- Email: [email protected]
FDEXTR 2.1: A new version of a program for the finite-difference solution of the coupled-channel Schrödinger equation using the Richardson extrapolation
ComputerPhysics Communications ELSEVIER
Computer Physics Communications 113 (1998) 105-107
FDEXTR 2.1" A new version of a program for the finite-difterence solution of the coupled-channel SchriSdinger equation using the Richardson extrapolation A.G. Abrashkevich i, D.G. Abrashkevich Chemical Physics Theory Group. Department of Chemistry. Universi~ of Toronto. Toronto, Canada MSS 3H6
Received 10 February
1998
Abstract A FORTRAN program is presented which solves the Sturm-Liouville problem for a system of coupled second-order differential equations by the finite difference method of the second order using the iterative Richardson extrapolation of the difference eigensolutions on a sequence of doubly condensed meshes. The same extrapolational procedure and error estimations are applied to the eigenvalues and eigenfunctions. Zero-value (Dirichlet) or zero-gradient (Neumann) boundary conditions are considered. (~) 1998 Elsevier Science B.V. Keyword~: Richardson extrapolation: Finite difference method: Schrodinger equation; Eigensolutions:Ordinary differential equations: Atomic: Molecular: Chemical physics
NEW VERSION SUMMARY
Catalogue identifier of previous version: ACVG
~tle of program." FDEXTR: version number: 2. I
Does the new version supersede the original pragram: yes
Catalogue identifier: ADIC
Computer for which the new version is designed and others on which it is operable: Computers: DEC 3000 ALPHA AXP Model 800, IBM RS/6000 Model 320H, Sun-EIc, HP 715, SGI Origin 2000, SGI Indigo R4000; Installations: Department of Chemistry, University of
Program Summary URL: hnp://www.cpc.cs.qub.ac.uk/cpc/summaries/ADlC Program obtainable from: CPC Program Library. Queen's Univer-
sity of Belfast, N. Ireland
Toronto,Canada: Departmentof Chemical Physics, The Weizmann Institute of Science, Israel: Computing Center of the Weizmann Institute of Science, Israel
Licensing provisions: none
Operating system.s under which the new version has been tested:
Refirrence in CPC to previous version: A.G. Abrd.shkevich, D.G. Abraxhkevich, Comput. Phys. Commun. 82 (1994) 209 (version
Digital UNIX 4.0 (DEC). AIX 3.2.5 (IBM), SunOs 4.1.2 (Sun), HP/UX 9.01 (HP). Irix 4.05 and 6.4 (SGI)
l.l) t
Corresponding author; e.mail: [email protected].
0010-4655/98/$19.00 (~) 1998 Elsevier Science B.V. All fights reserved, PII SO010-4655 ( 98 ) 0003 3-2
106
A.G. Abraxhlaevich. D G. Abrashkevich/Computer Physics Communication.* 113 (1998) 105-107
Programming language used: FORTRAN Memory required to execute with ~.'pi¢'al data: depends on the number of equations, the number of mesh points and the number of eigensolutions required; the test run requires 844 KR N,. eJf bits in a word" 64 No. ,Jpn~ce.*s,r.* used: one Hat the code been vecmri.$ed? no Overlay .strucnlre" none Peripheral.* uxed: line printer, scratch disc store No. re/bytes in distributed prognon, incheding test clara, on'.: 52783
the second order of accuracy is solved by the subspace iteration method 131. The same extrapolational procedure and error estimations are applied to the eigenvalues and eigenfunctions.
Snmma~. eJf revisions (i) The code has been modified and optimized to compile under the FORTRAN 90 compiler. (it) The SSPACE routine 131 used in the previous version of the FDEXTR 141 for the solution of the generalized eigenvalue problem has been replaced with the widely available standard routine F02FJF from the NAG Program Library I I I. This routine is designed for solution of generulized eigenvalue problems for large symmetric banded matrices. The procedure chooses a vector subspace of the full solution space and iterates upon successive solutions in the subspace. Subroutines FACTRS. DOT. IMAGE. MONIT. PROD and SOLVE have been added to the program. The function of each routine is briefly described below: Subroutine F02FJF I I I finds eigenvalues and eigenvectots of a real sparse generalized symmetric eigenvalue problem using the subspace iteration method 13 I. - Subroutine FACTRS calculates the L (D) L "r factoriralion of stiffness matrix. - DOUBLE PRECISION function DOT computes the generalized dot product wrB: for given n element vectors z and w and mass matrix B. Subroutine IMAGE solves the positive-definite system of equations Aw = Bz for w. where : is a given n element vector, and A and B are the stiffness and mass matrices. respectively. -
Distribution fi,rmat: ASCII Libra," routines used: F02FJF I I I Keyw, rds: Richardson extrapolation, finite difference method. Schfodinger equation, eigensolutions, ordinary differential equalions, atomic, molecular, chemical physics
-
Nature ~¢physical problem Coupled second-order differential equations of the form
t
""
J
progress of the F02FJF program. Subroutine PROD evaluates product of the two vectors stored in compact form. Subroutine SOLVE solves a system of linear algebraic equations AX = B for X using L (D) L v factorization. (iii) A new parameter NSITV has been added to the list of input program parameters. It specifies the number of simultaneous iteration vectors tO be used. -
with boundary conditions
-
Y(a) = 0
or
Y(b) = 0
or
d y ( x ) [,:,~ = 0. dx dY(x) = O. dx t=J,
are solved. Here ,~ is an eigenvalue. ¥ ( x ) is an eigenvector. Q ( x ) is. an symmetric potential matrix, and P = ~'1. where I is the unit matrix and ~" is a some constant (usually ~' = Ii2/21z or I ). Such systems of coupled differential equations usually arise in atomic, molecular and chemical physics calculations ader ~parating the scattering (radial) coordinate from the rest of variables in the multidimensional Sch~odinger equation. The main purpose of the present paper is to present an iterative extrapJlational procedure for high-precision calculation of the approximate eigensolufions of the coupled-channel Schr'odinger equation.
Method of solution The coupled differential equations are solved by the finitedifference method of the .second order on a sequence of doubly condoned meshes with the iterative Richardson extrapolation of the difference eigensoluti'ons 121 The generalized algebraic eigenvalue problem A Y = 3 B Y arising from the replacement of the differential problem by a symmetric difference scheme of
(iv) Output has been cleaned up and reduced.
Restrh'tilm~on the ('onq~le.rity o] tile problem The computer memory requirements depend on the number of equations to be solved, the number of grid points chosen, and the number of eigensolutions required. Restrictions due to dinvension sizes may be easily alleviated by altering PARAMETER stalements (see Long Write-Up and listing of 141 for details). The user must also supply a subroutine which evaluates the leotential matrix Q ( x ) at a given x. Typwal running time The running time depends critically upon (a) the number of coupied differential equations" ( b ) t h e number of required eigensolulions; (c) the number of mesh points on the interval la.bl. The test run which accompanies this paper took 1.67 s on the DEC 3000 ALPHA AXP Model 8(,~0.
A.G. Abrashkevich. D.G. Abrtt~hkevwh/C, mp,ter Physics Communication.~ 113 (199R) 105-107
Re/*erem'es I 11 NAG Fortran Library Manual, Mark 15 (The Numerical AIgorithm.~ Group Limited, Oxford. 1991 ). l21 A.G. Abra.~hkevich, D.G. Abra.~hkevich. Comput. Phys. Cornmun. 82 (1994) 193.
107
131 K.J. Bathe, Finite Element Procedurc-~ in Engineering Analysis (Prentice-Hall, Englewood Cliff.k, NJ. 1982). 141 A.G. Abr~hkevich. D.G. Abraxhkevich, Compm. Phys. Coramun. 82 (1994) 209.
Computer Physics Communications 113 (1998) 105-107
FDEXTR 2.1" A new version of a program for the finite-difterence solution of the coupled-channel SchriSdinger equation using the Richardson extrapolation A.G. Abrashkevich i, D.G. Abrashkevich Chemical Physics Theory Group. Department of Chemistry. Universi~ of Toronto. Toronto, Canada MSS 3H6
Received 10 February
1998
Abstract A FORTRAN program is presented which solves the Sturm-Liouville problem for a system of coupled second-order differential equations by the finite difference method of the second order using the iterative Richardson extrapolation of the difference eigensolutions on a sequence of doubly condensed meshes. The same extrapolational procedure and error estimations are applied to the eigenvalues and eigenfunctions. Zero-value (Dirichlet) or zero-gradient (Neumann) boundary conditions are considered. (~) 1998 Elsevier Science B.V. Keyword~: Richardson extrapolation: Finite difference method: Schrodinger equation; Eigensolutions:Ordinary differential equations: Atomic: Molecular: Chemical physics
NEW VERSION SUMMARY
Catalogue identifier of previous version: ACVG
~tle of program." FDEXTR: version number: 2. I
Does the new version supersede the original pragram: yes
Catalogue identifier: ADIC
Computer for which the new version is designed and others on which it is operable: Computers: DEC 3000 ALPHA AXP Model 800, IBM RS/6000 Model 320H, Sun-EIc, HP 715, SGI Origin 2000, SGI Indigo R4000; Installations: Department of Chemistry, University of
Program Summary URL: hnp://www.cpc.cs.qub.ac.uk/cpc/summaries/ADlC Program obtainable from: CPC Program Library. Queen's Univer-
sity of Belfast, N. Ireland
Toronto,Canada: Departmentof Chemical Physics, The Weizmann Institute of Science, Israel: Computing Center of the Weizmann Institute of Science, Israel
Licensing provisions: none
Operating system.s under which the new version has been tested:
Refirrence in CPC to previous version: A.G. Abrd.shkevich, D.G. Abraxhkevich, Comput. Phys. Commun. 82 (1994) 209 (version
Digital UNIX 4.0 (DEC). AIX 3.2.5 (IBM), SunOs 4.1.2 (Sun), HP/UX 9.01 (HP). Irix 4.05 and 6.4 (SGI)
l.l) t
Corresponding author; e.mail: [email protected].
0010-4655/98/$19.00 (~) 1998 Elsevier Science B.V. All fights reserved, PII SO010-4655 ( 98 ) 0003 3-2
106
A.G. Abraxhlaevich. D G. Abrashkevich/Computer Physics Communication.* 113 (1998) 105-107
Programming language used: FORTRAN Memory required to execute with ~.'pi¢'al data: depends on the number of equations, the number of mesh points and the number of eigensolutions required; the test run requires 844 KR N,. eJf bits in a word" 64 No. ,Jpn~ce.*s,r.* used: one Hat the code been vecmri.$ed? no Overlay .strucnlre" none Peripheral.* uxed: line printer, scratch disc store No. re/bytes in distributed prognon, incheding test clara, on'.: 52783
the second order of accuracy is solved by the subspace iteration method 131. The same extrapolational procedure and error estimations are applied to the eigenvalues and eigenfunctions.
Snmma~. eJf revisions (i) The code has been modified and optimized to compile under the FORTRAN 90 compiler. (it) The SSPACE routine 131 used in the previous version of the FDEXTR 141 for the solution of the generalized eigenvalue problem has been replaced with the widely available standard routine F02FJF from the NAG Program Library I I I. This routine is designed for solution of generulized eigenvalue problems for large symmetric banded matrices. The procedure chooses a vector subspace of the full solution space and iterates upon successive solutions in the subspace. Subroutines FACTRS. DOT. IMAGE. MONIT. PROD and SOLVE have been added to the program. The function of each routine is briefly described below: Subroutine F02FJF I I I finds eigenvalues and eigenvectots of a real sparse generalized symmetric eigenvalue problem using the subspace iteration method 13 I. - Subroutine FACTRS calculates the L (D) L "r factoriralion of stiffness matrix. - DOUBLE PRECISION function DOT computes the generalized dot product wrB: for given n element vectors z and w and mass matrix B. Subroutine IMAGE solves the positive-definite system of equations Aw = Bz for w. where : is a given n element vector, and A and B are the stiffness and mass matrices. respectively. -
Distribution fi,rmat: ASCII Libra," routines used: F02FJF I I I Keyw, rds: Richardson extrapolation, finite difference method. Schfodinger equation, eigensolutions, ordinary differential equalions, atomic, molecular, chemical physics
-
Nature ~¢physical problem Coupled second-order differential equations of the form
t
""
J
progress of the F02FJF program. Subroutine PROD evaluates product of the two vectors stored in compact form. Subroutine SOLVE solves a system of linear algebraic equations AX = B for X using L (D) L v factorization. (iii) A new parameter NSITV has been added to the list of input program parameters. It specifies the number of simultaneous iteration vectors tO be used. -
with boundary conditions
-
Y(a) = 0
or
Y(b) = 0
or
d y ( x ) [,:,~ = 0. dx dY(x) = O. dx t=J,
are solved. Here ,~ is an eigenvalue. ¥ ( x ) is an eigenvector. Q ( x ) is. an symmetric potential matrix, and P = ~'1. where I is the unit matrix and ~" is a some constant (usually ~' = Ii2/21z or I ). Such systems of coupled differential equations usually arise in atomic, molecular and chemical physics calculations ader ~parating the scattering (radial) coordinate from the rest of variables in the multidimensional Sch~odinger equation. The main purpose of the present paper is to present an iterative extrapJlational procedure for high-precision calculation of the approximate eigensolufions of the coupled-channel Schr'odinger equation.
Method of solution The coupled differential equations are solved by the finitedifference method of the .second order on a sequence of doubly condoned meshes with the iterative Richardson extrapolation of the difference eigensoluti'ons 121 The generalized algebraic eigenvalue problem A Y = 3 B Y arising from the replacement of the differential problem by a symmetric difference scheme of
(iv) Output has been cleaned up and reduced.
Restrh'tilm~on the ('onq~le.rity o] tile problem The computer memory requirements depend on the number of equations to be solved, the number of grid points chosen, and the number of eigensolutions required. Restrictions due to dinvension sizes may be easily alleviated by altering PARAMETER stalements (see Long Write-Up and listing of 141 for details). The user must also supply a subroutine which evaluates the leotential matrix Q ( x ) at a given x. Typwal running time The running time depends critically upon (a) the number of coupied differential equations" ( b ) t h e number of required eigensolulions; (c) the number of mesh points on the interval la.bl. The test run which accompanies this paper took 1.67 s on the DEC 3000 ALPHA AXP Model 8(,~0.
A.G. Abrashkevich. D.G. Abrtt~hkevwh/C, mp,ter Physics Communication.~ 113 (199R) 105-107
Re/*erem'es I 11 NAG Fortran Library Manual, Mark 15 (The Numerical AIgorithm.~ Group Limited, Oxford. 1991 ). l21 A.G. Abra.~hkevich, D.G. Abra.~hkevich. Comput. Phys. Cornmun. 82 (1994) 193.
107
131 K.J. Bathe, Finite Element Procedurc-~ in Engineering Analysis (Prentice-Hall, Englewood Cliff.k, NJ. 1982). 141 A.G. Abr~hkevich. D.G. Abraxhkevich, Compm. Phys. Coramun. 82 (1994) 209.