4. Predictor - Corrector Methods

In section 1,2 during the course of the discussion on the Euler's Algorithm, the approximated value of

$$\int\limits_{x_k}^{x_n+h}f(x,y(x))$$

is $hf(x_k,y(x_k))$. On the other hand we could have also considered its approximate value by

$$\frac{h}{2}\{f(x_k,y(x_k))+f(x_{k+1},y(x_{k+1}))\}$$

we could have thought of it to solve the IVP (numerically) by defining the approximations

$$Y_{k+1}=Y_k+\frac{h}{2}\{f(x_k,Y_k)+f(x_{k+1},Y_{k+1})\}$$ 4.1

with $Y_0=y+0$ and $k=0,1,2,...n-1$. An equation of the type (4.1) for $Y_k$ is called an implicit equation for $Y_{k+i}$. On many occasions solving of (4.1) could be too tough and so we so resort to (numerical) approximate value for $Y_{k+1}$. Note here we are concentrating more on computing $Y_{k+1}$, for a fixed $k$ between 0 and n-1. To start with we let $Y^0 k=0$ and let $Y_1^0=y_0$ and define , for k=0,1,2...

$$Y^{k+1}=Y_m+\frac{h}{2}\{f(x_0,Y_m)+f(x_1,Y^k_{m+1})\}$$ 4.2

Essentially we are trying to iterate for $Y_1$. We need to step this iteration at some stage and the "find value " of $Y_i^{k=1}$ is designated as $Y_1$. One method of stopping the iteration is when

$$(Y_1^{k+1}-Y_1^k)/Y_1^{k+1}$$

is "small" (small have means that the absolute value of the ratio is lesser then 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 (4.1) allows us to define

$$Y^0_{m+1}=Y_m$$

and define

$$Y^{k+1}_{m+1}=Y_m+\frac{h}{2}\{f(x_m,Y_m)+f(x_{m+1},Y_{m+1})\}$$

for k=0,1,2...and m=0,1,2,...n.
The iterated values $Y^0_{m+1},Y^1_{m+1},...$ are called the inner iterations for $Y_{m+1}$. Some more terminologies: Normally an explicit method like Euler's method or R-K methods are known as open type method or algorithm. They are readily available for computation and the starters are known. On the other hand implicit method as described by (4.2) 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 $Y^0_{m+1}$ for (4.2) 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 $Y^0_{m+1}$ and correct (it by iteration) to obtain $Y_{m+1}$, for this reason such methods are called Predictor-corrector method (in short called PC-method). Again, we repeat have that PC method needs some condition to step the inner iterations, usually they are
(a)the number M of iterations (called the tolerance on the number of
iterations)
(b)a bound on the relative error (called the tolerance in the relation error.)
As for as condition (a) is concerned, it simply says we do not wish to iterate beyond M iterations, while the condition (b) says that keep iterating, till the relative error is small, no matter how many iterations needed. On occasions many use ... a1 and b1 to stop the inner iterations, while even leads to early termination. With these preliminaries we state the predictor-corrector algorithm.