Elemental 0.78 documentation

Convex optimization

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Convex optimization

LogBarrier

Uses a careful calculation of the log of the determinant in order to return the log barrier of a Hermitian positive-definite matrix A, \(-\log(\mbox{det}(A))\).

typename Base<F>::type LogBarrier(UpperOrLower uplo, const Matrix<F>& A)
typename Base<F>::type LogBarrier(UpperOrLower uplo, const DistMatrix<F>& A)
typename Base<F>::type LogBarrier(UpperOrLower uplo, Matrix<F>& A, bool canOverwrite=false )
typename Base<F>::type LogBarrier(UpperOrLower uplo, DistMatrix<F>& A, bool canOverwrite=false )

LogDetDivergence

The log-det divergence of a pair of \(n \times n\) Hermitian positive-definite matrices \(A\) and \(B\) is

\[D_{ld}(A,B) = \mbox{tr}(A B^{-1}) -\log(\mbox{det}(A B^{-1})) - n,\]

which can be greatly simplified using the Cholesky factors of \(A\) and \(B\). In particular, if we set \(Z = L_B^{-1} L_A\), where \(A=L_A L_A^H\) and \(B=L_B L_B^H\) are Cholesky factorizations, then

\[D_{ld}(A,B) = \| Z \|_F^2 - 2 \log(\mbox{det}(Z)) - n.\]
typename Base<F>::type LogDetDivergence(UpperOrLower uplo, const Matrix<F>& A, const Matrix<F>& B)
typename Base<F>::type LogDetDivergence(UpperOrLower uplo, const DistMatrix<F>& A, const DistMatrix<F>& B)

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