Programs in Physics & Physical Chemistry
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|Manuscript Title: A program to set up systems of orthogonal polynomials.|
|Authors: U. Opik|
|Program title: ORTHOGONAL POLYNOMIALS|
|Catalogue identifier: AAXL_v1_0|
Distribution format: gz
|Journal reference: Comput. Phys. Commun. 46(1987)263|
|Programming language: Fortran.|
|Computer: ICL SERIES 39 LEVEL 80.|
|Operating system: VME.|
|Keywords: General purpose, Function, Orthogonal polynomials, Weight function, Three-term recurrence, Relation, Clenshaw's algorithm, Numerical integration, Gaussian quadrature, Chebyshev expansions.|
Nature of problem:
To set up a system of parameter-dependent orthogonal polynomials for use by other programs (to be published later) in computations of heavy- particle (hydrogenic ion+ bare nucleus) collision cross sections by the method of Morrison and Opik. Such polynomials may prove to be of use in various other kinds of physical problems.
The coefficients in the standard three-term recurrence relation are computed in succession for polynomials of degree 1,2,3, ... by formulae whose derivation is given in full in the Long Write-up.
The interval in which the polynomials are orthogonal must be either 0<= x <= b (b>0) or 0<= x < infinity, i.e. the interval -infinity< x < infinity is excluded.
The computation is divided into well-defined stages, called tasks; the sequence in which they are executed is largely under the control of the user. COMMON storage is not used except for diagnostic purposes, and it should be easy to use a subroutine, together with the subroutines called by it, as a module out of the context of this program.
To compute the necessary data on the polynomials for one value of the parameter u (see Long Write-up, for the definitions): up to 4th degree (i.e. N=5) ab initio, 0.218 -+0.0001; up to 4th degree from previously fitted curves, 0.032 -+0.0001; up to 14th degree ab initio, 4.04 _+0.02.
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