Abstract
The goal of testing is to determine whether an implementation linear operatorA conforms to a specification linear operatorS within a given error bound for all elements from an input setF. Suppose that an upper boundK on the norm of the difference ofS andA is given a priori. Then it is shown that in general any finite number of tests is inconclusive both in the worst case and on the average. However, the testing problem is still decidable in the limit for an arbitraryK; there is an algorithm of an infinite sequence of test-and-guess such that all but finitely many guesses are correct. On the other hand, if the error bound is relaxed for weak conformance then finite tests suffice even in the worst case and tight lower and upper bounds on the number of tests are derived. The test set is universal; it only depends on the set of valid inputsF. Furthermore, the test elements are on the boundary ofF. Two examples are used to illustrate the approaches and the paper is concluded with comments on two related problems: computation and verification.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
V. D. Agrawal and S. C. Seth,Test Generation for VLSI Chips, Computer Society Press, Washington, DC, 1988.
N. S. Bakhvalov,On the optimality of linear methods for operator approximation in convex classes of functions, USSR Comput. Math. Math. Phys., 11 (1971), pp. 244–249.
S. Ben-David,Can finite samples detect singularities of real-valued functions? Proc. Twenty-fourth Ann. ACM Symp. on the Theory of Computing, pp. 390–399, Victoria, 1992.
T. M. Cover,On determining the rationality of the mean of a random variable, The Annals of Statistics, 1 (1973), No. 5, pp. 862–871.
C. de Boor,On writing an automatic integration algorithm, in Mathematical Software, J. R. Rice, Ed., Academic Press, New York, pp. 201–209, 1971.
A. D. Friedman and P. R. Menon,Fault Detection in Digital Circuits, Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1971.
S. Gal and C. A. Micchelli,Optimal sequential and non-sequential procedures for evaluating a functional, Appl. Anal. 10 (1980), pp. 105–120.
W. M. Gentleman and S. B. Marovich,More on algorithms that reveal properties of floating point arithmetic units, Comm. ACM, 17, No. 5 (1974), pp. 276–277.
F. C. Hennie,Fault detecting experiments for sequential circuits, Proc. Fifth Ann. Symp. Switching Circuit Theory and Logical Design, pp. 95–110, Princeton, NJ, 1964.
R. Karpinski, Byte, pp. 223–235, February 1985.
B. W. Kernighan and P. J. Plauger,The Elements of Programming Style, McGraw-Hill Book Company, 2nd ed, 1978.
D. Lee and H. Woźniakowski,Testing nonlinear operators, Numerical Algorithms, 9 (1995), pp. 319–342.
D. Lee and M. Yannakakis,Testing finite state machines: state identification and verification, IEEE Trans. on Computers, 43 (1994), No. 3, pp. 306–320.
J. N. Lyness and J. J. Kaganove,Comments on the nature of automatic quadrature routines, ACM Trans. Math. Software, 2 (1976), No. 1, pp. 65–81.
M. A. Malcolm,Algorithms to reveal properties of floating-point arithmetic, Comm. ACM, 15 (1972), pp. 949–951.
E. F. Moore,Gedanken-experiments on sequential machines, Automata Studies, Annals of Mathematics Studies, No. 34 (1956), pp. 129–153, Princeton University Press, Princeton, NJ.
E. Novak and H. Woźniakowski,Relaxed verification for continuous problems, J. of Complexity, 89 (1992), pp. 124–152.
A. Papageorgiou and G. W. Wasilkowski,Average complexity of multivariate problems, J. of Complexity, 6 (1990), pp. 1–23.
B. N. Parlett,The Symmetric Eigenvalue Problem, Prentice-Hall, Englewood Cliffs, NJ, 1980.
J. R. Rice,A metaalgorithm for adaptive quadrature, J. ACM, 22 (1975), pp. 61–82.
N. L. Schryer,A test of a computer's floating-point arithmetic unit, AT&T Bell Laboratories Tech. Memo. No. 81-11274-1, 1981.
J. F. Traub, G. W. Wasilkowski, and H. Woźniakowski,Information-based Complexity, Academic Press, Inc., New York, 1988.
J. F. Traub and H. Woźniakowski,A General Theory of Optimal Algorithms, Academic Press, Inc., New York, 1980.
N. N. Vakhania,Probability Distributions on Linear Spaces, North-Holland, Amsterdam, 1981.
G. W. Wasilkowski,Information of varying cardinality, J. of Complexity, 2 (1986), pp. 204–228.
H. Woźniakowski,Complexity of verification and computation for IBC problems, J. of Complexity, 8 (1992), pp. 93–123, 1992.
M. Yannakakis and D. Lee,Testing finite state machines: fault detection, J. of Computer and System Sciences. 50 (1995), No. 2, pp. 209–227.
Author information
Authors and Affiliations
Additional information
This work was done while consulting at AT&T Bell Laboratories, and is partially supported by the National Science Foundation grant IRI-92-12597 and the Air Force Office of Scientific Research 91-0347.
Rights and permissions
About this article
Cite this article
Lee, D., Woźniakowski, H. Testing linear operators. Bit Numer Math 35, 331–351 (1995). https://doi.org/10.1007/BF01732608
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01732608