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}});
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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
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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)
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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
----------------------------------------------------------------------------------------------------------------------------
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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
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