Bulletin of the American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Help
ISSN 1088-9485(e) ISSN 0273-0979(p)

On the efficiency of algorithms of analysis

Author(s): Steve Smale
Journal: Bull. Amer. Math. Soc. 13 (1985), 87-121.
MSC (1980): Primary 65-02
MathSciNet review: 799791
Retrieve article in: PDF

References:

I. Adler, R. M. Karp and R. Shamir, A simplex variant solving an m x d linear program in O(min (m, Report UCB CSD 83/158, Computer Science Division, University of California, Berkeley (December 1983).

I. Adler and W. Megiddo, A simplex algorithm whose average number of steps is bounded between two quadratic functions of the smaller dimension, Report, Dec. 1983.

A. Aho, J. Hopcraft and J. Ullman, The design and analysis of computer algorithms, Addison-Wesley, Reading, Mass., 1979.

K. Atkinson, An introduction to numerical analysis, Wiley, N. Y., 1978. MR 504339

B. Barna, Über die Divergenzpunkte des Newtonschen Verfahrens zur Bestimmung von Wurzeln Algebraischer Gleichungen. II, Publ. Math. Debrecen 4 (1956), 384-397. MR 78956

P. Blanchard, Complex analytic dynamics on the Riemann sphere, Bull. Amer. Math. Soc. (N.S.) 11 (1984), 85-141. MR 741725

L. Blum and M. Shub, Evaluating rational functions: Infinite precision is finite cost and tractable on average, SIAM J. Comput. (to appear). MR 837590

K. H. Borgwardt, The average number of pivot steps required by the simplex-method is polynomial, Z. Oper. Res. 26 (1982), 157-177. MR 686603

J. Curry, On the periodic behaviour of Newton's Method for a family of real cubic polynomials, preprint, 1983.

J. Curry, L. Garnett and D. Sullivan, On the iteration of a rational function: Computer experiments with Newton's method, Comm. Math. Phys. 91 (1983), 267-277. MR 723551

G. Dantzig, Linear programming and extensions, Princeton Univ. Press, Princeton, N.J., 1963. MR 201189

B. Dejon and P. Henrici, Constructive aspects of the fundamental theorem of algebra, Wiley, N. Y., 1969. MR 253555

J. Demmel, A numerical analyst's Jordan canonical form, Thesis, Univ. of California, Berkeley, 1983.

R. Dorfman, P. Samuelson and R. Solow, Linear programming and economic analysis, McGraw-Hill, N. Y., 1958. MR 128543

A. Douady, Systèmes dynamiques holomorphes, Séminaire Bourbaki, No. 599, 1982. MR 728980

C. Eaves and H. Scarf, The solution of systems of piecewise linear equations, Math. Oper. Res. 1 (1976), 1-27. MR 445792

D. Elworthy, Gaussian measures on Banach spaces and manifolds, Global Analysis and its Applications, Vol. II, International Atomic Energy Agency, Vienna, 1974, pp. 151-166. MR 464297

P. Fatou, Sur les equations fonctionnelles, Bull. Soc. Math. France 47, 48 (1919-1920), 161-211; 33-94; 208-314. MR 1504792

G. Forsyth and C. Moler, Computer solution of linear algebraic systems, Prentice-Hall, Englewood Cliffs, N. J., 1967. MR 219223

H. Goldstine, A history of numerical analysis, from the 16th through the 19th century, Springer-Verlag, Berlin and New York, 1977. MR 484905

J. Guckenheimer, Endomorphisms of the Riemann sphere, Global Analysis (Chern and Smale, eds.), Amer. Math. Soc., Providence, R. I., 1970, pp. 95-123. MR 274740

W. Hayman, Multivalent functions, Cambridge Univ. Press, Cambridge, 1958. MR 108586

P. Henrici, Applied and computational complex analysis, Wiley, N. Y., 1977. MR 453984

E. Hille, Analytic function theory. II, Ginn, Boston, 1962. MR 201608

G. Julia, Mémoire sur l'itération des fonctions rationnelles, J. Math. Pures Appl. 4 (1918), 47-245.

E. Kostlan, Statistical complexity of numerical linear algebra, Thesis, Univ. of Calif. Berkeley, 1985.

H. Kuhn, Z. Wang and Senlin Xu, On the cost of computing roots of polynomials, Math. Programming 28 (1984), 156-163. MR 733207

H. H. Kuo, Gaussian measures in Banach spaces, Lecture Notes in Math., vol. 463, Springer-Verlag, Berlin and New York, 1975. MR 461643

S. Lang, Algebra, Addison-Wesley, Reading, Mass., 1963. MR 197234

C. Martin and R. Hurley, Newton's algorithm and chaotic dynamical systems, SIAM J. Math. Anal. 16 (1984), 238-252.

M. Mehta, Random matrices and the statistical theory of energy levels, Academic Press, N. Y., 1967. MR 220494

A. Ocneanu, On the stability of large linear systems (to appear).

A. Ostrowski, Solutions of equations in Euclidean and Banach spaces, Academic Press, N. Y., 1973. MR 359306

H. O. Peitgen, D. Saupe and F. v. Haeseler, Cayley's problem and Julia sets, Math. Intelligencer 6 (1984), 11-20. MR 738904

J. Renegar I, On the complexity of a piecewise linear algorithm for approximating roots of complex polynomials, Math. Programming (to appear). MR 796428

J. Renegar II, On the cost of approximating all roots of a complex polynomial, Math. Programming (to appear). MR 796429

J. Renegar III, On the efficiency of Newton's method in approximating all zeros of a system of complex polynomials, preprint, Colorado State Univ., 1984.

D. Saari and J. Urenko, Newton's method, circle maps, and chaotic motion, Amer. Math. Monthly 91 (1984), 3-17. MR 729188

G. Saunders, Iteration of rational functions of one complex variable and basins of attractive fixed points, Thesis, Univ. of California, Berkeley, 1984.

R. Shamir, The efficiency of the simplex method: A survey, preprint, Univ. of California, Berkeley, 1984. MR 885847

M. Shub, The geometry and topology of dynamical systems and algorithms for numerical problems, notes prepared for lectures given at D.D.4 Peking University, Beijing, China, Aug.-Sept. 1983.

M. Shub and S. Smale I, Computational complexity: On the geometry of polynomials and a theory of cost, Part I, Ann. Sci. École Norm. Sup. (4) (to appear). MR 1213484

M. Shub and S. Smale II, Computational complexity: On the geometry of polynomials and a theory of cost: Part II, SIAM J. Computing (to appear). MR 822199

S. Smale I, Differentiable dynamical systems, The Mathematics of Time, Springer-Verlag, New York and Berlin, 1980. MR 607330

S. Smale II, A convergent process of price adjustment and global Newton methods, J. Math. Econom. 3 (1976), 107-120. MR 411577

S. Smale III, The fundamental theorem of algebra and complexity theory, Bull. Amer. Math. Soc. (N. S.) 4 (1981), 1-36. MR 590817

S. Smale IV, On the average number of steps in the simplex method of linear programming, Math. Programming 27 (1983), 241-262. MR 725621

S. Smale V, The problem of the average speed of the Simplex method, Mathematical Programming, The State of the Art (Bonn, 1982) (Bachem et al., eds.), Springer-Verlag, Berlin and New York, 1983. MR 717413

D. Sullivan, Quasiconformal homeomorphisms and dynamics III: Topological conjugacy classes of analytic endomorphisms, Ann. of Math, (to appear). MR 934326

M. J. Todd, Polynomial expected behavior of a pivoting algorithm for linear complementary and linear programming problems, Technical Report No. 595, School of Operations Research and Industrial Engineering, Cornell University, Ithaca, New York, 1983.

J. Traub and H. Wozniakowski, Information and computation, in Vol. 23, Advances in Computers, vol. 23 (M. C. Yovits, ed.), Academic Press, N. Y., 1984. MR 772573

A. Vershik and P. Sporyshev, An estimate of the average number of steps in the simplex method and problems in asymptotic integral geometry, Soviet Math. Dokl. 28 (1983). (Russian)

J. Von Neumann, Collected works, vol. V (A. Taub, ed.), MacMillan, N. Y., 1963.

J. Wilkinson I, The algebraic eigenvalue problem, Oxford Univ. Press, Oxford, 1965. MR 184422

J. Wilkinson II, Rounding errors in algebraic processes, Prentice-Hall, Englewood Cliffs, N. J., 1973. MR 161456

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|