# Numerical Analysis homework help.

• March 29th 2009, 06:49 PM
Alicia
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.
• March 30th 2009, 06:34 AM
chisigma
Question#1

The ‘probability of false alarm’ is given by…

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

… and setting $p=6$ is…

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

The problem is to find $\eta$ for which is $p_{FA}=.001,p_{FA}=.01,p_{FA}=.1$. 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 $f(\eta)=0$ via the iterative procedure…

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

In our case is…

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

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

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

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

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

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

Kind regards

$\chi$ $\sigma$