
# Homework 6

Please answer the following questions in complete sentences in a typed manuscript and submit the solution on blackboard by on April 11th at noon. (These will be back before the midterm.)

## Problem 0: Homework checklist

• Please identify anyone, whether or not they are in the class, with whom you discussed your homework. This problem is worth 1 point, but on a multiplicative scale.

• Make sure you have included your source-code and prepared your solution according to the most recent Piazza note on homework submissions.

## Problem 1: Gautschi Exercise 4.1

The following sequences converge to $0$ as $n\to\infty$:

• $v_n = n^{-10}$
• $w_n = 10^{-n}$
• $x_n = 10^{-n^2}$
• $y_n = n^{10} 3^{-n}$
• $z_n = 10^{-3 \cdot 2^n}$

Indicate the type of convergence for each sequence in terms of

• Sublinear
• Linear
• Superlinear
1. Using any method we've seen to solve a scalar nonlinear equation (bisection, false position, secant), develop a routine to compute $\sqrt{x}$ using only addition, subtraction, multiplication, and division (and basic control structures) to numerical precision. (Use double-precision.)
2. Compare the results of your method to the Matlab/Julia/Python function sqrt. Comment on any differences that surprise you.
Consider the problem $f(x) = (1/2) x - \sin x = 0$. The only positive real root is located in $[ 1/2 \pi, \pi ]$. Compare the performance of bisection, false position, and the secant method in terms of the number of function evaluations to compute the solution to $7$ and full machine precision. For all these methods, use the boundary points $[a,b] = [ 1/2 \pi, \pi ]$ (or use those as the first two points for secant).