Predictor-Corrector Methods

In Sections $1$ and $2$ , during the course of the discussion on the Euler's algorithm, the value of $$\int\limits_{x_k}^{x_{k+1}}$$ has been approximated to $h f(x_k, y(x_k))$ . On the other hand, we could have also considered its approximate value by $$\frac{h}{2} \bigl( f(x_k, y(x_k)) + f(x_{k+1}, y(x_{k+1})) \bigr)$$ . We could have thought of it to solve the IVP (numerically) by defining the approximations

$$y_{k+1} = y_k + \frac{h}{2} \bigl( f(x_k, y_k) + f(x_{k+1}, y_{k+1}) \bigr) {\mbox{ for }} k = 0,1, 2, \ldots, n-1$$ (14.4.15)

with the initial value $y(a) = y_0$ . An equation of the type () for $y_k$ is called an implicit equation for $y_{k+1}$ . On many occasions solving equation () could be too tough and so we resort to (numerical) approximate value for $y_{k+1}$ . Note that we are concentrating more on computing $y_{k+1}$ , for a fixed $k$ integer lying between 0 and $n-1$ . To start with, we let $y_k^0 = 0, y_1^0 = y_0$ and for $k = 0,1,2,\ldots, n-1$ , we define

$$y^{k+1} = y_m + \frac{h}{2} \bigl( f(x_0, y_m) + f(x_1, y_{m+1}^k) \bigr).$$ (14.4.16)

Essentially, we are trying to iterate for $y_1$ . we need to stop this iteration at some stage and then ``find the value" of $y_i^k$ and designate it as $y_1$ . One method of stopping the iteration is when

$$\frac{ y_1^{k+1} - y_1^k}{y_1^{k+1}}$$

is ``small" (small have means that the absolute value of the ratio is lesser than an assigned (previously) small number). we repeat the process with $y_1$ in place of $y_0$ and $x_1$ in place of $x_0$ . In general, () allows us to recursively define

$$y^0_{m+1} = y_m, \;\; {\mbox{ and }} y_{m+1}^{k+1} = y_m + \frac{h}{2} \bigl( f(x_m, y_m) + f(x_{m+1}, y_{m+1}) \bigr)$$

for $k = 0, 1, 2, \ldots$ and $m = 0,1, 2, \ldots, n$ . The iterated values $y_{m+1}^0, y_{m+1}^1, \ldots$ are called THE INNER ITERATIONS for $y_{m+1}$ .

Some more terminologies:
Normally, an explicit method like the Euler's method or the R-K methods are known as open type methods or algorithms. They are readily available for computation and the starters are known. On the other hand, implicit method as described by () is called closed type. Many a times, it may happen that the starters for the (approximate solution) for closed type method is obtained from the open type one. The starter $y_{m+1}^0$ for () is also familiarly known as a Predictor whereas the value $y_{m+1}$ (so computed) is called a corrector. In short, we predict the value $y_{m+1}^0$ and correct it (by iteration) to obtain $y_{m+1}$ . For this reason such methods are called PREDICTOR-CORRECTOR MEHTODS, (in short PC methods). Again, we repeat that PC methods need some condition to step the inner iterations, usually they are:

1. the number $M$ of iterations (called the tolerance on the number of iterations),
2. a bound on the relative error (called the tolerance on the relative error).

As for as condition  is concerned, it simply says we do not wish to iterate beyond $M$ iterations, while condition  says that keep iterating till the relative error is small, no matter how many iterations are needed. The number $M$ is known as the tolerance. On occasions many use both the conditions  and  to stop the inner iterations, which even leads to early termination.

With these preliminaries, we state the Predictor-Corrector algorithm.