Calculus/Euler's Method

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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[edit]

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.

← Optimization Calculus Differentiation/Applications of Derivatives/Exercises →
Euler's Method