SpeedMapping.jl implements Alternating cyclic extrapolations: a new and fast algorithm for accelerating optimization algorithms. It may be used for a large class of problems requiring a solution to the mapping F(x) = x. It also performs multivariate optimization often faster than L-BFGS or the nonlinear conjugate gradient method, especially with box-constraints. It will be useful in statistics, computer science, physics, biology or economics and many other fields.