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Thread: Numerical Analysis homework help.

  1. #1
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    Numerical Analysis homework help.

    Hello, Can someone tell me how to go about getting the following questions started i dont need to know the answer but at the same time I havent the slightest clue about what they are asking me to do. Please see attachment thanks much.
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  2. #2
    MHF Contributor chisigma's Avatar
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    Question#1

    The ‘probability of false alarm’ is given by…

    $\displaystyle p_{FA}= \int_{\eta}^{\infty} \frac{x^{\frac{p}{2}-1}\cdot e^{-\frac{x}{2}}}{\Gamma (\frac{p}{2})\cdot 2^{\frac{p}{2}}}\cdot dx$ (1)

    … and setting $\displaystyle p=6$ is…

    $\displaystyle p_{FA}= \int_{\eta}^{\infty} \frac{x^{2}\cdot e^{-\frac{x}{2}}}{2^{4}}\cdot dx= \frac{1}{2}\cdot \int_{\eta}^{\infty} t^{2}\cdot e^{-t}\cdot dt = e^{-\frac{\eta}{2}}(1+\frac{\eta}{2} + \frac{\eta^{2}}{8})$ (2)

    The problem is to find $\displaystyle \eta$ for which is $\displaystyle p_{FA}=.001,p_{FA}=.01,p_{FA}=.1$. The method suggested in textbook is the bisection method, which is very inaccurate and requires long and tedious computation. Much better is the Newton-Raphson method, which solves an equation in the form $\displaystyle f(\eta)=0$ via the iterative procedure…

    $\displaystyle \eta_{n+1}= \eta_{n}- \frac {f(\eta_{n})}{f^{'}(\eta_{n})}$ (3)

    In our case is…

    $\displaystyle f(\eta)= e^{-\frac{\eta}{2}}\cdot (1+\frac{\eta}{2} + \frac{\eta^{2}}{8}) - p_{FA}$

    $\displaystyle f^{'}(\eta) = -\frac {1}{16}\cdot \eta^{2}\cdot e^{-\frac{\eta}{2}}$ (4)

    In all cases the initial value is $\displaystyle \eta_{0} = 10$…

    For $\displaystyle p_{FA}=.1$ in $\displaystyle 4$ iterations we obtain the ‘stable value’ $\displaystyle \eta \approx 10.64464$…

    For $\displaystyle p_{FA}=.01$ in $\displaystyle 6$ iterations we obtain the ‘stable value’ $\displaystyle \eta \approx 16.81189$…

    For $\displaystyle p_{FA}=.001$ in $\displaystyle 8$ iterations we obtain the ‘stable value’ $\displaystyle \eta \approx 22.45774$…

    Kind regards

    $\displaystyle \chi$ $\displaystyle \sigma$
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