The trapezoidal rule ½(f(*a*) + f(*b*))(*b* - *a*) for integration
is much better than using only the left end-point with a formula like
f(*a*)(*b* - *a*) because it samples the value of
the function at two points. If the function is reasonably smooth,
then the interpolation of these two points is appropriate. Similarly, Euler's method only
samples the slope at the left end-point, that is, the initial point
(*t*_{0}, *y*_{0}). Heun's method is an
attempt to use information about the slope at both end points to
find the average slope.

Like Euler, the eu in Heun sounds like oi, that is, Heun rhymes with *coin*.

# Background

Useful background for this topic includes:

- 7. Taylor Series
- 14.1 Euler's Method

# References

# Interactive Maplet

Copyright ©2005 by Douglas Wilhelm Harder. All rights reserved.