Enclosure method for the solution of linear ODEs with polynomial coefficients

Numerical Functional Analysis and Optimization

Volumn 20, Issue 7, 1999, Pages 779-803

  Neher, Markus ^a  

^a UNIVERSITY OF KARLSRUHE   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

COMPUTATIONAL GEOMETRY; ERROR ANALYSIS; ERROR CORRECTION; INITIAL VALUE PROBLEMS; LINEAR EQUATIONS; POLYNOMIALS; PROBLEM SOLVING;

ENCLOSURE METHODS; STAGGERED CORRECTION TECHNIQUES;

ORDINARY DIFFERENTIAL EQUATIONS;

EID: 0032594370     PISSN: 01630563     EISSN: None     Source Type: Journal    
DOI: 10.1080/01630569908816923     Document Type: Article

References (26)

1. G. Alefeld and J. Herzberger, "Introduction to Interval Computations", Academic Press, New York, 1983.
2. M. Abramowitz and I. A. Stegun, "Handbook of Mathematical Functions", Dover Publ. Inc., New York, 1972.
3. G. Birkhoff and G.-C. Rota, "Ordinary Differential Equations", Ginn and Company, Boston, 1962.
4. L. Collatz, "The Numerical Treatment of Differential Equations", Springer, Berlin, 1959.
5. G. F. Corliss, Survey of Interval Algorithms for Ordinary Differential Equations, Appl. Math. Comput. 31 (1989), 112-120.
6. G. F. Corliss, Guaranteed Error Bounds for Ordinary Differential Equations, in "Theory and Numerics of Ordinary and Partial Differential Equations" (M. Ainsworth, J. Levesley, W. A. Light, and M. Marletta, Eds.), pp. 1-75, Clarendon Press, Oxford, 1995.
7. G. F. Corliss and R. Rihm, Validating an A Priori Enclosure Using High-Order Taylor Series, in "Scientific Computing and Validated Numerics" (G. Alefeld, A. Frommer, and B. Lang, Eds.), pp. 228-238, Akademie-Verlag, Berlin, 1996.
8. R. Courant and D. Hilbert, "Methods of Mathematical Physics, Vol. 1", Interscience Publishers, New York, 1953.
9. R. Klatte, U. Kulisch, M. Neaga, D. Ratz, and C. Ullrich, "PASCAL-XSC - Language Reference with Examples", Springer, Berlin, 1992.
10. R. Klatte, U. Kulisch, C. Lawo, M. Rauch, and A. Wiethoff, "C-XSC, A C++ Class Library for Extended Scientific Computing", Springer, Berlin, 1993.
11. U. Kulisch and W. L. Miranker, "Computer Arithmetic in Theory and Practice", Academic Press, New York, 1981.
12. R. Lohner, Enclosing the Solutions of Ordinary Initial- and Boundary-Value Problems, in "Computer Arithmetic: Scientific Computation and Programming Languages" (E. Kaucher, U. Kulisch, and C. Ullrich, Eds.), pp. 255-286, Teubner, Stuttgart, 1987.
13. R. Lohner, "Einschließung der Lösung gewöhnlicher Anfangs- und Randwertaufgaben und Anwendungen", Dissertation, Universität Karlsruhe, 1988.
14. R. Lohner, Computation of Guaranteed Solutions of Ordinary Initial and Boundary Value Problems, in "Computational Ordinary Differential Equations" (J. R. Cash and I. Gladwell, Eds.), pp. 425-435, Clarendon Press, Oxford, 1992.
15. R. Lohner, Interval Arithmetic in Staggered Correction Format, in "Scientific Computing with Automatic Result Verification" (E. Adams and U. Kulisch, Eds.), pp. 301-321, Academic Press, San Diego, 1993.
16. R. E. Moore, The Automatic Analysis and Control of Error in Digital Computation Based on the Use of Interval Numbers, in "Error in Digital Computation, Vol. I" (L. B. Rall, Ed.), pp. 61-130, John Wiley and Sons, New York, 1965.
17. R. E. Moore, Automatic Local Coordinate Transformations to Reduce the Growth of Error Bounds in Interval Computation of Solutions of Ordinary Differential Equations, in "Error in Digital Computation, Vol. II" (L. B. Rall, Ed.), pp. 103-140, John Wiley and Sons, New York, 1965.
18. R. E. Moore, "Interval Analysis", Prentice Hall, Englewood Cliffs, N.J., 1966.
19. M. Neher, Inclusion of Eigenvalues and Eigenfunctions of the Sturm-Liouville Problem, in "Computer Arithmetic and Enclosure Methods" (L. Atanassova, J. Herzberger, Eds.), pp. 401-408, Elsevier, Amsterdam, 1992.
20. R. Rihm, Interval Methods for Initial Value Problems in ODES, in "Topics in Validated Computations" (J. Herzberger, Ed.), pp. 173-207, Elsevier, Amsterdam, 1994.
21. J. Schröder, Funktionalanalytische Herleitung von Fehlerabschätzungen und ihre praktische Durchführung auf Rechenanlagen, ZAMM 40 (1960), T27-T37.
22. J. Schröder, Fehlerabschätzungen mit Rechenanlagen bei gewöhnlichen Differentialgleichungen erster Ordnung, Numer. Math. 3 (1961), 39-61.
23. H. J. Stetter, Sequential Defect Correction for High-Accuracy Floating-Point Arithmetic, in "Numerical Analysis", pp. 186-202, Lecture Notes in Mathematics, Vol. 1066, Springer, Berlin, 1984.
24. H. J. Stetter, Validated Solution of Initial Value Problems for ODE, in "Computer Arithmetic and Self-Validating Numerical Methods" (C. Ullrich, Ed.), pp. 171-186, Academic Press, San Diego, 1990.
25. W. Walter, "Differential and Integral Inequalities", Springer, Heidelberg, 1970.
26. W. V. Walter, FORTRAN-XSC: A Portable Fortran 90 Module Library for Accurate and Reliable Scientific Computing, in "Validation Numerics - Theory and Applications" (R. Albrecht, G. Alefeld, and H. J. Stetter, Eds.), pp. 265-285, Computing Supplementum, Vol. 9, Springer, Wien, 1993.