Ceng 375 Numerical Computing
Midterm
Nov 16, 2005 10.40-12.30
Good Luck!

1 (20 Pts)

i

Under what conditions can parallel processing not be used to speed up a computation?

ii

How many iterations of bisection will be required to attain an accuracy of $10^{-4}$ if the starting interval is $[a,b]$?

2 (20 Pts) The function $f(x)=4x^3-1-e^{x^2/2}$ has values of zero near $x=1.0$ and $x=3.0$.

iii

What is the derivative of $f(x)$?

iv

If you begin Newton's method at $x=2$, which root is reached? How many iterations to achieve an error less than $10^{-5}$?

3 (20 Pts) Solve this system by Gaussian elimination with pivoting

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

v

How many row interchanges are needed?

vi

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

vii

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 $[-3,7,-2]^T$

viii

Solve the second item again but use only three significant digits of precision.

4 (20 Pts) Consider the linear system

$$\begin{array}{r} 7x_1-3x_2+4x_3=6\\ -3x_1+2x_2+6x_3=2\\ 2x_1+5x_2+3x_3=-5\\ \end{array}$$

ix

Solve this system with the Jacobi method. First rearrange to make it dioganally dominant if possible. Use $[0,0,0]$ as the starting vector.

x

Repeat with Gauss-Seidel method. Compare with Jacobi method.

5 (20 Pts) For the given data points;

$$\begin{array}{rr} x & y \\ \hline 2.1 & -12.4 \\ 4.1 & 7.3 \\ 7.1 & 10.1 \\ \end{array}$$

1. Write out the Lagrangian polynomial from this table

   xi

   confirm that it reproduces the $y$'s for each $x$-value.

   xii

   interpolate with it to estimate $y$ at $x=3$.

   xiii

   extrapolate with it to estimate $y$ at $x=8$.

2. Suppose in previous item that the $y$-value for $x=4.1$ is mistakenly entered as $7.2$ rather than $7.3$. Repeat the previous item with this incorrect value. How much difference does this make?
3. Expand the Lagrangian polynomials in the previous items to get the quadratics in the form $ax^2+bx+c$. How different are the values for $a$,$b$, and $c$?

6 (20 Pts) For the given data points;

$$\begin{array}{rr} x & y \\ \hline 1 & 1.06 \\ 2 & 1.12 \\ 3 & 1.34 \\ 5 & 1.78 \\ \end{array}$$

xiv

construct the divided-difference table.

xv

interpolate for $x=4$.

xvi

extrapolate for $x=5.5$.