(M)(X) gamma(v/2 + 1)*Determinant((Matrix([[-1, -2]]) + X)*(Matrix([ [-1], [-2]]) + X.T) + Matrix([[1]]))**(-v/2 - 1)/(pi**1.0*gamma(v/2)*Determinant(Matrix([[1]]))**1.0*Determinant(Matrix([ [1, 0], [0, 1]]))**0.5) References ========== .. [1] https://en.wikipedia.org/wiki/Matrix_t-distribution )rDr_r rfr")r6rrrrr-s&&&&& r'MatrixStudentTr1/sdX/4(()/:.$''(8.$''(8  @D f0$ 77r)N)2mathrsympy.core.basicrsympy.core.numbersrsympy.core.singletonr&sympy.functions.elementary.exponentialr'sympy.functions.special.gamma_functionsrsympy.core.sympifyr r sympy.matricesr r r rrrrrrsympy.stats.rvrrrrrrrsympy.externalrrrfrhrrrrrrrprr~rr"r1rQr)r'r<s"!"6>0***KKK( )X6)XX#P#PLPP>&Q&QR     '~'`#%0#%J-Ld"%,"%H*>^+1+Z06j.&!3.&d38r)