Hints:

Problem 1

Typo: it should say "in the case of a tie pick the number with d₄=0"

Problem 2

Note: Even with format long g, different machine numbers are sometimes displayed in the same way. E.g., x = 1 + 1e-15 is displayed as 1, but typing num2bin(x) or x-1 shows that it is a different machine number from 1.

Problem 3

3(a): Let f(c) = (-b + sqrt(...))/2, then find f'(c) by taking the derivative with respect to c. Show that the condition number is y1/(y1-y2) and evaluate this for (i),(ii),(iii).
3(b): The file qeq1.m should look like this:
  function [y1,y2] = qeq1(b,c)
  y1 = ... ;
  y2 = ... ;
To find the errors of your computed solution use in your main code
  y1 = ... ; y2 = ... ;
  b = -y1-y2; c = y1*y2;
  [y1hat, y2hat] = qeq1(b,c)
and then compute the relative errors of y1hat, y2hat compared to y1, y2.

Problem 4

4(a): Let y1 = exp(x) and y2 = exp(-x). First find an upper bound for the relative error of the computed values y1hat, y2hat. Then use the error propagation formula for c = a - b (see the PDF file about error propagation below).
4(b): Use the remainder term of the Taylor series to get an upper bound for |f(x)-f₄(x)|/|f(x)|.
4(c): linspace(a,b,n) gives n equidistant numbers in the interval [a,b] (including the endpoints), e.g.,
x = linspace(0,20*pi,1000); plot(x,x.*sin(x))