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{2}
A = A_h + a_2 h^2 + a_4 h^4 + a_6 h^6 + O(h^8).
\end{equation}


Here, \(h\) should be thought of as a small positive number, and \(a_1\), \(a_2\), \(a_4\), and \(a_6\) are fixed nonzero,
but otherwise unknown, numbers.
\begin{enumerate}

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


\item
Use Eq. (2) to derive an approximation \(B_h\) to \(A\) for which the error \(A - B_h = O(h^4)\) and which
uses only your program and some simple algebra.

\end{enumerate}