scoreEquations(...,CovarianceMatrix=>...) is set to false by default. If b is true, scoreEquations gives an additional output: the covariance matrix with rational entries in the same variables as the ideal of score equations.
i1 : G = mixedGraph(digraph {{1,2},{1,3},{2,3},{3,4}},bigraph {{3,4}}); |
i2 : R=gaussianRing(G); |
i3 : U = matrix{{6, 10, 1/3, 1}, {3/5, 3, 1/2, 1}, {4/5, 3/2, 9/8, 3/10}, {10/7, 2/3,1, 8/3}}; 4 4 o3 : Matrix QQ <--- QQ |
i4 : (J,Sigma)=scoreEquations(R,U,CovarianceMatrix=>true); |
i5 : Sigma o5 = | p_(1,1) | l_(1,2)p_(1,1) | l_(1,2)l_(2,3)p_(1,1)+l_(1,3)p_(1,1) | l_(1,2)l_(2,3)l_(3,4)p_(1,1)+l_(1,3)l_(3,4)p_(1,1) ---------------------------------------------------------------------------------------------------------------------------- l_(1,2)p_(1,1) l_(1,2)^2p_(1,1)+p_(2,2) l_(1,2)^2l_(2,3)p_(1,1)+l_(1,2)l_(1,3)p_(1,1)+l_(2,3)p_(2,2) l_(1,2)^2l_(2,3)l_(3,4)p_(1,1)+l_(1,2)l_(1,3)l_(3,4)p_(1,1)+l_(2,3)l_(3,4)p_(2,2) ---------------------------------------------------------------------------------------------------------------------------- l_(1,2)l_(2,3)p_(1,1)+l_(1,3)p_(1,1) l_(1,2)^2l_(2,3)p_(1,1)+l_(1,2)l_(1,3)p_(1,1)+l_(2,3)p_(2,2) l_(1,2)^2l_(2,3)^2p_(1,1)+2l_(1,2)l_(1,3)l_(2,3)p_(1,1)+l_(1,3)^2p_(1,1)+l_(2,3)^2p_(2,2)+p_(3,3) l_(1,2)^2l_(2,3)^2l_(3,4)p_(1,1)+2l_(1,2)l_(1,3)l_(2,3)l_(3,4)p_(1,1)+l_(1,3)^2l_(3,4)p_(1,1)+l_(2,3)^2l_(3,4)p_(2,2)+l_(3,4 ---------------------------------------------------------------------------------------------------------------------------- l_(1,2)l_(2,3)l_(3,4)p_(1,1)+l_(1,3)l_(3,4)p_(1,1) l_(1,2)^2l_(2,3)l_(3,4)p_(1,1)+l_(1,2)l_(1,3)l_(3,4)p_(1,1)+l_(2,3)l_(3,4)p_(2,2) l_(1,2)^2l_(2,3)^2l_(3,4)p_(1,1)+2l_(1,2)l_(1,3)l_(2,3)l_(3,4)p_(1,1)+l_(1,3)^2l_(3,4)p_(1,1)+l_(2,3)^2l_( )p_(3,3)+p_(3,4) l_(1,2)^2l_(2,3)^2l_(3,4)^2p_(1,1)+2l_(1,2)l_(1,3)l_(2,3)l_(3,4)^2p_(1,1)+l_(1,3)^2l_(3,4)^2p_(1,1)+l_(2,3 ---------------------------------------------------------------------------------------------------------------------------- | | 3,4)p_(2,2)+l_(3,4)p_(3,3)+p_(3,4) | )^2l_(3,4)^2p_(2,2)+l_(3,4)^2p_(3,3)+2l_(3,4)p_(3,4)+p_(4,4) | 4 4 o5 : Matrix (frac(QQ[l ..l , l , l , p , p , p , p , p ])) <--- (frac(QQ[l ..l , l , l , p , p , p , p , p ])) 1,2 1,3 2,3 3,4 1,1 2,2 3,3 4,4 3,4 1,2 1,3 2,3 3,4 1,1 2,2 3,3 4,4 3,4 |