Functional iteration

From the point of view of a numerical mathematician, which we adopted in Chapters 1–5, the solution of ordinary differential equations is all about analysis – i.e. convergence, order, stability and an endless progression of theorems and proofs. The outlook of Chapter 6 parallels that of a software engineer, being concerned with the correct assembly of computational components and with choosing the step sequence dynamically. Computers, however, are engaged neither in analysis nor in algorithm design but in the real work concerned with solving ODEs, and this consists in the main of the computation of (mostly nonlinear) algebraic systems of equations.

Why not – and this is a legitimate question – use explicit methods, whether multistep or Runge–Kutta, thereby dispensing altogether with the need to calculate algebraic systems? The main reason is computational cost. This is obvious in the case of stiff equations, since, for explicit time-stepping methods, stability considerations restrict the step size to an extent that renders the scheme noncompetitive and downright ineffective. When stability questions are not at issue, it often makes very good sense to use explicit Runge–Kutta methods. The accepted wisdom is, however, that, as far as multistep methods are concerned, implicit methods should be used even for non-stiff equations since, as we will see in this chapter, the solution of the underlying algebraic systems can be approximated with relative ease.

Let us suppose that we wish to advance the (implicit) multistep method (2.8) by a single step.