Euler's method allows us to find approximates values forat a point given the equation
and
Sometimes this equation cannot be solved analytically, and Euler's method is a quick method to find values of
numerically. We start by defining a step size
We assume
is constant on the interval
then
hence
Givenand a step size
we can find
etc.
Example: Givenwith
and a step size of
find
For this simple example we can find the exact answer.
From this we findan error of less than 1%.
If the step size is reduced we would expect a more accurate estimate ofIn general if the step size is halved the error is reduced by a factor of 4. If the step size is reduced by a factor of
the error is reduced by a factor of
The number of calculations increases by a factor of k, and there is a balance to be struck between accuracy and the number of calculations.