Checking approximate computations of polynomials and functional equations

SIAM Journal on Computing

Volumn 31, Issue 2, 2001, Pages 550-576

  Ergün, Funda ^a   Kumar, S Ravi ^b   Rubinfeld, Ronitt ^c  

^a CASE WESTERN RESERVE UNIVERSITY   (United States)

^b IBM ALMADEN RESEARCH CENTER   (United States)

^c NEC LABORATORIES AMERICA   (United States)

Author keywords

Approximate testing; Functional equations; Polynomials; Program testing; Property testing

Indexed keywords

ALGORITHMS; FUNCTIONS; THEOREM PROVING;

PROGRAM TESTING;

POLYNOMIALS;

EID: 0036248033     PISSN: 00975397     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0097539798337613     Document Type: Article

References (35)

1. J. ACZEL, Lectures on Functional Equations and Their Applications, Academic Press, New York, 1966.
2. M. ALBERT AND J.A. BAKER, Functions with bounded n-th differences, Ann. Polon. Math., 43 (1983), pp. 93-103.
3. S. AR, M. BLUM, B. CODENOTTI, AND P. GEMMELL, Checking approximate computations over the reals, in Proceedings of the 25th ACM Symposium on Theory of Computing, 1993, pp. 786-795.
4. S. ARORA, C. LUND, R. MOTWANI, M. SUDAN, AND M. SZEGEDY, Proof verification and hardness of approximation problems, J. ACM, 45 (1998), pp. 501-555.
5. S. ARORA AND S. SAFRA, Probabilistic checking of proofs: A new characterization of NP, J. ACM, 45 (1998), pp. 70-122.
6. L. BABAI, L. FORTNOW, AND C. LUND, Nondeterministic exponential time has two-prover interactive protocols, Comput. Complexity, 1 (1991), pp. 3-40.
7. M. BLUM AND S. KANNAN, Designing programs that check their work, J. ACM, 42 (1995), pp. 269-291.
8. M. BLUM, M. LUBY, AND R. RUBINFELD, Self-testing/correcting with applications to numerical problems, J. Comput. System Sci., 47 (1993), pp. 549-595.
9. M. BLUM AND H. WASSERMAN, Software reliability via run-time result-checking, J. ACM, 44 (1997), pp. 826-849.
10. M. BLUM AND H. WASSERMAN, Reflections on the Pentium division bug, IEEE Trans. Comput., 45 (1996), pp. 385-393.
11. E. CASTILLO AND M.R. RUIZ-COBO, Functional Equations and Modeling in Science and Engineering, Marcel Dekker, New York, 1992.
12. P.W. CHOLEWA, The stability problem for a generalized Cauchy type functional equation, Rev. Roumaine Math. Pures Appl., 29 (1984), pp. 457-460.
13. W.J. CODY, Performance evaluation of programs related to the real gamma function, ACM Trans. Math. Software, 17 (1991), pp. 46-54.
14. W.J. CODY AND I. STOLTZ, The use of Taylor series to test accuracy of function programs, ACM Trans. Math. Software, 17 (1991), pp. 55-63.
15. n - 1) and its applications, Ann. Polon. Math., 22 (1969), pp. 189-198.
16. G.L. FORTI, An existence and stability theorem for a class of functional equations, Stochastica, 4 (1980), pp. 23-30.
17. G.L. FORTI, Hyers-Ulam stability of functional equations in several variables, Aequationes Math., 50 (1995), pp. 143-190.
18. G.L. FORTI AND J. SCHWAIGER, Stability of homomorphisms and completeness, C.R. Math. Rep. Acad. Sci. Canada, 11 (1989), pp. 215-220.
19. Z. GAJDA, Local stability of the functional equation characterizing polynomial functions, Ann. Polon. Math., 52 (1990), pp. 119-137.
20. P. GEMMELL, R. LIPTON, R. RUBINFELD, M. SUDAN, AND A. WIGDERSON, Self-testing/correcting for polynomials and for approximate functions, in Proceedings of the 23rd ACM Symposium on Theory of Computing, 1991, pp. 32-42.
21. D.H. HYERS, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. USA, 27 (1941), pp. 222-224.
22. M. KIWI, F. MAGNIEZ, AND M. SANTHA, Approximate testing with relative error, in Proceedings of the 31st ACM Symposium on Theory of Computing, 1999, pp. 51-60.
23. S.R. KUMAR AND D. SIVAKUMAR, On self-testing without the generator bottleneck, in Proceedings of the 15th Annual Conference on Foundations of Software Technology and Theoretical Computer Science, Lecture Notes in Comput. Sci. 1026, Springer-Verlag, Berlin, 1995, pp. 248-262.
24. R. LIPTON, New directions in testing, in Proceedings of the DIMACS Workshop on Distributed Computing and Cryptography, 1991, pp. 191-202.
25. G.G. LORENTZ, Approximation of Functions, Holt, Rinehart and Winston, New York, 1966.
26. C. LUND, The Power of Interaction, Ph.D. Thesis, University of Chicago, Chicago, IL, 1991.
27. S. MAZUR AND W. ORLICZ, Grundlegende Eigenschaften der Polynomischen Operationen, Studia Math., 5 (1934), pp. 50-68.
28. R. RUBINFELD, On the robustness of functional equations, SIAM J. Comput., 28 (1999), pp. 1972-1997.
29. R. RUBINFELD AND M. SUDAN, Self-testing polynomial functions efficiently and over rational domains, in Proceedings of the Third Annual ACM-SIAM Symposium on Discrete Algorithms, Orlando, FL, 1992, pp. 23-43.
30. R. RUBINFELD AND M. SUDAN, Robust characterizations of polynomials and their applications to program testing, SIAM J. Comput., 25 (1996), pp. 252-271.
31. F. SKOF, Sull'approssimazione delle applicazioni localmente δ-additive, Atti della Accademia delle Scienze di Torino, 117 (1983), pp. 377-389.
32. M. SUDAN, private communication, University of California, Berkeley, 1991.
33. A.F. TIMAN, Theory of Approximation of Functions of a Real Variable, Pergamon Press, New York, 1963.
34. E.C. TITCHMARSH, The Theory of Functions, Oxford University Press, 1947.
35. F. VAINSTEIN, Algebraic Methods in Hardware/Software Testing, Ph.D. Thesis, Boston University, Boston, MA, 1993.