Midpoint method
![](/Images/godic/202501/27/Midpoint_method_illustration0605.png")
![](/Images/godic/202501/27/Numerical_integration_illustration,_step=1.svg0605.png")
![](/Images/godic/202501/27/Numerical_integration_illustration_step=0.25.svg0605.png")
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,\dots](/Images/godic/202501/27/22c1001034b1cc6e20b711d6331513bb0605.png")
Here, ![h](/Images/godic/202501/27/2510c39011c5be704182423e3a695e910605.png")
is the step size — a small positive number, ![t_{n}=t_{0}+nh,](/Images/godic/202501/27/1e1cd4de55b6fd2731a3ec0d2cf672120605.png")
and ![y_{n}](/Images/godic/202501/27/0aac89cc5848912240b16f540cc5a6740605.png")
is the computed approximate value of ![y(t_{n}).](/Images/godic/202501/27/3e1d03a57cf40327e22556b4c53f4eec0606.png")
The explicit midpoint method is also known as the modified Euler method, the implicit method is the most simple collocation method, and, applied to Hamiltionian dynamics, a symplectic integrator.