Computer-assisted proofs in analysis and programming in logic: A case study

SIAM Review

Volumn 38, Issue 4, 1996, Pages 565-604

  Koch, Hans ^a   Schenkel, Alain ^b   Wittwer, Peter ^b  

^a UNIVERSITY OF TEXAS AT AUSTIN   (United States)

^b UNIVERSITY OF GENEVA   (Switzerland)

Author keywords

Certified computing; Computer assisted proofs; Interval analysis; Programming in Prolog; Validated numerics

Indexed keywords

BOUNDARY CONDITIONS; COMPUTER AIDED ANALYSIS; DIGITAL ARITHMETIC; MATHEMATICAL MODELS; PROLOG (PROGRAMMING LANGUAGE); THEOREM PROVING;

CERTIFIED COMPUTING; FEIGENBAUM EQUATION; INTERVAL ANALYSIS; VALIDATED NUMERICS;

LOGIC PROGRAMMING;

EID: 0030396365     PISSN: 00361445     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036144595284180     Document Type: Article

References (45)

1. [A] O. ABERTH, Precise Numerical Analysis, W. C. Brown Publishers, Dubuque, IA, 1988.
2. A. CELLETTI, Construction of librational invariant tori in the spin-orbit problem, J. Appl. Math. Phys. (ZAMP), 45 (1993), p. 61.
3. [CC] A. CELLETTI AND L. CHIERCHIA, Construction of analytic KAM surfaces and effective stability bounds, Comm. Math. Phys., 118 (1988), pp. 119-161.
4. [CM] W. F. CLOCKSIN AND C. S. MELLISH, Programming in Prolog, 3rd revised and extended ed., Springer-Verlag, Berlin, 1987.
5. [E] H. EPSTEIN, Fixed points of composition operators, in Nonlinear Evolution and Chaotic Phenomena, G. Gallavotti and P. Zweifel, eds., NATO ASI Series B: Phys., 176, 1987, pp. 71-100.
6. [EKW1] J.-P. ECKMANN, H. KOCH, AND P. WITTWER, Existence of a fixed point of the doubling transformation for area-preserving maps of the plane, Phys. Rev. A, 26 (1982), pp. 720-722.
7. [EKW2] _, A computer-assisted proof of universality for area-preserving maps, Mem. Amer. Math. Soc., 47 (1984), pp. 1-121.
8. [EMO] J.-P. ECKMANN, A. MALASPINAS, AND S. OLIFFSON-KAMPHORST, A software tool for analysis in function spaces, in Computer Aided Proofs in Analysis, K. Meyer and D. Schmidt, eds., The IMA Volumes in Mathematics, 28, 1991, pp. 147-166.
9. [EW1] J.-P. ECKMANN AND P. WITTWER, Computer Methods and Borel Summability Applied to Feigenbaum's Equation, Lecture Notes in Physics, 227, Springer-Verlag, Berlin, Heidelberg, New York, 1985.
10. [EW2] _, A complete proof of the Feigenbaum conjectures, J. Statist. Phys., 46 (1987), pp. 455-477.
11. [FL] C. FEFFERMANN AND R. DE LA LLAVE, Relativistic stability of matter, I, Revista Matemática Iberoamericana, 2/1,2 (1986), pp. 119-213.
12. [FS] C. FEFFERMANN AND L. SECO, Aperiodicity of the Hamiltonian flow in the Thomas-Fermi potential, Revista Matemática Iberoamericana, 9/3 (1993), pp. 409-551.
13. [HHTZ] B. HASSARD, S. P. HASTINGS, W. C. TROY, AND J. ZHANO, A computer proof that the Lorentz equations have "chaotic" solutions, Appl. Math. Lett., 7 (1994), pp. 79-83.
14. [IEEE] ANSI/IEEE STANDARDS, The IEEE Standard for Binary Floating-Point Arithmetic, ANSI/IEEE Std. 754-1985. (Can be ordered from IEEE Standard Board, 345 East 47th Street, New York, NY 10017, USA.)
15. [KP] R. S. MACKAY AND I. C. PERCIVAL, Converse KAM: Theory and practice, Comm. Math. Phys., 98 (1985), pp. 469-512.
16. [KSW] H. KOCH, A. SCHENKEL, AND P. WITTWER, Computer-assisted proofs in analysis and programming in logic: A case study. A TEXfile containing the computer program is archived at mp-arc@math.utexas.edu, preprint 94-394.
17. [KW1] H. KOCH AND P. WITTWER, A non-Gaussian renormalization group fixed point for hierarchical scalar lattice field theories, Comm. Math. Phys., 106 (1986), pp. 495-532.
18. [KW2] _, Rigorous computer-assisted renormalization group analysis, in VIIIth International Congress on Mathematical Physics, M. Mebkhout and R. Sénéor, eds., World Scientific, 1986.
19. [KW3] _, Computing bounds on critical indices, in Non-Linear Evolution and Chaotic Phenomena, G. Gallavotti and P. Zweifel, eds., NATO ASI Series B: Phys., 176, 1987, pp. 269-277.
20. [KW4] _, The unstable manifold of a nontrivial RG fixed point, Canad. Math. Soc. Conference Proc., 9 (1988), pp. 99-105.
21. [KW5] _, On the renormalization group transformation for scalar hierarchical models, Comm. Math. Phys., 138 (1991), pp. 537-568.
22. [KW6] _, A nontrivial renormalization group fixed point for the Dyson-Baker hierarchical model, Comm. Math. Phys., 164 (1994), pp. 627-647.
23. [KW7] _, Bounds on the Zeros of a Renormalization Group Fixed Point, Preprint Austin/Geneva, 1995, mp_arc 95-202.
24. [La1] O. E. LANFORD, Remarks on the accumulation of period-doubling bifurcations, in Mathematical Problems in Theoretical Physics, Lecture Notes in Physics, 116, 1979, pp. 340-342.
25. [La2] _, Computer-assisted proof of the Feigenbaum conjectures, Bull. Amer. Math. Soc. (N.S.), 6 (1982), pp. 427-434.
26. [La3] _, Computer-assisted proofs in analysis, Physica, 124 A (1984), pp. 465-470.
27. [La4] _, A shorter proof of the existence of the Feigenbaum fixed point, Comm. Math. Phys., 96 (1984), pp. 521-538.
28. [Lla] R. DE LA LLAVE, Computer assisted proofs of stability of matter, in Computer Aided Proofs in Analysis, K. Meyer and D. Schmidt, eds., The IMA Volumes in Mathematics, 28, 1991, pp. 116-126.
29. [Llo] J. W. LLOYD, Foundations of Logic Programming, 2nd extended ed. Springer-Verlag, Berlin, 1987.
30. [LPA] LOGIC PROGRAMMING ASSOCIATES LTD., Studio 4, The Royal Victoria Patriotic Building, Trinity Road, London SW18 3SX, England.
31. [LR1] R. DE LA LLAVE AND D. RANA, Accurate bounds in K.A.M. theory, in VIIIth International Congress on Mathematical Physics, M. Mebkhout and R. Sénéor, eds., World Scientific, 1986.
32. [LR2] _, Accurate strategies far small denominator problems, Bull. Amer. Math. Soc., 22 (1990), pp. 85-90.
33. [LR3] _, Accurate strategies in K.A.M. problems and their implementation, in Computer Aided Proofs in Analysis, K. Meyer and D. Schmidt, eds., The IMA Volumes in Mathematics, 28, 1991, pp. 127-146.
34. [Me] B. D. MESTEL, A Computer Assisted Proof of Universality for Cubic Critical Maps of the Circle with Golden Mean Rotation Number, Ph.D. thesis, Math. Dept., University of Warwick, 1985.
35. K. MISCHAIKOW AND M. MROZEK, Chaos in the Lorentz equations: A computer-assisted proof, Bull. Amer. Math. Soc., 32 (1995), pp. 66-72.
36. [Mr] M. MROZEK, Toplogical Invariants, Multivalued Maps and Computer Assisted Proofs in Dynamics, Instytut Informatyki Uniwersytet Jagiellonski, Kraków, 1995, preprint.
37. [Ra] D. RANA, Proof of Accurate Upper and Lower Bounds for Stability Domains in Small Denominator Problems, Ph.D. thesis, Princeton University, Princeton, NJ, 1989.
38. [RNC] T. RAGE, A. NEUMAIER, AND C. SCHLIER, Rigorous verification of chaos in a molecular model, Phys. Rev. E, 50 (1994), pp. 2682-2688.
39. [Ro] P. Ross, Advanced Prolog: Techniques and Examples, Addison-Wesley, Reading, MA, 1989.
40. [Se1] L. SECO, Lower Bounds for the Ground State Energy of Atoms, Thesis, Princeton University, Princeton, NJ, 1989.
41. [Se2] _, Computer assisted lower bounds for atomic energies, in Computer Aided Proofs in Analysis, K. Meyer and D. Schmidt, eds., The IMA Volumes in Mathematics, 28, 1991, pp. 241-251.
42. [SP] SWEDISH INSTITUTE OF COMPUTER SCIENCE, P.O. Box 1263, S-164 28 Kista, Sweden.
43. [SS] L. STERLING AND E. SHAPIRO, The Art of Prolog: Advanced Programming Techniques, The Massachusetts Institute of Technology, 1988.
44. [St] A. STIRNEMANN, Existence of the Siegel disc renormalization fixed point, Nonlinearity, 7 (1994), pp. 959-974.
45. [XSC] E. ADAMS AND E. KULISCH, EDS, Scientific Computing with Automatic Result Verification, Applications in the Engineering Sciences, Academic Press, San Diego, CA, 1993.