Extrapolation Techniques

• The errors of a central-difference approximation to $f'(x)$ were of $O(h^2)$. In effect, suggests that the errors are proportional to $h^2$ although that is true only in the limit as $h \rightarrow 0$. Unless $h$ is quite large, we can assume the proportionality.
• So, from two computations with $h$ being half as large in the second, we can estimate the proportionality factor, $C$. For example, in Table 2;
  

  h	Approximation
  0.05	4.15831
  0.025	4.16361

  
  If errors were truly $C(h^2)$, we can write two equations:
  
  $$\begin{array}{l} True~value=4.15831+C(0.05^2)\\ True~value=4.16361+C(0.025^2)\\ \end{array}$$
  
  
  from which we can solve for the true value, eliminating the unknown constant $C$, getting;
  
  $$\begin{array}{ll} True~value& =4.16361+(1/3)*(4.16361 - 4.15831)\\ & =4.16538\\ \end{array}$$
  
  
  which is very close to the exact value for $f'(1.9)$, 4.165382.
• You can easily derive the general formula for improving the estimate, when errors decrease by $O(h^n)$
  

  $$\begin{array}{cl} Better&=more+ (1/(2^n-1))*(more-less)\\ estimate& accurate\\ \end{array}$$ (4)

  
  
  where more and less in the last term are the two estimates at $h_1$ and $h_2=h_1/2$. More accurate is the estimate at the smaller value of $h$ and $n$ is the power of $h$ in the order of the errors.
• As example, apply this to values from Table 1 which were from forward-difference approximations. Here the errors are $O(h)$.
  

  h	Approximation
  0.05	4.05010
  0.025	4.10955

  
  
  Using Eq. 4, we have
  
  $$\begin{array}{cl} Better~estimate&=4.10955+(1/(2^1-1))(4.10955-4.05010)\\ &=4.16900\\ \end{array}$$
  
  
  which shows considerable improvement but not as good as from the central differences.