"Leapfrog" methods for numerical solution of differential equations, sinewave generation, and magnitude approximation

Numerical integration by alternating updates between two variables-the "leapfrog" technique-has remarkably useful properties, which we trace across three widely different applications. For a dynamic physical system, alternation of position and velocity updates helps us to contrive a second-order numerical integration method twice as accurate as the more complex predictor-corrector method. Next, we specialize this method to get a simple incremental sine-cosine waveform generator which is exact except for rounding-error accumulation and competitive with Bresenham (1965) type generators for graphics applications. Finally, me further adapt the waveform generator to serve as a generalized vector-magnitude-approximation method which includes one of the best "two-line" methods as a special case.