Ceng 375 Numerical Computing
Midterm
Nov 10, 2010 14.40-16.30
Good Luck!

1. (10 pts) A three digit, decimal machine which rounds all intermediate calculations, calculates the value of
   
   $$f(x) = x^2-6x + 8~ for~ x = 1.99~as~\overline{f}(1.99) = 0.0600$$
   
   
   What are the forward and backward errors error associated with this calculation?
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
2. (10 pts) Derive the Newton's method formula using a Taylor series of $f(x)$.
3. (20 pts) Use Muller's method to find the root of
   
   $$f(x) = x^3 - x -2$$
   
   

   Figure 1: Plot of the function, $x^3-x-2$.

   
   
   Start with $x_2=1.0$, $x_0=1.2$, and $x_1=1.4$ and find $x_3$ and $x_4$ (two iterations).
4. (30 pts) Consider the function:
   
   $$f(x) = sin(x)-4*x+2$$
   
   

   Figure 2: Plot of the function, $sin(x)-4*x+2$.

   

   i

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

   ii

   Estimate the error in your answer to part i (Use more than four significant figures).

   iii

   Approximately how many iterations of the bisection method would have been required to achieve the same error of part ii? (Hint: if the value in part ii is negative, take absolute value of it.)

5. (30 pts) Consider the linear system ($Ax=b$);
   
   $$A=\left[ \begin{array}{rrrr} 1 & 3 & 1 & 1 \\ 2 & 5 & 2 &... ...egin{array}{r} 6 \\ 2 \\ 4 \\ 3 \\ \end{array} \right]$$
   
   

   iv

   Solve this system by Gaussian elimination with pivoting. How many row interchanges are needed?

   v

   What is the value of determinant?

   vi

   Obtain the $LU$ decomposition of the system.

   vii

   Repeat without any row interchanges (only for the first item). Do you get the same results? Why?