Ceng 375 Numerical Computing
Midterm
Nov 17, 2006 12.40-14.30
Good Luck!

1. (30 Pts) Consider the function:
   
   $$f(x) = cos(x)- 2x=0$$
   
   

   i

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

   ii

   Estimate the error in your answer to part i.

   iii

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

2. (30 Pts) Consider the function $f(x)$, on $[0, 1]$, defined by
   
   $$f(x) = \sqrt{x}-cos(x)$$
   
   

   iv

   Describe the general working of a bracketing method. What are the assumptions for this family of methods? Are these assumptions satisfied for $f(x)$?

   v

   Describe how the secant method determine a smaller sub-interval containing a root.

   vi

   What is the advantages and disadvantages of the secant method?

   vii

   Apply the secant method to $f(x)$, to reduce the size of the original interval $[0, 1]$, twice.

3. (25 Pts) Find the $LU$ factorization of
   
   $$A=\left[ \begin{array}{rrrr} 1 & 3 & 1 & 1 \\ 2 & 5 & 2 & 2 \\ -1 &-3 &-3 & 5 \\ 1 & 3 & 2 & 2 \\ \end{array} \right]$$
   
   
   by Gaussian elimination (without pivoting). Clearly show how you get the entries of $L$ and $U$.
   
4. (25 Pts) Consider solving the following linear system by the Jacobi method.
   
   $$\begin{array}{r} 4x_1+x_2=5\\ x_1+5x_2=6\\ \end{array}$$
   
   

   viii

   Write down the Jacobi iteration formula for this problem given initial guess $x^{(0)}= 0.1$.

   ix

   Assume that the error (vector) at iteration $k$ is denoted by $e^{(k)}$ and that $\vert\vert e^{(0)}\vert\vert=0.9$. How many iterations do we need before $\vert\vert e^{(k)}\vert\vert\leq 10^{-4}$ ?