Midpoint Method

The Midpoint (or Improved Polygon) Method is another modification of Euler’s method, which uses Euler’s method to predict a value at the mid-point of the interval.

The $y$ value at the midpoint is:

$$y_{i+1/2}=y_i+f(x_i,y_i)\frac{h}{2}$$

And the slope at the midpoint is:

$$y_{i+1/2}^{\prime }=f(x_{i+1/2},y_{i+1/2})$$

So the $y_{i+1}$ value is:

$$y_{i+1}=y_i+f\left(x_{i+1/2},y_{i+1/2}\right)h$$