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Complexity theory and numerical analysis

  • Steve Smale (a1)

Complexity theory of numerical analysis is the study of the number of arithmetic operations required to pass from the input to the output of a numerical problem.

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Blum L., Cucker F., Shub M. and Smale S. (1997), Complexity and Real Computation, Springer. To appear. Referred to as BCSS (1997).
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Dedieu J.-P. (1997 b), Condition number analysis for sparse polynomial systems. Preprint.
Dedieu J.-P. (1997 c), ‘Condition operators, condition numbers and condition number theorem for the generalized eigenvalue problem’, Linear Algebra Appl. To appear.
Dedieu J.-P. (1997 d), ‘Estimations for separation number of a polynomial system’, J. Symbolic Computation. To appear.
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Renegar J. (1987 a), ‘On the efficiency of Newton's method in approximating all zeros of systems of complex polynomials’, Math, of Oper. Research 12, 121148.
Renegar J. (1987 b), ‘On the worst case arithmetic complexity of approximating zeros of polynomials’, J. Complexity 3, 90113.
Renegar J. (1996), ‘Condition numbers, the Barrier method, and the conjugate gradient method’, SIAM J. Optim. To appear.
Renegar J., Shub M. and Smale S., eds (1997), Proceedings of the Summer Seminar on ‘Mathematics of Numerical Analysis: Real Number Algorithm’, AMS Lectures in Applied Mathematics, AMS, Providence, RI.
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Shub M. (1993), On the work of Steve Smale on the theory of computation, in From Topology to Computation: Proceedings of the Smalefest (Hirsch M., Marsden J. and Shub M., eds), Springer, pp. 443455.
Shub M. and Smale S. (1985), ‘Computational complexity: on the geometry of polynomials and a theory of cost I’, Ann. Sci. École Norm. Sup. 18, 107142.
Shub M. and Smale S. (1986), ‘Computational complexity: on the geometry of polynomials and a theory of cost II’, SIAM J. Comput. 15, 145161.
Shub M. and Smale S. (1993 a), ‘Complexity of Bézout's theorem I: geometric aspect’, J. Amer. Math. Soc. 6, 459501. Referred to as Bez I.
Shub M. and Smale S. (1993 b), Complexity of Bézout's theorem II: volumes and probabilities, in Computational Algebraic Geometry (Eyssette F. and Galligo A., eds), Vol. 109 of Progress in Mathematics, pp. 267285. Referred to as Bez II.
Shub M. and Smale S. (1993 c), ‘Complexity of Bézout's theorem III: condition number and packing’, J. Complexity 9, 414. Referred to as Bez III.
Shub M. and Smale S. (1994), ‘Complexity of Bézout's theorem V: polynomial time’, Theoret. Comput. Sci. 133, 141164. Referred to as Bez V.
Shub M. and Smale S. (1996), ‘Complexity of Bézout's theorem IV: probability of success; extensions’, SIAM J. Numer. Anal. 33, 128148. Referred to as Bez IV.
Smale S. (1976), ‘A convergent process of price adjustment and global Newton method’, J. Math. Economy 3, 107120.
Smale S. (1981), ‘The fundamental theorem of algebra and complexity theory’, Bull. Amer. Math. Soc. 4, 136.
Smale S. (1985), ‘On the efficiency of algorithms of analysis’, Bull. Amer. Math. Soc. 13, 87121.
Smale S. (1986), Newton's method estimates from data at one point, in The Merging of Disciplines: New Directions in Pure, Applied, and Computational Mathematics (Ewing R., Gross K. and Martin C., eds), Springer, pp. 185196.
Smale S. (1987 a), Algorithms for solving equations, in Proceedings of the International Congress of Mathematicians, AMS, Providence, RI, pp. 172195.
Smale S. (1987b), ‘On the topology of algorithms I’, J. Complexity 3, 8189.
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Acta Numerica
  • ISSN: 0962-4929
  • EISSN: 1474-0508
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