Predictor-corrector methods

The predictor-corrector methods form a large class of general methods for numerical integration of ordinary differential equations. As an illustration, consider Milne's method Milne49 for the first-order equation File:Hepa img841.gif initial value y(x₀)=y₀. Define

Then by Simpson's rule ( Numerical Integration, Quadrature),

Because

this corrector equation is an implicit equation for y_n+1; if h is sufficiently small, and if a first approximation for y_n+1 can be found, the equation is solved simply by iteration, i.e. by repeated evaluations of the right hand side. To provide the first approximation for y_n+1, an explicit predictor formula is needed, e.g. Milne's formula

The need for a corrector formula arises because the predictor alone is numerically unstable; it gives spurious solutions growing exponentially. Milne's predictor uses four previous values of y, hence extra starting formulae are needed to find y₁, y₂ and y₃ when y₀ is given.

The starting problem is a weakness of predictor-corrector methods in general; nevertheless they are serious competitors to Runge-Kutta methods. For details Numerov's Method and Numerov's Method and, , Wong92 or Press95.