Midpoint method
In numerical analysis, a branch of applied mathematics, the midpoint method is a one-step method for numerically solving the differential equation, y ′ ( t ) = f ( t , y ( t ) ) , y ( t 0 ) = y 0 . {\displaystyle y'(t)=f(t,y(t)),\quad y(t_{0})=y_{0}.} The explicit midpoint method is given by the formula the implicit midpoint method by for n = 0 , 1 , 2 , … {\displaystyle n=0,1,2,\dots } Here, h {\displaystyle h} is the step size — a small positive number, t n = t 0 + n h , {\displaystyle t_{n}=t_{0}+nh,} and y n {\displaystyle y_{n}} is the computed approximate value of y ( t n ) .