Felipe Cucker
Skip Abstract Section
Abstract
This paper has two agendas. One is to develop the foundations of round-off in computation. The other is to describe an algorithm for deciding feasibility for polynomial systems of equations and inequalities together with its complexity analysis and its round-off properties. Each role reinforces the other.
- ALLGOWER, E., AND GEORG, K. 1990. Numerical Continuation Methods. Springer-Verlag, New York.]] Google Scholar
- BALCAZAR, J., D{AZ, J., AND GABARRO, J. 1988. Structural complexity I. In EATCS Monographs on Theoretical Computer Science, vol. 11. Springer-Verlag, New York.]] Google Scholar
- BASU, S., POLLACK, R., AND ROY, M.-F. 1994. On the combinatorial and algebraic complexity of quantifier elimination. In Proceedings of the 35th Annual IEEE Symposium on Foundations of Computer Science. IEEE Computer Society Press, Los Alamitos, Calif., pp. 632-641.]]Google Scholar
- BLUM, L., CUCKER, F., SHUB, M., AND SHALE, S. 1998. Complexity and Real Computation. Springer-Verlag, New York.]] Google Scholar
- BLUM, L., SHUB, M., AND SHALE, S. 1989. On a theory of computation and complexity over the real numbers: NP-completeness, recursive functions and universal machines. Bull, AMS 21, 1-46.]]Google Scholar
- CAMPBELL, S., AND MEYER, C. 1979. Generalized Inverses of Linear Transformations. Pitman, London.]]Google Scholar
- COLLINS, G. 1975. Quantifier elimination for real closed fields by cylindrical algebraic decomposi- tion. In Lecture Notes in Computer Science, vol. 33. Springer-Verlag, New York, pp. 134-183.]] Google Scholar
- CUCKER, F., MONTAI#A, J., AND PARDO, L. 1995. Models for parallel computations with real numbers. In Number Theoretic and Algebraic Methods in Computer Science, I. Shpardinski, ed. World Scientific, River Edge, N.J., pp. 53-63.]]Google Scholar
- DEDmU, J.-P. 1996. Approximate solutions of numerical problems, condition number analysis and condition number theorem. In The Mathematics of Numerical Analysis, J. Renegar, M. Shub, and S. Smale, eds. Lectures in Applied Mathematics, vol. 32. American Mathematical Society, Providence, R.I., pp. 263-283.]]Google Scholar
- DEDmU, J.-P., AND SHUB, M. 1997. Multihomogeneous Newton methods. Preprint.]]Google Scholar
- GRAHAM, R., KNUTH, D., AND PATASHNIK, O. 1989. Concrete Mathematics. Addison-Wesley, Reading, Mass.]] Google Scholar
- GRIGORmV, D., AND VOROBJOV, N. 1988. Solving systems of polynomial inequalities in subexponential time. J. Symb. Comput. 5, 37-64.]] Google Scholar
- HEINTZ, J., ROY, M.-F., AND SOLERNO, P. 1990. Sur la complexit6 du principe de Tarski- Seidenberg. Bull. Soc. Math. France 118, 101-126.]]Google Scholar
- HIGHAM, N. 1996. Accuracy and Stability of Numerical Algorithms. SIAM, New York.]] Google Scholar
- MEGIDDO, N. 1993. A general NP-completeness theorem. In From Topology to Computation: Proceedings of the Smalefest, M. Hirsch, J. Marsden, and M. Shub, eds. Springer-Verlag, New York, pp. 432-442.]]Google Scholar
- PAPADIMITRIOU, C. 1994. Computational Complexity. Addison-Wesley, Reading, Mass.]]Google Scholar
- RENEGAR, J. 1992. On the computational complexity and geometry of the first-order theory of the reals. Part I. J. Symb. Comput. 13, 255-299.]] Google Scholar
- SNUB, M. 1994. Mysteries of mathematics and computation. Math. Int. 16, 10-15.]]Google Scholar
- SHUB, M., AND SHALE, S. 1993a. Complexity of Bezout's theorem I: Geometric aspects. J. AMS. 6, 459-501.]]Google Scholar
- SHUB, M., AND SHALE, S. 1993b. Complexity of Bezout's theorem II: Volumes and probabilities. In Computational Algebraic Geometry, F. Eyssette and A. Galligo, eds. Progress in Mathematics, vol. 109. Birkhfiuser, Boston, Mass., pp. 267-285.]]Google Scholar
- SHUB, M., AND SHALE, S. 1993c. Complexity of Bezout's theorem III: Condition number and packing. J. Complex. 9, 4-14.]] Google Scholar
- SHUB, M., AND SHALE, S. 1994. Complexity of Bezout's theorem V: Polynomial time. Theoret. Comput. Sci. 133, 141-164.]] Google Scholar
- SHUB, M., AND SHALE, S. 1996. Complexity of Bezout's theorem IV: Probability of success; extensions. SIAM J. Numer. Anal. 33, 128-148.]] Google Scholar
- SHALE, S. 1986. Newton's method estimates from data at one point. In The Merging of Disciplines: New Directions in Pure, Applied, and Computational Mathematics, R. Ewing, K. Gross, and C. Martin, eds. Springer-Verlag, New York.]]Google Scholar
- SHALE, S. 1997. Complexity theory and numerical analysis. In Acta Numerica. Cambridge University Press, Cambridge, England, pp. 523-551.]]Google Scholar
- TARSKI, A. 1951. A Decision Method for Elementary Algebra and Geometry. University of California Press.]]Google Scholar
- TREFETHEN, L., AND BAU, D. III 1997. Numerical Linear Algebra. SIAM, Philadelphia, Pa.]]Google Scholar
- TURING, A. 1948. Rounding-off errors in matrix processes. Quart. J. Mech. Appl. Math. 1, 287-308.]]Google Scholar
- VON NEUMANN, J., AND GOLDSTINE, H. 1947. Numerical inverting matrices of high order. Bull. AMS. 53, 1021-1099.]]Google Scholar
- VON NEUMANN, J., AND GOLDSTINE, U. 1951. Numerical inverting matrices of high order, II. Proc. AMS. 2, 188-202.]]Google Scholar
- WILKINSON, J. 1963. Rounding Errors in Algebraic Processes. Prentice-Hall, Englewood Cliffs, New Jersey.]] Google Scholar
- WILKINSON, J.1971. Modern error analysis. SIAM Rev. 13, 548-568.]]Google Scholar
- W/JTHRICH, H. 1976. Ein Entscheidungsverfahren ftir die Theorie der reell-abgeschlossenen K6rper. In Komplexitiit von Entscheidungsproblemen, E. Specker and V. Strassen, eds. Lecture Notes in Computer Science, vol. 43. Springer-Verlag, New York, pp. 138-162.]] Google Scholar
Index Terms
- Complexity estimates depending on condition and round-off error
Recommendations
A study of errors, error-proneness, and error diagnosis in Cobol
This paper provides data on Cobol error frequency for correction of errors in student-oriented compilers, improvement of teaching, and changes in programming language. Cobol was studied because of economic importance, widespread usage, possible error-...
A web-based multi-user error correction, analysis, and feedback system
DIWEB'07: Proceedings of the 7th WSEAS international conference on Distance learning and web engineeringTaking C programming language learning as an example, this paper describes the development of a Web-based multi-user error correction, analysis, and feedback system. This system reinvented traditional paper-based error feedback and error correction in ...
Error analysis of a derivative-free algorithm for computing zeros of holomorphic functions
We consider the quadrature method developed by Kravanja and Van Barel (Computing 63(1):69-91, 1999) for computing all the zeros of a holomorphic function that lie inside the unit circle. The algorithm uses only the function values and no (first or ...
Comments