Calculus/Euler's Method

Euler's Method is a method for estimating the value of a function based upon the values of that function's first derivative.

The general algorithm for finding a value of $y(x)$ is:

Template:Calculus/Def where f is $y^{\prime }(x)$ . In other words, the new value, $y_{n+1}$ , is the sum of the old value $y_n$ and the step size $\mathrm{\Delta }{x}_{\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{p}}$ times the change, $f(x_n,y_n)$ .

You can think of the algorithm as a person traveling with a map: Now I am standing here and based on these surroundings I go that way 1 km. Then, I check the map again and determine my direction again and go 1 km that way. I repeat this until I have finished my trip.

The Euler method is mostly used to solve differential equations of the form

Examples

A simple example is to solve the equation:

Template:Calculus/Def This yields $f=y^{\prime }=x+y$ and hence, the updating rule is: Template:Calculus/Def Step size $\mathrm{\Delta }{x}_{\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{p}}=0.1$ is used here.

The easiest way to keep track of the successive values generated by the algorithm is to draw a table with columns for $n,x_n,y_n,y_{n+1}$ .

The above equation can be e.g. a population model, where y is the population size and x is time.