Section 15.1 of Stewart discusses the details of one method of
numerical inegration, *Euler's Method*. The program
*Numerical Methods* in the folder *MacMath 9.2* lets
you explore this method as well as two more refined ones.

All these methods have two kinds of errors associated. For step sizes
small but not too small, each method has a characteristic order p
so that the error (by some fixed time) in computed solutions is
typically
for some constant *C*. This part is
called truncation error.
However for step sizes too small, roundoff error causes further
reduction of step size to cause a random walk, or worse, and
consequently no improvement in accuracy. By adjusting the
number of bits used for numerical computation
in the program *Numerical Methods* you can
control at which stepsize truncation error predominates, and when
roundoff does.

For the equation *y*'=*x* with *y*(0)=1, here are some explicit suggestions
to look at how the dependence of the error at *t*=1 depends on
choices of method, stepsize, and rounding technique.

**Euler Method 64 Bit:**- Try step sizes of , , and . Can you see a truncation error approximately proportional to ?
**Midpoint Euler Method 64 Bit:**- Try step sizes of and . Can you see a truncation error approximately proportional to ?
**Midpoint Euler Method 16 Bit Trunc round:**- Try step sizes of , , and . Can you see a roundoff error approximately proportional to ?
**Midpoint Euler Method 16 Bit Trunc down:**- Try step
sizes of
,
,
and
.
Can you
see a roundoff error approximately proportional to
?