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Manuscript Title: A program for the predictor-corrector Numerov method.
Authors: W.E. Baylis, S.J. Peel
Program title: PCNUM
Catalogue identifier: AARJ_v1_0
Distribution format: gz
Journal reference: Comput. Phys. Commun. 25(1982)21
Programming language: Fortran.
Computer: IBM 370/3031.
Operating system: OSMVT UNDER VM.
RAM: 10K words
Word size: 32
Keywords: Coupled second-order Differential equation, Nonlinear equations, Predictor-corrector Method, Numerov, Calculations scattering, General purpose, Nuclear physics, Theoretical methods.
Classification: 4.3, 17.16.

Nature of problem:
Matrix differential equations of the form y''(x) = f(x,y(x)) are solved, including homogeneous equations such as arise in quantum scattering theory.

Solution method:
A predictor-corrector version of the Numerov method is applied. Step size is adjusted to maintain a specified accuracy, and solutions to homogeneous differential equations can be stabilized by orthogonalization.

Restrictions:
The matrices Y and F are assumed to be real matrices with dimensions not exceeding 8*8 (they are not necessarily square). These restrictions may may be removed at the cost of additional storage by adding appropriate COMPLEX*16 type declarations on the one hand, and by increasing array dimensions on the other. Although F may be a nonlinear function of Y, it is assumed that the jth column of F depends only on components in the corresponding column of Y. As with other Numerov algorithms, the equations to be solved need to be written so that there are no first- order derivatives.

Unusual features:
The error monitored is the relative error in the local change of the independent variable, and this is compared with the specified error limit to determine whether to halve, maintain or double the step size. Unlike other predictor-corrector methods, no special starting procedure is required. Solutions to linear problems can be stabilized to ensure linear independence. The stabilization is accomplished by periodic orthogonalization of the columns of Y with the frequency of application determined by the amount of change caused by the orthogonalization.

Running time:
Test runs with 4 channels (4*4 matrices) requiring evaluation at about 2500 points and stabilization, took 6 s to compile and 20 s to run on the IBM 370/3031.