High precision series solutions of differential equations: Ordinary and regular singular points of second order ODEs ?
A subroutine for a very-high-precision numerical solution of a class of ordinary differential equations is provided. For a given evaluation point and equation parameters the memory requirement scales linearly with precision P P , and the number of algebraic operations scales roughly linearly with P P when P P becomes sufficiently large. We discuss results from extensive tests of the code, and how one, for a given evaluation point and equation parameters, may estimate precision loss and computing time in advance. Program summary Program title: seriesSolveOde1 Catalogue identifier: AEMW_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEMW_v1_0.html Program obtainable from: CPC Program Library, Queens University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 991 No. of bytes in distributed program, including test data, etc.: 488116 Distribution format: tar.gz Programming language: C++ Computer: PCs or higher performance computers. Operating system: Linux and MacOS RAM: Few to many megabytes (problem dependent). Classification: 2.7, 4.3 External routines: CLN  Class Library for Numbers [1] built with the GNU MP library [2], and GSL  GNU Scientific Library [3] (only for time measurements). Nature of problem: The differential equation Solution method : The solution ?(z) ? ( z ) , and optionally ??(z) ? ? ( z ) , is evaluated at the point z z by executing the recursion Additional comments: The code of the main algorithm is in the file seriesSolveOde1.cc, which #include the file checkForBreakOde1.cc. These routines, and the programs using them, must #include the file seriesSolveOde1.cc. Running time:  On a Linux PC that is a few years old, at y = 10 to an accuracy of P=200 P = 200 decimal digits, evaluating the ground state wavefunction of the anharmonic oscillator (with the eigenvalue known in advance); (cf. Eq. (6)) takes about 2ms, and about 40min at an accuracy of P = 100 000 decimal digits. References: [1] B.Haible and R.B.Kreckel, CLN  Class Library for Numbers, http://www.ginac.de/CLN/ [2] T.Granlund and collaborators, GMP  The GNU Multiple Precision Arithmetic Library, http://gmplib.org/ [3] M. Galassi etal., GNU Scientific Library Reference Manual (3rd Ed.), ISBN 0954612078., http://www.gnu.org/software/gsl/
regular singular points second order odes ordinary points frobenius method