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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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E. Allgower and K. Georg (1990), Numerical Continuous Methods, Springer.

E. Allgower and K. Georg (1993), Continuation and path following, in Acta Numerica, Vol. 2, Cambridge University Press, pp. 164.

O. Axelsson (1994), Iterative Solution Methods, Cambridge University Press.

S. Batterson (1994), ‘Convergence of the Francis shifted QR algorithm on normal matrices’, Linear Algebra Appl. 207, 181195.

S. Batterson and D. Day (1992), ‘Linear convergence in the shifted QR algorithm’, Math. Comp. 59, 141151.

S. Batterson and J. Smillie (1989), ‘The dynamics of Rayleigh quotient iteration’, SIAM J. Numer. Anal. 26, 624636.

S. Batterson and J. Smillie (1990), ‘Rayleigh quotient iteration for nonsymmetric matrices’, Math. Comp. 55, 169178.

D. Bini and V. Pan (1994), Polynomial and Matrix Computations, Birkhäuser, Basel.

L. Blum , F. Cucker , M. Shub and S. Smale (1996), ‘Complexity and real computation: a manifesto’, Int. J. Bifurcation and Chaos 6, 326. Referred to as the Manifesto.

L. Blum , M. Shub and S. Smale (1989), ‘On a theory of computation and complexity over the real numbers: NP-completeness, recursive functions and universal machines’, Bull. Amer. Math. Soc. 21, 146. Referred to as BSS (1989).

R. Brockett (1973), in Geometric Methods in Systems Theory, Proceedings of the NATO Advanced Study Institute ( D. Mayne and R. Brockett , eds), D. Reidel, Dordrecht.

W. Brownawell (1987), ‘Bounds for the degrees in the Nullstellensatz’, Annals of Math. 126, 577591.

J. J. M. Cuppen (1981), ‘A divide and conquer method for the symmetric tridiagonal eigenproblem’, Numer. Math. 36, 177195.

J.-P. Dedieu (1997 c), ‘Condition operators, condition numbers and condition number theorem for the generalized eigenvalue problem’, Linear Algebra Appl. To appear.

J.-P. Dedieu (1997 d), ‘Estimations for separation number of a polynomial system’, J. Symbolic Computation. To appear.

J. Demmel (1987), ‘On condition numbers and the distance to the nearest ill-posed problem’, Numer. Math. 51, 251289.

J. J. Dongarra and D. C. Sorensen (1987), ‘A fully parallel algorithm for the symmetric eigenvalue problem’, SIAM J. Sci. Statist. Comput. 8, 139154.

Q. Du , M. Jin , T. Y. Li and Z. Zeng (1997 a), ‘The quasi-Laguerre iteration’, Math. Comp. To appear.

C. Eckart and G. Young (1936), ‘The approximation of one matrix by another of lower rank’, Psychometrika 1, 211218.

A. Edelman (1988), ‘Eigenvalues and condition numbers of random matrices’, SIAM J. Matrix Anal. Appl. 9, 543556.

A. Edelman and E. Kostlan (1995), ‘How many zeros of a random polynomial are real?’, Bull. Amer. Math. Soc. 32, 138.

G. Golub and C. van Loan (1989), Matrix Computations, Johns Hopkins University Press.

M. R. Hestenes and E. Stiefel (1952), ‘Method of conjugate gradients for solving linear systems’, J. Res. Nat. Bur. Standards 49, 409436.

M. Hirsch and S. Smale (1979), ‘On algorithms for solving f(x) = 0’, Comm. Pure Appl. Math. 32, 281312.

W. Hoffman and B. N. Parlett (1978), ‘A new proof of global convergence for the tridiagonal QL algorithm’, SIAM J. Numer. Anal. 15, 929937.

R. Kellog , T. Li and J. Yorke (1976), ‘A constructive proof of Brouwer fixed-point theorem and computational results’, SIAM J. Numer. Anal. 13, 473483.

M. Kim (1988), ‘On approximate zeros and rootfinding algorithms for a complex polynomial’, Math. Comp. 51, 707719.

E. Kostlan (1988), ‘Complexity theory of numerical linear algebra’, J. Comput. Appl. Math. 22, 219230.

E. Kostlan (1991), ‘Statistical complexity of dominant eigenvector calculation’, J. Complexity 7, 371379.

E. Kostlan (1993), On the distribution of the roots of random polynomials, in From Topology to Computation: Proceedings of the Smalefest ( M. Hirsch , J. Marsden and M. Shub , eds), Springer, pp. 419431.

G. Malajovich (1994), ‘On generalized Newton algorithms: quadratic convergence, path-following and error analysis’, Theoret. Comput. Sci. 133, 6584.

J. M. McNamee (1993), ‘A bibliography on roots of polynomials’, J. Comput. Appl. Math. 47(3), 391394.

J. Milnor (1964), On the Betti numbers of real varieties, in Proceedings of the Amer. Math. Soc., Vol. 15, pp. 275280.

C. Neff (1994), ‘Specified precision root isolation is in NC, J. Comput. System Sci. 48, 429463.

C. Neff and J. Reif (1996), ‘An efficient algorithm for the complex roots problem’, J. Complexity 12, 81115.

A. Ostrowski (1958), ‘On the convergence of Rayleigh quotient iteration for the computation of the characteristic roots and vectors, I’, Arch. Rational Mech. Anal. 1, 233241.

V. Pan (1997), ‘Solving a polynomial equation: some history and recent progress’, SIAM Review. To appear.

E. A. Rakhmanov , E. B. Saff and Y. M. Zhou (1994), ‘Minimal discrete energy on the sphere’, Mathematical Research Letters 1, 647662.

J. Renegar (1987 a), ‘On the efficiency of Newton's method in approximating all zeros of systems of complex polynomials’, Math, of Oper. Research 12, 121148.

J. Renegar (1987 b), ‘On the worst case arithmetic complexity of approximating zeros of polynomials’, J. Complexity 3, 90113.

J. Renegar (1996), ‘Condition numbers, the Barrier method, and the conjugate gradient method’, SIAM J. Optim. To appear.

J. R. Rice (1966), ‘A theory of condition’, SIAM J. Numer. Anal. 3, 287310.

M. Shub (1993), On the work of Steve Smale on the theory of computation, in From Topology to Computation: Proceedings of the Smalefest ( M. Hirsch , J. Marsden and M. Shub , eds), Springer, pp. 443455.

M. Shub and S. Smale (1985), ‘Computational complexity: on the geometry of polynomials and a theory of cost I’, Ann. Sci. École Norm. Sup. 18, 107142.

M. Shub and S. Smale (1986), ‘Computational complexity: on the geometry of polynomials and a theory of cost II’, SIAM J. Comput. 15, 145161.

M. Shub and S. Smale (1993 a), ‘Complexity of Bézout's theorem I: geometric aspect’, J. Amer. Math. Soc. 6, 459501. Referred to as Bez I.

M. Shub and S. Smale (1993 b), Complexity of Bézout's theorem II: volumes and probabilities, in Computational Algebraic Geometry ( F. Eyssette and A. Galligo , eds), Vol. 109 of Progress in Mathematics, pp. 267285. Referred to as Bez II.

M. Shub and S. Smale (1993 c), ‘Complexity of Bézout's theorem III: condition number and packing’, J. Complexity 9, 414. Referred to as Bez III.

M. Shub and S. Smale (1994), ‘Complexity of Bézout's theorem V: polynomial time’, Theoret. Comput. Sci. 133, 141164. Referred to as Bez V.

M. Shub and S. Smale (1996), ‘Complexity of Bézout's theorem IV: probability of success; extensions’, SIAM J. Numer. Anal. 33, 128148. Referred to as Bez IV.

S. Smale (1976), ‘A convergent process of price adjustment and global Newton method’, J. Math. Economy 3, 107120.

S. Smale (1981), ‘The fundamental theorem of algebra and complexity theory’, Bull. Amer. Math. Soc. 4, 136.

S. Smale (1985), ‘On the efficiency of algorithms of analysis’, Bull. Amer. Math. Soc. 13, 87121.

S. Smale (1987b), ‘On the topology of algorithms I’, J. Complexity 3, 8189.

S. Smale (1990), ‘Some remarks on the foundations of numerical analysis’, SIAM Review 32, 211220.

J. Steele and A. Yao (1982), ‘Lower bounds for algebraic decision trees’, Journal of Algorithms 3, 18.

R. Thom (1965), Sur l'homologie des variétés algébriques réelles, in Differential and Combinatorial Topology ( S. Cairns , ed.), Princeton University Press.

X. Wang (1993), Some results relevant to Smale's reports, in From Topology to Computation: Proceedings of the Smalefest ( M. Hirsch , J. Marsden and M. Shub , eds), Springer, pp. 456465.

H. Wozniakowski (1977), ‘Numerical stability for solving non-linear equations’, Nu-mer. Math. 27, 373390.

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Acta Numerica
  • ISSN: 0962-4929
  • EISSN: 1474-0508
  • URL: /core/journals/acta-numerica
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