article

Algorithm 810: The SLEIGN2 Sturm-Liouville Code

Editor: Ron Boisvert Authors:

P. B. Bailey,

W. N. Everitt,

A. ZettlAuthors Info & Claims

ACM Transactions on Mathematical Software (TOMS), Volume 27, Issue 2

Pages 143 - 192

https://doi.org/10.1145/383738.383739

Published: 01 June 2001 Publication History

Get Access

Abstract

The SLEIGN2 code is based on the ideas and methods of the original SLEIGN code of 1979. The main purpose of the SLEIGN2 code is to compute eigenvalues and eigenfunctions of regular and singular self-adjoint Sturm-Liouville problems, with both separated and coupled boundary conditions, and to approximate the continuous spectrum in the singular case. The code uses some new algorithms, which we describe, and has a driver program that offers a user-friendly interface. In this paper the algorithms and their implementations are discussed, and the class of problems to which each algorithm applied is identified.

Supplementary Material

GZ File (810.gz)

Software for "The SLEIGN2 Sturm-Liouville Code"

Download
160.96 KB

References

[1]

BAILEY, P. B. 1966. Sturm-Liouville eigenvalues via a phase function. SIAM J. Appl. Math. 14, 242-249.

Google Scholar

[2]

BAILEY, P. B. 1976. SLEIGN: An eigenvalue-eigenfunction code for Sturm-Liouville problems. Rep. SAND77-2044. Sandia National Laboratories, Livermore, CA.

Google Scholar

[3]

BAILEY, P. B. 1978. A slightly modified Prufer transformation useful for calculating Sturm-Liouville eigenvalues. J. Comput. Phys. 29, 2, 306-310.

Google Scholar

[4]

BAILEY, P. B. 1997. On the approximation of eigenvalues of Sturm-Liouville problems by those of suitably chosen regular ones. In Spectral Theory and Computational Methods of Sturm-Liouville Problems, Proceedings of the 1996 Barratt Lectures. Lecture Notes in Pure and Applied Mathematics (Knoxville, TN), D. Hinton and P. W. Schaefer, Eds. Marcel Dekker, Inc., New York, NY, 171-182.

Google Scholar

[5]

BAILEY,P.B.,EVERITT,W.N.,WEIDMANN, J., AND ZETTL, A. 1993. Regular approximation of singular Sturm-Liouville problems. Results Math. 23, 1, 3-22.

Google Scholar

[6]

BAILEY,P.B.,EVERITT,W.N.,AND ZETTL, A. 1991a. Computing eigenvalues of singular Sturm-Liouville problems. Results Math. 20, 391-423.

Google Scholar

[7]

BAILEY,P.B.,EVERITT,W.N.,AND ZETTL, A. 1996. Regular and singular Sturm-Liouville problems with coupled boundary conditions. Proc. Roy. Soc. Edinburgh 126A, 505-514.

Google Scholar

[8]

BAILEY,P.B.,EVERITT,W.N.,AND ZETTL, A. 1995. On the numerical computation of the spectrum of singular Sturm-Liouville problems. NSF Final Rep. for Grant DMS-9106470.

Google Scholar

[9]

BAILEY,P.B.,GARBOW,B.S.,KAPER,H.G.,AND ZETTL, A. 1991b. Eigenvalue and eigenfunction computations for Sturm-Liouville problems. ACM Trans. Math. Softw. 17,4 (Dec.), 491-499.

Crossref

Google Scholar

[10]

BAILEY,P.B.,GARBOW,B.S.,KAPER,H.G.,AND ZETTL, A. 1991c. ALGORITHM 700: A FORTRAN software package for Sturm-Liouville problems. ACM Trans. Math. Softw. 17,4 (Dec.), 500-501.

Crossref

Google Scholar

[11]

BAILEY,P.B.,GORDON,M.K.,AND SHAMPINE, L. F. 1976. Solving Sturm-Liouville eigenvalues problems. Rep. SAND76-0560. Sandia National Laboratories, Livermore, CA.

Google Scholar

[12]

BAILEY,P.B.,GORDON,M.K.,AND SHAMPINE, L. F. 1978. Automatic solution of the Sturm-Liouville problem. ACM Trans. Math. Softw. 4, 3 (Sept.), 193-208.

Crossref

Google Scholar

[13]

BIRKHOFF,G.AND ROTA, G.-C. 1969. Ordinary Differential Equations. Blaisdell Publishing Co., London, UK.

Google Scholar

[14]

CIARLET, P., SCHULTZ, M., AND VARGA, R. 1968. Numerical methods for high-order accuracy for non linear boundary value problems; III: Eigenvalue problems. Numer. Math. 12, 120-133.

Google Scholar

[15]

CODDINGTON,E.A.AND LEVINSON, N. 1995. Theory of Ordinary Differential Equations. McGraw-Hill, London, UK.

Google Scholar

[16]

EASTHAM,M.S.P.,KONG, Q., WU, H., AND ZETTL, A. 1999. Inequalities among eigenvalues of Sturm-Liouville problems. J. Ineq. Appl. 3, 1, 25-43.

Google Scholar

[17]

EVERITT, W. N. 1982. On the transformation theory of ordinary second-order symmetric differential equations. Czech. Math. J. 32 (107), 2, 275-306.

Google Scholar

[18]

EVERITT,W.N.,KWONG,M.K.,AND ZETTL, A. 1983. Oscillations of eigenfunctions of weighted regular Sturm-Liouville problems. J. London Math. Soc. 27, 2, 106-120.

Google Scholar

[19]

EVERITT,W.N.,MOLLER, M., AND ZETTL, A. 1997. Discontinuous dependence of the n-th Sturm-Liouville eigenvalue. In Proceedings of the International Conference on General Inequalities 7, C. Bandle, W. N. Everitt, L. Losonczi, and W. Walter, Eds. International Series of Numerical Mathematics. Birkhauser-Verlag, Basel, Switzerland, 147-150.

Google Scholar

[20]

EVERITT,W.N.,MOLLER, M., AND ZETTL, A. 1999. Sturm-Liouville problems and discontinous eigenvalues. Proc. Roy. Soc. Edinburgh 129A, 707-716.

Google Scholar

[21]

KELLER, H. B. 1968. Numerical Methods for Two-Point Boundary Value Problems. Ginn-Blaisdell, Waltham, MA.

Google Scholar

[22]

KELLER, H. B. 1976. Numerical Solution of Two Point Boundary Value Problems. SIAM, Philadelphia, PA.

Google Scholar

[23]

KONG, Q., WU, H., AND ZETTL, A. 1997. Dependence of eigenvalues on the problem. Math. Nachr. 188, 173-201.

Google Scholar

[24]

KONG, Q., WU, H., AND ZETTL, A. 1999. Dependence of the n-th Sturm-Liouville eigenvalue on the problem. J. Differ. Eq. 156, 238-354.

Google Scholar

[25]

LITTLEWOOD, J. E. 1966. On linear differential equation sof the second order with a strongly oscillating coefficient of y. J. London Math. Soc. 41, 627-638.

Google Scholar

[26]

MARLETTA, M. 1991. Certification of Algorithm 700: Numerical tests of the SLEIGN software for Sturm-Liouville problems. ACM Trans. Math. Softw. 17, 4 (Dec.), 481-490.

Crossref

Google Scholar

[27]

MCLEOD, J. B. 1968. Some examples of wildly oscillating potentials. J. London Math. Soc. 43, 647-654.

Google Scholar

[28]

NAIMARK, M. A. 1968. Linear Differential Operators: II. Frederick Ungar, New York, NY. Translated from the 2nd Russian edition.

Google Scholar

[29]

PRUESS,S.AND FULTON, C. T. 1993. Mathematical software for Sturm-Liouville problems. ACM Trans. Math. Softw. 19, 3 (Sept.), 360-376. Also available as NSF Final Report for Grants DMS88 and DMS88-00839.

Crossref

Google Scholar

[30]

PRYCE, J. D. 1986. The NAG Sturm-Liouville codes and some applications. NAG Newslett. 3, 4-26.

Google Scholar

[31]

PRYCE, J. D. 1993. Numerical Solution of Sturm-Liouville Problems. Monographs on Numerical Analysis. Oxford University Press, Oxford, UK.

Google Scholar

[32]

PRYCE, J. D. 1999. A test package for Sturm-Liouville solvers. ACM Trans. Math. Softw. 25, 1 (Mar.). This issue

Crossref

Google Scholar

[33]

TITCHMARSH, E. 1962. Eigenfunction Expansions Associated with Second Order Differential Equations, I. 2nd ed. Oxford University Press, Oxford, UK.

Google Scholar

[34]

WEIDMANN, J. 1987. Spectral Theory of Ordinary Differential Operators. Springer Lecture Notes in Mathematics, vol. 1258. Springer-Verlag, Vienna, Austria.

Crossref

Google Scholar

[35]

ZETTL, A. 1993. Computing continous spectrum. In Proceedings of the International Symposium on Ordinary Differential Equations and Applications: Trends and Developments in Ordinary Differential Equations. World Scientific Publishing Co, Inc., Singapore, 393-406.

Google Scholar

[36]

ZETTL, A. 1997. Sturm-Liouville problems. In Spectral Theory and Computational Methods of Sturm-Liouville Problems, Proceedings of the 1996 Barratt Lectures. Lecture Notes in Pure and Applied Mathematics (Knoxville, TN), D. Hinton and P. W. Schaefer, Eds. Marcel Dekker, Inc., New York, NY, 1-104.

Google Scholar

[37]

ZETTL, A. 2001. Sturm-Liouville theory. Department of Mathematical Sciences, Northern Illinois University, DeKalb, IL. Preprint.

Google Scholar

Cited By

View all

Suo J(2025)Dependence of eigenvalues of fourth-order Sturm-Liouville problems on canonical boundary conditionsJournal of Mathematical Analysis and Applications10.1016/j.jmaa.2024.128890543:2(128890)Online publication date: Mar-2025
https://doi.org/10.1016/j.jmaa.2024.128890
Karjanto NSadhani P(2024)Unraveling the Complexity of Inverting the Sturm–Liouville Boundary Value Problem to Its Canonical FormMathematics10.3390/math1209132912:9(1329)Online publication date: 26-Apr-2024
https://doi.org/10.3390/math12091329
Suo JShi ZWei Z(2024)DEPENDENCE OF EIGENVALUES ON THE REGULAR FOURTH-ORDER STURM-LIOUVILLE PROBLEMJournal of Applied Analysis & Computation10.11948/2023042514:5(2788-2807)Online publication date: 2024
https://doi.org/10.11948/20230425
Show More Cited By

Index Terms

Algorithm 810: The SLEIGN2 Sturm-Liouville Code
1. Mathematics of computing
  1. Mathematical analysis
    1. Differential equations
      1. Ordinary differential equations
  2. Mathematical software
2. Software and its engineering
  1. Software notations and tools
    1. General programming languages
      1. Language types

Recommendations

Algorithm 827: irbleigs: A MATLAB program for computing a few eigenpairs of a large sparse Hermitian matrix

irbleigs is a MATLAB program for computing a few eigenvalues and associated eigenvectors of a sparse Hermitian matrix of large order n. The matrix is accessed only through the evaluation of matrix-vector products. Working space of only a few n-vectors ...
A unitary Hessenberg QR-based algorithm via semiseparable matrices

In this paper, we present a novel method for solving the unitary Hessenberg eigenvalue problem. In the first phase, an algorithm is designed to transform the unitary matrix into a diagonal-plus-semiseparable form. Then we rely on our earlier adaptation ...
M-matrix asymptotics for Sturm-Liouville problems on graphs

We consider a system formulation for Sturm-Liouville operators with formally self-adjoint boundary conditions on a graph. An M-matrix associated with the boundary value problem is defined and related to the matrix Prufer angle associated with the system ...

Reviews

Reviewer: Charles Raymond Crawford

SLEIGN2 is the latest version of a package that has been developed over the last 35 years to solve a wide variety of Sturm-Liouville eigenvalue problems. The bulk of this paper covers the analytic background of the general problem. This includes the general equation, endpoint classification, notations for the definition of boundary conditions and their classification, initial values, self-adjoint problems, and finally, classification of the spectrum. Although these sections duplicate material from many ordinary differential equations (ODE) texts, they are a useful compact reference for terminology and notation. The final parts of the paper discuss computational methods. According to the authors, SLEIGN2 can deal with a wider variety of problems than can other easily available packages for Sturm-Liouville problems. These new problem classes are those with: (1) any type of coupled self-adjoint regular boundary conditions; (2) any type of separated self-adjoint singular boundary conditions; (3) any type of coupled self-adjoint singular boundary conditions; (4) initial values at regular and limit-circle endpoints. Two sections of the paper are devoted to examples of results from SLEIGN2 compared with those of two other packages. Online Computing Reviews Service

Access critical reviews of Computing literature here

Become a reviewer for Computing Reviews.

Comments

Information & Contributors

Information

Published In

cover image ACM Transactions on Mathematical Software

ACM Transactions on Mathematical Software Volume 27, Issue 2

June 2001

154 pages

ISSN:0098-3500

EISSN:1557-7295

DOI:10.1145/383738

Editor:
Ron Boisvert
National Institute of Standards and Technology, Gaithersburg, MD

Issue’s Table of Contents

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 01 June 2001

Published in TOMS Volume 27, Issue 2

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Badges

Author Tags

Qualifiers

Article

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

103
Total Citations
View Citations
1,081
Total Downloads

Downloads (Last 12 months)19
Downloads (Last 6 weeks)1

Reflects downloads up to 12 Nov 2024

Other Metrics

View Author Metrics

Citations

Cited By

View all

Suo J(2025)Dependence of eigenvalues of fourth-order Sturm-Liouville problems on canonical boundary conditionsJournal of Mathematical Analysis and Applications10.1016/j.jmaa.2024.128890543:2(128890)Online publication date: Mar-2025
https://doi.org/10.1016/j.jmaa.2024.128890
Karjanto NSadhani P(2024)Unraveling the Complexity of Inverting the Sturm–Liouville Boundary Value Problem to Its Canonical FormMathematics10.3390/math1209132912:9(1329)Online publication date: 26-Apr-2024
https://doi.org/10.3390/math12091329
Suo JShi ZWei Z(2024)DEPENDENCE OF EIGENVALUES ON THE REGULAR FOURTH-ORDER STURM-LIOUVILLE PROBLEMJournal of Applied Analysis & Computation10.11948/2023042514:5(2788-2807)Online publication date: 2024
https://doi.org/10.11948/20230425
Hardy YZinsou B(2024)CANONICAL FORMS FOR BOUNDARY CONDITIONS OF SELF-ADJOINT DIFFERENTIAL OPERATORSJournal of Applied Analysis & Computation10.11948/2022007314:4(1854-1868)Online publication date: 2024
https://doi.org/10.11948/20220073
Farzana HKumar Bhowmik SAlim M(2024)Bernstein collocation technique for a class of Sturm-Liouville problemsHeliyon10.1016/j.heliyon.2024.e2888810:7(e28888)Online publication date: Apr-2024
https://doi.org/10.1016/j.heliyon.2024.e28888
Zhang MLi KJu P(2024) Discontinuity of the th eigenvalue for a vibrating beam equation Applied Mathematics Letters10.1016/j.aml.2024.109146156(109146)Online publication date: Oct-2024
https://doi.org/10.1016/j.aml.2024.109146
Gesztesy FLittlejohn LPiorkowski MStanfill J(2024)The Jacobi Operator on $$(-1,1)$$ and Its Various m-FunctionsComplex Analysis and Operator Theory10.1007/s11785-024-01576-418:7Online publication date: 17-Sep-2024
https://doi.org/10.1007/s11785-024-01576-4
Zheng JLi KZheng Z(2024)Spectral properties for discontinuous Dirac system with eigenparameter‐dependent boundary conditionMathematical Methods in the Applied Sciences10.1002/mma.10364Online publication date: 30-Aug-2024
https://doi.org/10.1002/mma.10364
Zheng JLi KZheng Z(2023)EIGENVALUES OF STURM-LIOUVILLE PROBLEMS WITH EIGENPARAMETER DEPENDENT BOUNDARY AND INTERFACE CONDITIONSMathematical Modelling and Analysis10.3846/mma.2023.1709428:3(374-392)Online publication date: 4-Sep-2023
https://doi.org/10.3846/mma.2023.17094
Li KZhang MZheng Z(2023)Dependence of eigenvalues of Dirac system on the parametersStudies in Applied Mathematics10.1111/sapm.12567150:4(1201-1216)Online publication date: 20-Feb-2023
https://doi.org/10.1111/sapm.12567
Show More Cited By

View Options

Get Access

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Cited By

Index Terms

Recommendations

Algorithm 827: irbleigs: A MATLAB program for computing a few eigenpairs of a large sparse Hermitian matrix

A unitary Hessenberg QR-based algorithm via semiseparable matrices

M-matrix asymptotics for Sturm-Liouville problems on graphs

Reviews

Access critical reviews of Computing literature here