FGfun {TRES}

The Objective function and its gradient

Description

Calculates the objective function and its gradient for estimating the M-envelope of span(U), where M is positive definite and U is positive semi-definite.

Usage

FGfun(Gamma, M, U)

Arguments

Gamma

\Gamma matrix in the envelope objective function. A p-by-u matrix.

M

The p-by-p positive definite matrix M in the envelope objective function.

U

The p-by-p positive semi-definite matrix U in the envelope objective function.

Details

The generic objective function F(\Gamma) and its gradient G(\Gamma) are listed below for estimating M-envelope of span(U). For the detailed description, see Cook, R. D., & Zhang, X. (2016).

F(\Gamma)=\log|\Gamma^T M \Gamma|+\log| \Gamma^T(M+U)^{-1}\Gamma|

G(\Gamma) = dF/d \Gamma = 2 M \Gamma (\Gamma^T M \Gamma)^{-1} + 2 (M + U)^{-1} \Gamma (\Gamma^T (M + U)^{-1} \Gamma)^{-1}

Value

F

The value of the objective function at Gamma.

G

The value of the gradient function at Gamma.

References

Cook, R.D. and Zhang, X., 2016. Algorithms for envelope estimation. Journal of Computational and Graphical Statistics, 25(1), pp.284-300.