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

1. ## 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.

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

Remember that
$\displaystyle (I-S)^{-1}=I+S+S^2+\ldots, ||S||<1$

so the function can be written as

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

Its Frechet derivative easily now equals

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

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

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