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   {\bf ``Analysis'' portion of Atkinson's Problem 6.6 \\
        Gordon Wade.
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The problem given in the book says to implement the Euler method for
solving the initial value problem 
\begin{eqnarray*}
      y' &=& f(x,y), \\
  y(x_0) &=& Y_0,
\end{eqnarray*}
to test it with $x\in[0,4]$, $Y_0=1$, and 
\[
  f(x,y) = x^2 - y, 
\]
and to ``analyze the results''.

The implementation is carried out elsewhere, in Matlab, and the results 
are in a Matlab ``diary'' file. 

I interpret ``analyze the results'' to mean ``see if we observe 
numerically what the theory predicts''. In particular, we should see
\[
   e_{max} = \max_{n=1:N} |e_n| \leq Bh
\]
for some constant $B$, independent of $h$.
Here is what we see in the Matlab diary file:
\begin{eqnarray*}
   h=0.25 &\mbox{ yields }& e_{max} = 0.2392, \\
   h=0.125   &\mbox{yields}& e_{max} =  0.1189, \\
   h=0.0625  &\mbox{yields}& e_{max} = 0.05926.
\end{eqnarray*}
Setting the ``observed'' value of $B$ to be $\tilde{B} = e_{max}/h$, 
we get
\begin{eqnarray*}
   h=0.25 &\mbox{ yields }&  \tilde{B} = 0.9568 \\
   h=0.125   &\mbox{yields}& \tilde{B} = 0.9511 \\
   h=0.0625  &\mbox{yields}& \tilde{B} = 0.9481
\end{eqnarray*}
So it does indeed seem that $\tilde{B}\approx B$ is constant as theory 
predicts, with
$\tilde{B}\approx 0.95$.


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