Ceng 375 Numerical Computing
Final
Jan 14, 2005 09.40-11.30
Good Luck!

1 (20 Pts)

IV

In Newton's method the approximation $x_{n+1}$ to a root of $f(x) = 0$ is computed from the approximation $x_n$ using the equation

$$x_{n+1}=x_n -\frac{f(x_n)}{f'(x_n)}$$

Derive the above formula, using a Taylor series of $f(x)$.

V

Consider the function:

$$f(x) = 5x-e^{-x}$$

i

Show that this function has a simple root in the interval $0 < x < 1$

ii

Estimate this root using two iterations of the Secant Method.

iii

Estimate the error in your answer to part ii.

2 (20 Pts) Solve this system by Gaussian elimination with pivoting

$$\left[ \begin{array}{rrrr} 1 &-2 &4&6\\ 8 &-3 &2&2\\ -1 &10 &2&4\\ \end{array} \right]$$

vi

How many row interchanges are needed?

vii

Repeat without any row interchanges. Do you get the same results?

viii

You could have saved the row multipliers and obtained a $LU$ equivalent of the coefficient matrix. Use this $LU$ to solve but with right-hand sides of $[1,-3,5]^T$

3 (25 Pts)

ix

A function $f_{app}(x)$ is to be used as an approximation to a set of data $(x_i, f_i)$ with $i = 0, 1, 2,\ldots,N$. Suppose further that the function $f_{app}(x)$ depends on two parameters $a$ and $b$. Provide full details of how the parameters $a$ and $b$ can be determined by a Least Squares Method.

x

Using the result of the previous item, obtain the normal equations for the function $f_{app}(x)= a+b\sqrt{x}$. Do not attempt to solve these equations.

4 (20 Pts)

xi

Find the Fourier coefficients for $f(x)=x^3$ if it is periodic and one period extends from $x=-1$ to $x=2$.

xii

Write the Fourier series for this function.

5 (25 Pts) Consider the following table of data

$x_i$	$f_i$
0.0000	0.0000
0.2000	0.5879
0.4000	1.0637
0.6000	1.3927
0.8000	1.5573
1.0000	1.5575
1.2000	1.4091

xiii

Approximate $\int_0^{1.2} f(x)$dx using the Trapezoidal Rule and a step size of $h = 0.6$.

xiv

Approximate $\int_0^{1.2} f(x)$dx using the Trapezoidal Rule and a step size of $h = 0.2$.

xv

Estimate the error in your answer to previous item.
Hint: Use the procedure to estimate the proportionality factor, $C$.