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Manuscript Title: FDEXTR, a program for the finite-difference solution of the coupled- channel Schrodinger equation using Richardson extrapolation.
Authors: A.G. Abrashkevich, D.G. Abrashkevich
Program title: FDEXTR
Catalogue identifier: ACVG_v1_0
Distribution format: gz
Journal reference: Comput. Phys. Commun. 82(1994)209
Programming language: Fortran.
Computer: IBM RS/6000 Model 320H.
Operating system: AIX3.3.3, ULTRIX 4.2A, DEC OSF/1 V1.2, OPEN VMS.
RAM: 492K words
Word size: 64
Peripherals: disc.
Keywords: General purpose, Richardson extrapolation, Finite difference method, Schrodinger equation, Eigensolutions, Ordinary Differential equations, Atomic, Molecular, Chemical physics.
Classification: 4.3.

Nature of problem:
Coupled second-order differential equations of the form
                 
                  d**2                                                   
              [-P ---- + Q(x)]Y(x) = lambda Y(x), x in [a,b],            
                  dx**2   
with boundary conditions
                                               
                                        dY(x) |                          
                         Y(a) = 0   or  ----- |    = 0,                  
                                         dx   |x=a                       
                                                                         
                                        dY(x) |                          
                         Y(b) = 0   or  ----- |    = 0,                  
                                         dx   |x=b                       
                                                                     
are solved. Here lambda is an eigenvalue, Y(x) is an eigenvector, Q(x) is a symmetric potential matrix and P = cI, where I is the unit matrix and c is a certain constant (usually c = hbar**2/2mu or 1). Such systems of coupled differential equations usually arise in atomic, molecular and chemical physics calculations after separating the scattering (radial) coordinate from the rest of the variables in the multidimensional Schrodinger equation. The main purpose of the present paper is to present an iterative extrapolational procedure for high- precision calculation of the approximate eigensolutions of the coupled- channel Schrodinger equation.

Solution method:
The coupled differential equations are solved by the finite-difference method of second order on a sequence of doubly condensed meshes with iterative Richardson extrapolation of the difference eigensolutions [1]. The generalized algebraic eigenvalue problem A Y = lambda B Y arising after the replacement of the differential problem by a symmetric difference scheme of the second order of accuracy is solved by the subspace iteration method [2] using the SS-PACE program [2]. The same extrapolational procedure and error estimations are applied to the eigenvalues and eigenfunctions.

Restrictions:
The computer memory requirements depend on the number of euations to be solved, the number of grid points chosen, and the number of eigensolutions required. Restrictions due to dimension sizes may be easily alleviated by altering PARAMETER statements (see Long Write-Up and listing for details). The user must also supply a subroutine which evaluates the potential matrix Q(x) at a given x.

Running time:
The running time depends critically upon: a) the number of coupled differential equations; b) the number of required eigensolutions; c) the number of mesh points on the interval [a,b]. The test run which accompanies this paper took 3.87 s on the IBM RS/6000 Model 320H.

References:
[1] A.G. Abrashkevich and D.G. Abrashkevich, Comp. Phys. Commun. (see the preceeding paper).
[2] K.-J. Bathe, Finite Element Procedures in Engineering Analysis (Prentice-Hall, Englewood Cliffs, New York, 1982).