The Optimization of Convergence for Chebyshev Polynomial Methods in an Unbounded Domain
Abstract
By using the method of steepest descent, I have compared the suitability of three different methods for solving problems in a semiinfinite or infinite domain using Chebyshev polynomials. Exponential mappings are uniformyly bad. Domain truncation and algebraic mapping both work well, but each is superior for a different category of problems. When the solution is an entire function, then domain truncation is best. It is always simpler to apply than algebraic mapping and always at least as accurate—much more accurate for functions which decay very rapidly. For singular functions, on the other hand, algebraic mapping is better because it is less sensitive to the mapping scale factor L, permitting a better compromise in resolving both the singularity and the exponential decay. For both types of problems, I give simple, explicit estimates of both the optimum choise of domain size or mapping factor and of the attainable accuracy. For the model entire function exp [ Az^{k}] on a semiinfinite interval [0, ∞], the optimum domain size is L = 0.896( {n}/{A}) ^{{1}/{k}} and the smallest attainable error is roughly e^{ n(0.896) k} where n is the number of Chebyshev polynomals used. Similar formulas for singular functions handled via algebraic mapping are given in (3.41) to (3.45) below.
 Publication:

Journal of Computational Physics
 Pub Date:
 January 1982
 DOI:
 10.1016/00219991(82)901024
 Bibcode:
 1982JCoPh..45...43B