# Computing the second order Frechet derivative of P(I-XP)^(-1)

• July 30th 2012, 07:40 PM
afdavid
Computing the second order Frechet derivative of P(I-XP)^(-1)
Hi, I am new to the forum. I work on control engineering and want to calculate the second order Frechet derivative of f(X)=P(I-XP)^(-1) with respect to X, X and P being positive semidefinite matrices. I tried to compute the first order one, which is -P(I-XP)^(-1)K(I-XP)^(-1), but have no idea on how to calculate the second order one... Thanks for help.
• February 4th 2015, 05:27 PM
Rebesques
Re: Computing the second order Frechet derivative of P(I-XP)^(-1)
Remember that
$(I-S)^{-1}=I+S+S^2+\ldots, ||S||<1$

so the function can be written as

$f(X)=P+PXP+PXPXP+\ldots...$

Its Frechet derivative easily now equals

$\langle Df(X),Y\rangle = PYP+PXPYP+PYPXP+\ldots$
$=P(I+XP+\ldots)Y(I+XP+\ldots)=P(I-XP)^{-1}Y(I-XP)^{-1}$
(also there's a minus in your formula that I can't relate to anything).

So, for the second derivative, differentiate $PYP+PXPYP+PYPXP+\ldots$