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In numerical analysis, a branch of applied mathematics, the midpoint method is a one-step method for numerically solving the differential equation,
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 . {\displaystyle y.} The explicit midpoint method is sometimes also known as the modified Euler method, the implicit method is the most simple collocation method, and, applied to Hamiltonian dynamics, a symplectic integrator. Note that the modified Euler method can refer to Heun's method, for further clarity see List of Runge–Kutta methods.