The Euler Method is a very simple method used for numerical solution of initial-value problems. Although there are much better methods in practise, it is a nice intuitive mechanism.

The objective is to find a solution to the equation

$$ \frac{dy}{dt} = f(t,y) $$

over a grid of points (equally spaced, in our case). Euler’s method uses the relation (basically a truncated form of Taylor’s approximation theorem with the error terms chopped off):

$$ y(t_{i+1}) = y(t_i) + h\frac{df(t_i, y(t_i))}{dt}$$

In R, we can express this iterative solution as:

euler <- function(dy.dx=function(x,y){}, h=1E-7, y0=1, start=0, end=1) { nsteps <- (end-start)/h ys <- numeric(nsteps+1) ys[1] <- y0 for (i in 1:nsteps) { x <- start + (i-1)*h ys[i+1] <- ys[i] + h*dy.dx(x,ys[i]) } ys }

Note that given the start and end points, and the size of each step, we figure out the number of steps. Inside the loop, we calculate each successive approximation.

An example using the difference equation

$$ \frac{df(x,y)}{dx} = 3x – y + 8$$

is:

dy.dx <- function(x,y) { 3*x - y + 8 } euler(dy.dx, start=0, end=0.5, h=0.1, y0=3) [1] 3.00000 3.50000 3.98000 4.44200 4.88780 5.31902