Exercises 12.6 Homework
1.
Find the order of error of the formula (12.3.2) (theoretical exercise).
2.
Rewrite the formula
to avoid the loss of significance when \(x\) is close to \(0\text{.}\) Evaluate both the original and rewritten formulas when \(x = 10^{-4}\text{.}\) How different are the results?
3.
A function \(f\) has been evaluated at the points \(0, 0.1, 0.2, \dots, 1\) (in Matlab notation, x = 0:0.1:1). Its values are
y = [1, 0.99, 0.96, 0.91, 0.85, 0.78, 0.7, 0.61, 0.53, 0.44, 0.37]
Write a script which plots this function together with its first derivative \(f'\) and its second derivative \(f''\text{.}\) Use symmetric difference formulas with \(h=0.1\text{.}\)
Expressions like y(3:end) - y(1:end-2) will be useful. Note that while the values of \(f\) can be plotted against the given vector x, the derivatives need to be plotted against x(2:end-1) because the symmetric formulas do not apply at the endpoints. Recall we can combine several functions in one plot command like plot(x, y, xx, yy).