4.2/

Extrapolation

1. What?
   
   • Extrapolation is used to
   • Extrapolation can be applied whenever
   • Suppose that for each number \(h \neq 0\) we have a formula \(N_1(h)\) that approximates an unknown constant \(\ \ \ \ \ \ \ \\), and that the truncation error involved with the approximation has the form,
     \(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\)
   • The truncation error is \(\ \ \ \ \ \ \ \ \ \ \ \ \ \\) \(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\)
     \(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\)
     and, in general,
     \(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\)
   • The object of extrapolation is
2. Why?
   
3. The \(\mathcal{O}(h)\) formula for approximating \(M\):
   

   The First Formula:

   The Second Formula:

4. The \(\mathcal{O}(h^2)\) approximation formula for M:
   
5. When to apply Extrapolation?
   
6. The \(\mathcal{O}(h^4)\) formula for approximating \(M\):
   
7. The \(\mathcal{O}(h^6)\) formula for approximating \(M\):
   
8. The \(\mathcal{O}(h^{2j})\) formula for approximating \(M\):
   
9. The Order the Approximations Generated:
   

Deriving n-point Formulas with Extrapolation

1. Deriving Five-point Formula: