# A survey of recent problems and results in analytic computational complexity

Invited Lectures

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## Abstract

We survey recent problems in analytic computational complexity. We deal primarily with the problem of solving nonlinear equations from complexity point of view. We summarize known results and we pose some open problems. The paper is concluded with new areas of further investigations in analytic computational complexity.

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## 6. Bibliography

- Brent [72]Brent, R.P., "The computational complexity of iterative methods for systems of nonlinear equations", Proc. Complexity Symposium, Yorktown Heights, N.Y., Plenum Press, New York.Google Scholar
- Brent [75]Brent, R.P., "A class of optimal-order zero-finding methods using derivative evaluations", in Analytic Computational Complexity, edited by J. F. Traub, Academic Press 1976, pp. 59–73.Google Scholar
- Brent, Winograd and Wolfe [73]Brent, R.P., Winograd, S., Wolfe, P., "Optimal iterative processes for root-finding," Numer.Math., 20, pp. 327–341.Google Scholar
- Jankowska [75]Jankowska, J., "Multivariate secant method", Ph.D.thesis, University of Warsaw, to appear in SIAM J. on Num.Anal. Google Scholar
- Kacewicz [75]Kacewicz, B., "Integrals with kernel in the solution of nonlinear equations in N dimensions", Computer Science Department Report, Carnegie-Mellon University, 1975.Google Scholar
- Kung and Traub [74]Kung, K.T., Traub, J.F., "Optimal order of one-point and multipoint iterations" J.Assoc.Comput.Mach. Vol. 21, No. 4, 1974, pp. 643–651.Google Scholar
- Meersman [76]Meersman, R., "Optimal use of information in certain iterative processes", in Analytic Computational Complexity, Academic Press 1976, pp. 109–125.Google Scholar
- Nowożyński and Wasilkowski [77]Nowożyński, K., Wasilkowski, G., "Global convergence of interpolatory iterations in the scalar case", to appearGoogle Scholar
- Traub and Woźniakowski [76a]Traub, J.F., Woźniakowski, H., "Strict lower and upper bounds on iterative computational complexity", in Analytic Computational Complexity, Academic Press 1976, pp. 15–34.Google Scholar
- Traub and Woźniakowski [76b]"Optimal radius of convergence of interpolatory iterations for operator equations", Dept. of Computer Science Report, Carnegie-Mellon University, 1976.Google Scholar
- Traub and Woźniakowski [76c]"Optimal linear information for the solution of nonlinear equations", Algorithms and Complexity, New Directions and Recent Results, edited by J.F. Traub, Academic Press 1976.Google Scholar
- Traub and Woźniakowski [77]"Convergence and complexity of interpolatory — Newton iteration in a Banach space", Computer Science Department Report, Carnegie-Mellon University, 1977.Google Scholar
- Woźniakowski [74]Woźniakowski, H., "Maximal stationary iterative methods for the solution of operator equations", SIAM J.Numer.Anal., 11, 5, pp. 934–949, 1974.CrossRefGoogle Scholar
- Woźniakowski [75]"Generalized information and maximal order of iteration for operator equations", SIAM J.Numer.Anal., 12,1,pp. 121–135, 1975.Google Scholar
- Woźniakowski [76]"Maximal order of multipoint iterations using n evaluations", in Analytic Computational Complexity, Academic Press 1976, pp. 75–107.Google Scholar
- Rice [76]Rice, J.R., "On the Computational Complexity of Approximation Operators II", in Analytic Computational Complexity, Academic Press, 1976, 191–205.Google Scholar
- Schultz [76]Schultz, M.M., "Complexity and Differential Equations", in Analytic Computational Complexity, Academic Press, 1976, 143–151.Google Scholar
- Winograd [76]Winograd, S., "Some Remarks on Proof Techniques in Analytic Complexity", in Analytic Computational Complexity, Academic Press, 1976, 5–15.Google Scholar
- Traub [73]Traub, J.F., "Complexity of Sequential and Parallel Numericall Algorithms", editor, Academic Press, 1973.Google Scholar
- Traub [76a]Traub, J.F., "Analytic Computational Complexity", editor, Academic Press, 1976.Google Scholar
- Traub [76b]Traub, J.F., "Algorithms and Complexity: Recent Results and New Directions", editor, Academic Press, 1976.Google Scholar

## Copyright information

© Springer-Verlag Berlin Heidelberg 1977