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:

$y_{n+1}=y_{n}+\Delta x_{\rm {step}}\cdot f(x_{n},y_{n}),$

where f is $y'(x)$ . In other words, the new value, $y_{n+1}$ , is the sum of the old value $y_{n}$ and the step size $\Delta x_{\rm {step}}$ 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

$y'=f(x,y),y(x_{0})=y_{0}$

Examples

A simple example is to solve the equation:

$y'=x+y,y(0)=1.$

This yields $f=y'=x+y$ and hence, the updating rule is:

$y_{n+1}=y_{n}+0.1(x_{n}+y_{n})$

Step size $\Delta x_{\rm {step}}=0.1$ 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.