Foster Morrison
06-29-2005, 10:01 AM
In the early days of computers some people suggested CAC (continuous numerical continuation, basically Taylor series in the independent variable) as a high precision method for solving ODEs numerically. The need for stable recursion formulas for higher derivatives severly limits the number of possible applications. More recently I had the idea of using Chebyshev approximation. To test the feasibility I ran some tests against simple, solvable problems: SHM and the Kepler orbit. Stability was indicated, but an eignevalue analysis is needed. After that, assuming stability is indicated, there are opportunities to adapt the method to various problems using sophisticated control logics rather than the minimal one adopted for the tests. Not only can step size be varied, but also the sampling rate and the orders; the decay rate of the coefficients provides a semi-analytic tool that can be incorporated into the control logic. Step sizes for the tests were much longer than what can be used for Runge-Kutta or traditional multi-step methods. Numerical Recipes subroutines adapted: those for Chebyshev approximation. Gill's version of Runge-Kutta was coded for the starting method. The NumRec eigenvalue subroutine can be used for stabilty analysis simply by taking the matrix operator for Chebyshev quadrature, call it Q, by using the positive definite matrix Q^T*Q. Extension of numerical orthogonal function methods to PDEs is another possible development. The (Fortran 77) source code and documentation is available in one zip file and MS-DOS executables in another. Send requests to turtle_hollow@sigmaxi.org.