Dense Output

Introduction

We solve numerically an initial value problem, IVP, for a first-order system of ordinary differential equations, ODEs. That is, we approximate the solution y(t) of

$$\displaystyle{y^{{\prime}}(t) = f(t,y(t)),\qquad t_{ 0} \leq t \leq t_{F}}$$

that has given initial value y(t₀). In the early days this was done with pencil and paper or mechanical calculator. A numerical solution then was a table of values, $y_{j} \approx y(t_{j})$, for mesh points t _j that were generally at an equal spacing or step size of h. On reaching t _n where we have an approximation y _n , we take a step of size h to form an approximation at $t_{n+1} = t_{n} + h$. This was commonly done with previously computed approximations and an Adams-Bashforth formula like

$$\displaystyle{ y_{n+1} = y_{n} + h\left [\dfrac{23} {12}f_{n} -\dfrac{16} {12}f_{n-1} + \dfrac{5} {12}f_{n-2}\right ]. }$$

(1)

Here $f_{j} = f(t_{j},y_{j}) \approx f(t_{j},y(t_{j})) = y^{{\prime}}(t_{j})$. The number of times the...