SLEIGN2 home page

Sturm-Liouville Problems

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PLEASE read this file and the intro.pdf file carefully before using the SLEIGN2 package for the first time.

There are six FORTRAN files in the SLEIGN2 package:

In addition there are two TEXT files:

and three PDF files:

All of these files can be downloaded by clicking on the links above.

To run one of the examples in the xamples.f file in an UNIX environment with a FORTRAN compiler, enter the following command:

f77 xamples.f drive.f sleign2.f -o xamples.x

This will create the executable file xamples.x and two object files drive.o and sleign2.o. Now run xamples.x whenever you want to work an example from the list of examples in the xamples.f file.

To run your own problem proceed as follows:

Step1: Enter the command
```
f77 makepqw.f -o makepqw.x
```
Step2: Run
```
makepqw.x
```
(This interactive program will ask you for a file name - this must end in .f - for example, if your chosen problem has the name bloggs then enter bloggs.f). This file, after makepqw.x has been run, will contain the subroutines for p,q,w and, if necessary, the functions u and v which are used to define singular limit-circle boundary conditions at one or both end-points.
Step3: Enter the command
```
f77 bloggs.f drive.f sleign2.f -o bloggs.x
```
(drive.f and sleign2.f can be replaced with drive.o and sleign2.o if these .o files are available to speed up the compilation; these .o files are created by the first compilation.)
Step4: Run
```
bloggs.x
```
(The user is asked to provide information for the code to run bloggs.x, for example boundary conditions, eigenvalue indexes, numerical tolerances, name for report file if so desired. See the autoinput.txt file for instructions on how to automate the input of the required information.)

Above, f77 can be replaced with f90 for users who wish to use a FORTRAN 90 compiler: all six .f files are FORTRAN 90 compatible.

Note that the above procedures may have to be modified for non UNIX environments, e.g. DOS or Apple.

Note also that in running xamples.x or bloggs.x the user has access to the interactive help device; at any point where the program halts type h for access; to return to the point in the program where help was accessed, type r. Additional information on the help device can be found in the intro.pdf file.

The file sepdr.f is a sample driver for separated regular and singular boundary conditions; coupdr.f is a sample driver program for coupled regular and singular boundary conditions. Experienced users who want to bypass the extensive user-friendly interface provided in drive.f and in makepqw.f and use their own driver program may wish to look at these two sample drivers; this also applies to users who want to call the subroutines within sleign2.f in their own programs.

All six of the FORTRAN .f files are supplied in single precision; to convert these files to double precision replace "REAL" by "DOUBLE PRECISION" throughout. In UNIX this can be effected using sed, or within the vi editor as follows:

:1,$ s/REAL/DOUBLE PRECISION

It is recommended that the user try the program in single precision and switch to double precision as required.

Additional information on the SLEIGN2 package can be found in the intro.pdf file.

A number of recent papers related to the SLEIGN2 package are also available:

P.B. Bailey, W.N. Everitt and A. Zettl, The SLEIGN2 Sturm-Liouville Code, ACM Trans. Math. Software, 21 (2001), 143-192.
A. Zettl, Sturm-Liouville Theory, Mathematical Surveys and Monographs, v. 121, American Mathematical Society, 2005.
P.B. Bailey, W.N. Everitt, J. Weidmann and A. Zettl, Regular approximations of singular Sturm-Liouville problems, Results in Mathematics, 22 (1993), 3-22.
E.S.P. Eastham, Q. Kong, H. Wu and A. Zettl, Inequalities among eigenvalues of Sturm-Liouville problems, J. Inequalities and Applications, 3 (1999), 25-43.
P.B. Bailey, W.N. Everitt and A. Zettl, Computing eigenvalues of singular Sturm-Liouville problems, Results in Mathematics, 20 (1991), 391-423.
A. Zettl, Computing continuous spectrum, World Scientific, 1994, 393-406. Proceedings of International Symposium, "Trends and developments in ordinary differential equations", edited by Y. Alavi and P.-F. Hsieh.
W.N. Everitt, A catalogue of Sturm-Liouville differential equations. In Sturm-Liouville Theory, Past and Present, p. 271-331, Birkhäuser Verlag, Basel 2005 (edited by W.O. Amrein, A.M. Hinz and D.B. Pearson.)
L. Kong, Q. Kong, H. Wu and A. Zettl, Regular approximations of singular Sturm-Liouville problems with limit-circle endpoints, Results in Mathematics, 45 (2004), 274-292.
Q. Kong, H. Wu and A. Zettl, Multiplicity of Sturm-Liouville eigenvalues, J. Computational and Applied Math., 171 (2004), 291-309.
Q. Kong, H. Wu and A. Zettl, Left-Definite Sturm-Liouville Problems, J. Differential Equations, 177 (2001), 1-26.
Q. Kong, Q. Lin, H. Wu and A. Zettl, A new proof of the inequalities among Sturm-Liouville eigenvalues, PanAmerican Math. J., 10 (2000), 1-11.
Q. Kong, H. Wu and A. Zettl, Geometric Aspects of Sturm-Liouville problems, I. Structures on spaces of boundary conditions, Proc. Roy. Soc. Edinburgh, 130A (2000), 561-589.
B.M. Brown, D.K.R. McCormack and A. Zettl, On a computer assisted proof of eigenvalues below the essential spectrum of the Sturm-Liouville problem, J. Computational and Applied Math., 125 (2000), 385-393.
Q. Kong, H. Wu and A. Zettl, Dependence of the n-th Sturm-Liouville eigenvalue on the problem, J. Differential Equations, 156 (1999), 328-354.
B.M. Brown, D.K.R. McCormack and A. Zettl, On the existence of an eigenvalue below the essential spectrum, Proc. Roy. Soc. London, 455A (1999), 2229-2234.
Q. Kong, H. Wu and A. Zettl, Inequalities among eigenvalues of singular Sturm-Liouville problems, Dynamical Systems and Applications, 8 (1999), 517-531.
W.N. Everitt, M. Möller and A. Zettl, Sturm-Liouville problems and discontinuous eigenvalues, Proc. Roy. Soc. Edinburgh, 129A (1999), 707-716.
A. Zettl, Sturm-Liouville Problems, spectral theory and computational methods of Sturm-Liouville problems, Marcel Dekker, 191 (1997), 1-104. Proceedings of the 1996 Knoxville Barrett Conference, edited by D. Hinton and P. Schaefer. (This is a survey/research article.)
Q. Kong, H. Wu and A. Zettl, Dependence of eigenvalues on the problem, Mathematische Nachrichten, 188 (1997), 173-201.
W.N. Everitt, C. Shubin, G. Stolz and A. Zettl, Sturm-Liouville problems with an infinite number of interior singularities, spectral theory and computational methods of Sturm-Liouville problems, Marcel Dekker, 191 (1997), 211-249. Proceedings of the 1996 Knoxville Barrett Conference, edited by D. Hinton and P. Schaefer.
P.B. Bailey, W.N. Everitt and A. Zettl, Regular and singular Sturm-Liouville problems with coupled boundary conditions, Proc. Roy. Soc. Edinburgh, 126A (1996), 505-514.
Q. Kong and A. Zettl, Dependence of eigenvalues of Sturm-Liouville problems on the boundary, J. Differential Equations, 126 no. 2 (1996), 389-407.
Q. Kong and A. Zettl, Eigenvalues of regular Sturm-Liouville problems, J. Differential Equations, 131 no. 1 (1996), 1-19.
M. Möller and A. Zettl, Differentiable dependence of eigenvalues of operators in Banach spaces, J. Operator Theory, 36 (1996), 335-355.

Authors: Paul Bailey, Norrie Everitt and Tony Zettl (with the assistance of Burt Garbow.)

Acknowledgments: The authors are grateful to their colleagues Howard Dwyer, Qingkai Kong and Hongyou Wu for help and advice at a number of stages in the development of this program. Some of the theoretical underpinnings of the algorithm for coupled boundary conditions were obtained jointly with Michael Eastham, Qingkai Kong and Hongyou Wu.

The authors thank Burt Garbow for expert assistance in preparing the FORTRAN content of the SLEIGN2 files.

A special thanks to Eric Behr for his help throughout the development of the code, for setting up the public access through the World Wide Web, and for informed advice. All suggestions, comments and criticisms are welcome; please send all comments to sl2math.niu.edu

Northern Illinois University, DeKalb, IL. 60115-2888, USA, and School of Mathematics, University of Birmingham, Birmingham B15 2TT, England, UK. 01 December, 2000.

This page is at http://www.math.niu.edu/SL2/ with a mirror at http://web.mat.bham.ac.uk/SL2/
Last revised: 8/15/2006

	P.B. Bailey, W.N. Everitt and A. Zettl, The SLEIGN2 Sturm-Liouville Code, ACM Trans. Math. Software, 21 (2001), 143-192.
	A. Zettl, Sturm-Liouville Theory, Mathematical Surveys and Monographs, v. 121, American Mathematical Society, 2005.
	P.B. Bailey, W.N. Everitt, J. Weidmann and A. Zettl, Regular approximations of singular Sturm-Liouville problems, Results in Mathematics, 22 (1993), 3-22.
	E.S.P. Eastham, Q. Kong, H. Wu and A. Zettl, Inequalities among eigenvalues of Sturm-Liouville problems, J. Inequalities and Applications, 3 (1999), 25-43.
	P.B. Bailey, W.N. Everitt and A. Zettl, Computing eigenvalues of singular Sturm-Liouville problems, Results in Mathematics, 20 (1991), 391-423.
	A. Zettl, Computing continuous spectrum, World Scientific, 1994, 393-406. Proceedings of International Symposium, "Trends and developments in ordinary differential equations", edited by Y. Alavi and P.-F. Hsieh.
	W.N. Everitt, A catalogue of Sturm-Liouville differential equations. In Sturm-Liouville Theory, Past and Present, p. 271-331, Birkhäuser Verlag, Basel 2005 (edited by W.O. Amrein, A.M. Hinz and D.B. Pearson.)
	L. Kong, Q. Kong, H. Wu and A. Zettl, Regular approximations of singular Sturm-Liouville problems with limit-circle endpoints, Results in Mathematics, 45 (2004), 274-292.
	Q. Kong, H. Wu and A. Zettl, Multiplicity of Sturm-Liouville eigenvalues, J. Computational and Applied Math., 171 (2004), 291-309.
	Q. Kong, H. Wu and A. Zettl, Left-Definite Sturm-Liouville Problems, J. Differential Equations, 177 (2001), 1-26.
	Q. Kong, Q. Lin, H. Wu and A. Zettl, A new proof of the inequalities among Sturm-Liouville eigenvalues, PanAmerican Math. J., 10 (2000), 1-11.
	Q. Kong, H. Wu and A. Zettl, Geometric Aspects of Sturm-Liouville problems, I. Structures on spaces of boundary conditions, Proc. Roy. Soc. Edinburgh, 130A (2000), 561-589.
	B.M. Brown, D.K.R. McCormack and A. Zettl, On a computer assisted proof of eigenvalues below the essential spectrum of the Sturm-Liouville problem, J. Computational and Applied Math., 125 (2000), 385-393.
	Q. Kong, H. Wu and A. Zettl, Dependence of the n-th Sturm-Liouville eigenvalue on the problem, J. Differential Equations, 156 (1999), 328-354.
	B.M. Brown, D.K.R. McCormack and A. Zettl, On the existence of an eigenvalue below the essential spectrum, Proc. Roy. Soc. London, 455A (1999), 2229-2234.
	Q. Kong, H. Wu and A. Zettl, Inequalities among eigenvalues of singular Sturm-Liouville problems, Dynamical Systems and Applications, 8 (1999), 517-531.
	W.N. Everitt, M. Möller and A. Zettl, Sturm-Liouville problems and discontinuous eigenvalues, Proc. Roy. Soc. Edinburgh, 129A (1999), 707-716.
	A. Zettl, Sturm-Liouville Problems, spectral theory and computational methods of Sturm-Liouville problems, Marcel Dekker, 191 (1997), 1-104. Proceedings of the 1996 Knoxville Barrett Conference, edited by D. Hinton and P. Schaefer. (This is a survey/research article.)
	Q. Kong, H. Wu and A. Zettl, Dependence of eigenvalues on the problem, Mathematische Nachrichten, 188 (1997), 173-201.
	W.N. Everitt, C. Shubin, G. Stolz and A. Zettl, Sturm-Liouville problems with an infinite number of interior singularities, spectral theory and computational methods of Sturm-Liouville problems, Marcel Dekker, 191 (1997), 211-249. Proceedings of the 1996 Knoxville Barrett Conference, edited by D. Hinton and P. Schaefer.
	P.B. Bailey, W.N. Everitt and A. Zettl, Regular and singular Sturm-Liouville problems with coupled boundary conditions, Proc. Roy. Soc. Edinburgh, 126A (1996), 505-514.
	Q. Kong and A. Zettl, Dependence of eigenvalues of Sturm-Liouville problems on the boundary, J. Differential Equations, 126 no. 2 (1996), 389-407.
	Q. Kong and A. Zettl, Eigenvalues of regular Sturm-Liouville problems, J. Differential Equations, 131 no. 1 (1996), 1-19.
	M. Möller and A. Zettl, Differentiable dependence of eigenvalues of operators in Banach spaces, J. Operator Theory, 36 (1996), 335-355.