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Thread: Hessian Matrix help

  1. #1
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    Hessian Matrix help

    I need to calculate the Hessian matrix for this function in "very simple terms":

    $\displaystyle f(x) = (Ax \cdot x)e^{||x||^{2}}$.

    I have no idea how to do it in "simple terms."

    If you could either help me get started or tell me the answer, I could do the rest.
    Last edited by tubetess123; Dec 9th 2011 at 02:52 PM.
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  2. #2
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    Re: Hessian Matrix help

    Let $\displaystyle g(x)=x^tAx, h(x)=e^{x^tx}$. Then for any vector $\displaystyle v$, and real number s, we have
    $\displaystyle g(x+sv)=g(x)+s v^t(A+A^t)x+s^2v^tAv$
    $\displaystyle h(x+sv)=e^{x^tx+2sx^tv+s^2v^tv}=h(x)e^{2sx^tv+s^2v ^tv}$
    =$\displaystyle h(x)[1+2sx^tv+s^2v^tv+(2sx^tv+s^2v^tv)^2/2+o(s^2)]$
    =$\displaystyle h(x)+2sh(x)x^tv+s^2h(x)[v^tv+2(x^tv)^2]+o(s^2)$
    Then the $\displaystyle s^2$ term of $\displaystyle f(x+sv)=g(x+sv)h(x+sv)$ is
    $\displaystyle g(x)h(x)[v^tv+2(x^tv)^2]+2v^t(A+A^t)xh(x)x^tv+v^tAvh(x)$
    =$\displaystyle v^t[ g(x)h(x)(I+2xx^t)+2h(x)(A+A^t)xx^t+h(x)A]v$
    So the hessian is $\displaystyle 2h(x)[g(x)I+2(g(x)I+A+A^t)xx^t+A]$
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