Lab 2

This lab will be mostly focused constructing more and more complicated functions to handle tasks that need to be repeated.

Evaluating the exponential function
Exercise 1
Exercise 2
Exercise 3
Exercise 4

Evaluating the exponential function

Consider the Taylor series expansion for $e^x$ about $x = 1$ :

$e^x = \sum_{n=0}^\infty \frac{x^n}{n!}.$

The absolute error for partial sums is given by

$\left| e^x - \sum_{n=1}^N \frac{x^n}{n!} \right| \leq \frac{e^{|x|} |x|^{N+1}}{N!}.$

Exercise 1

Write a function exp_frac(x) to compute the exponential function to a relative accuracy of 10e-15 for $0 \leq x \leq 1$ .

Create a plot of the absolute and relative error as $x$ increases. Use the semilogy() command to help with this visualzation.

Exercise 2

The exponential function has the important property that $e^{x/2} = (e^x)^2$ . Construct a function exp_series(x) to compute the exponential function for any number $x$ using exp_frac(y) where $0 < y < 1$ and $x = y 2^m$ for some $m$ .

Create a plot of the absolute and relative error as $x$ increases. Use the semilogy() command to help with this visualzation.

Exercise 3

Write a function bisection(f,a,b,TOL,Nmax) to find the root of a function f.

Exercise 4

Write a function log_bisect(x) to compute the natural logarithm of $x \geq 1$ using your exp_series() and bisection() functions. What challenges do you encounter?