Numerical Analysis

January 30, 2002

Due: February 6, 2002

Assignment#1: (50 points) All work should be your own. Remember all homework is due at the start of class on the 6th.

1. (5 pt.) Describe the most efficient way to evaluate

   $${ p(x) = 1 - \frac{x^6}{6!} + \frac{x^12}{12!} - \frac{x^{18}}{18!} + \frac{x^{24}}{24!} - \frac{x^{30}}{30!}}$$

   Assume that the coefficients have been previously computed and are stored. How many multiplications are necessary?

2. (10 pt.) Produce the linear and quadratic Taylor polynomials for the folowing functions. You must show all necessary steps in your calculations. DO NOT find the general form. Remember to use radians for trig functions and not degrees.
   1. $f(x) = \sqrt{x + 1}, a = 1$
   2. $f(x) = e^{\sin (x)}, a = \frac{\pi}{2}$

3. (5 pt.) Produce the general formula (including the error term) for a Taylor polynomial of degree $n$ for the folowing function. You must show all necessary steps in your calculations.
   1. $f(x) = \sin (2\pi x), a = 0$

4. (10 pt.) For the following functions find the value of $n$ such that
   $${ \vert f(x) - p_n (x)\vert \leq 0.00001}$$

   where $p_n (x)$ and its remainder are given in the book (p. 9) and given the specified $x$ values.

   1. $f(x) = \sin (x), -\pi \leq x \leq \pi$
   2. $f(x) = e^x, -0.5 \leq x \leq 0.5$

5. (5 pt.) Problem 2.1.1 (a & c)

6. (5 pt.) Problem 2.1.3 (a & c)

7. (5 pt.) Convert the 32-bit value 10101010101010101010101010101010 to a floating point number, assuming we are using IEEE Standard 754 storage. See the course website for a pointer to information about this.

8. (5 pt.) Problem 2.2.5 (no need to do 2.2.4 first)