a predictor-corrector method

Implicit methods are accurate and stable but awkward: their formula has the unknown next value on BOTH sides, so you cannot just compute it — you have to solve an equation for it. A predictor-corrector method sidesteps that by working in two moves. First a cheap, explicit method makes a rough guess for the next value (the prediction). Then that guess is plugged into the right-hand side of a better, implicit formula, which spits out a refined value (the correction). Guess, then improve — like sketching in pencil, then inking over the sketch.

Precisely: the PREDICTOR is an explicit formula (say Adams-Bashforth, or a plain Euler step) that produces an initial estimate y*_(n+1) using only already-known values. The CORRECTOR is an implicit formula (say Adams-Moulton, or the trapezoidal rule) that would normally need y_(n+1) on its right-hand side — but now you feed in the predictor's y* there, turning the implicit formula into a direct computation. Often you evaluate f once at the prediction and once after the correction; this evaluate-correct cycle is the PECE pattern. The corrector can even be applied more than once, each pass refining the value.

This is the everyday way the powerful implicit Adams methods get used without ever solving a nonlinear equation. The pairing also gives a free bonus: the gap between the predicted and corrected values is a cheap, built-in estimate of the local error, which is exactly what an adaptive step controller needs to decide whether to grow or shrink the step. Improved Euler is the simplest predictor-corrector of all — Euler predicts, the trapezoidal rule corrects. The idea scales all the way up to the multistep Adams-Bashforth-Moulton pairs used in production solvers.