Concluding Remarks

In our examples, we have been able to use the natural iteration scheme, but it is useful to know that a slight modification can yield convergence when this fails. With a linear integral equation, whose general form we write as

the standard iteration leading to the Neumann series is guaranteed to converge for any initial guess only if is less than the modulus of the smallest eigenvalue. But Bückner [14] has shown that if we use instead the iteration

with some suitable constant, we can achieve convergence for over a greater range of values; this reduces to the standard iteration if we set . As an example of this, consider the equation

which has the single eigenvalue , and the solution for all is

We begin with

and use the same means of testing accuracy as previously with

The procedure is much as used earlier, with

For the value , the usual iteration with converges.

The initial approximation is not very good.

But we easily improve on it.

Now try the same routine with .

First a look at the initial error.

Now matters are not improved by this iteration!

But if we use in the scheme instead of , convergence is restored.