Math 6620 • Analysis of Numerical Methods II

Suppose you have a program for calculating approximations to a quantity \(A\) which takes a value
of \(h\) as input and produces a value \(A_h\) as output. Suppose that
\begin{equation}
\tag{1}
A = A_h + a_1 h + a_2 h^2 + a_3 h^3 + a_4 h^4 + O(h^5).
\end{equation}


Here, \(h\) should be thought of as a small positive number, and \(a_1\), \(a_2\), \(a_3\), and \(a_4\) are fixed (but
unknown) numbers.
\begin{enumerate}

\item
What is the order of the approximation \(A_h\) for \(A\)?


\item
Use Eq. (1) to derive an approximation \(B_h\) to \(A\) for which the error \(A - B_h = O(h^3)\) and which
uses only your program and some simple algebra.

\end{enumerate}