Ceng 375 Numerical Computing
Midterm
July 18, 2005 09.00-11.00
Good Luck!

1 (20 Pts) 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 (25 Pts)Consider the function:

$$f(x) = cos(x)- 2x=0$$

iv

Use two iterations of Newton s method to estimate the root of this function between $x = 0.0$ and $x = 1.0$

v

Estimate the error in your answer to part i.

vi

Approximately how many iterations of the bisection method would have been required to achieve the same error?

3 (30 Pts) Solve this system by Gaussian elimination with pivoting

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

vii

How many row interchanges are needed?

viii

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

ix

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 $[1,-3,5]^T$

4 (25 Pts) Consider solving the following linear system by the Jacobi method.

$$\begin{array}{r} 4x_1+x_2=5\\ x_1+5x_2=6\\ \end{array}$$

x

Write down the Jacobi iteration formula for this problem given initial guess $x^{(0)}= 0$.

xi

Assume that the error (vector) at iteration $k$ is denoted by $e^{(k)}$ and that $\vert\vert e^{(0)}\vert\vert=1$. How many iterations do we need before $\vert\vert e^{(k)}\vert\vert\leq 10^{-4}$ ?