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

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    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.
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  2. #2
    Super Member Rebesques's Avatar
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    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
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