TABLE OF CONTENTS Our textbook function ode23tx is a simplified version of the function ode23 that is included with MATLAB. The algorithm is due to Bogacki and Shampine [9, 50]. The " 23" in the function names indicates that two simultaneous single-step formulas, one of second order and one of third order, are involved.
The method has three stages, but there are four slopes s i because, after the first step, the s 1 for one step is the s 4 from the previous step. The essentials are
The simplified pictures in Figure 7.1 show the starting situation and the three stages. We start at a point (t n, y n) with an initial slope s 1 = f(t n, y n) and an estimate of a good step size, h. Our goal is to compute an approximate solution y n+1 at t n+1 = t n + h that agrees with the true solution y( t n+1) to within the specified tolerances.
The first stage uses the initial slope s 1 to take an Euler step halfway across the interval. The function is evaluated there to get the second slope, s 2. This slope is used to take an Euler step three-quarters of the way across the interval. The function is evaluated again to get the third slope, s
