It is very common in mathematical physics and engineering to expand functions or solve differential equations by expansion into a Fourier basis or into a system of orthogonal polynomials. However, because these basis functions are well-behaved, convergence near singular points is slow--a famous example being the Gibbs' phenomena for Fourier series. If the form of the singularity is known, then one can explicitly take this into account using series rearrangement to dramatically improve the convergence of series solutions.
To take a specific example, consider the following identity,
where are Legendre polynomials. A plot comparing the left-hand side with a partial sum of the right-hand side reveals that the sum is slowly convergent.
Using a recurrence relation,
the generic term, , of the expression becomes
Re-indexing the summation, using pattern matching, and eliminating , we obtain
which means that
We can use this expression to accelerate the convergence of Legendre expansions that have a singularity at .
Incidentally, the above method for manipulating summations by focusing on the generic term can also be used to generate recurrence relations for differential equations. Consider the Legendre differential equation,
After using Function to substitute the generic term (using Function) of the series solution, into the differential equation,
we can generate the recurrence relation using pattern matching as follows.
Returning to our original problem, we multiply the numerator and denominator of the right-hand side of equation (1) by (to remove the singularity).
Then, from equation (0), we compute
Here are the comparative plots.
The convergence acceleration is quite dramatic. We have used only four terms in the accelerated sum, compared to 40 in the original expression! One reason for the acceleration is obvious: the coefficients of the in the accelerated sum are asymptotically like . The technique outlined here is easily generalized to other eigenfunction expansions.
Converted by Mathematica September 30, 1999 [Prev Page][Next Page]