Complexity theory and numerical analysis

  title={Complexity theory and numerical analysis},
  author={Stephen Smale},
  journal={Acta Numerica},
  pages={523 - 551}
  • S. Smale
  • Published 1 January 1997
  • Mathematics
  • Acta Numerica
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. 
Real Computations with Fake Numbers
The notions playing major roles in this theory of computation over the real numbers are described, with special emphasis on those which do not appear in discrete complexity theory -- and some of its results are reviewed.
Solving Linear Problems with Finite Precision III: Sharp Expectation Bounds
We prove an O(log n) bound for the expectation of the logarithm of the condition number K for the computation of optimizers of linear programs.
Smoothed Analysis of Moore-Penrose Inversion
It is proved that, asymptotically, the expected value of this condition number depends only on the elongation of the matrix and not on the center and variance of the underlying probability distribution.
A numerical algorithm for zero counting. II: Distance to ill-posedness and smoothed analysis
A Condition Number Theorem is shown that this condition number of zero counting for real polynomial systems equals the inverse of the normalized distance to the set of ill-posed systems (i.e., those having multiple real zeros).
Complexity estimates depending on condition and round-off error
This paper 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.
Characteristic polynomials of typical matrices are ill-conditioned
We prove that the expectation of the logarithm of the condition number of each of the zeros of the characteristic polynomial of a complex standard Gaussian matrix is $\Omega(n)$. This gives a
Approximate computation of pseudovarieties
Many other combinatorial properties of the size of the Dixon matrix and the structure of a given polynomial system are related to the support hull of the polynometric system and their projections along different dimensions.
On the condition of characteristic polynomials
We prove that the expectation of the logarithm of the condition number of each of the zeros of the characteristic polynomial of a complex standard Gaussian matrix is ${\Omega}(n)$. This may provide
A numerical algorithm for zero counting. IV: An adaptive speedup
An adaptive version of the algorithm for counting real roots of real polynomial systems of Cucker, Krick, Malajovich and Wschebor runs in expected time, while preserving the good properties of the original: numerically stable, highly parallelizable and good probabilistic run-time.


Some Remarks on the Foundations of Numerical Analysis
The problem of increasing the understanding of algorithms by considering the foundations of numerical analysis and computer science is considered and the legitimacy and importance of models of machines that accept real numbers is considered.
Computational Complexity: On the Geometry of Polynomials and a Theory of Cost: II
Traditional algorithms, Newton's method and higher order generalization due to Euler are dealt with and some understanding of this phenomenon of finding a zero of a complex polynomial is given.
Complexity theory of numerical linear algebra
Complexity of Bezout's Theorem V: Polynomial Time
Some Remarks on Bezout’s Theorem and Complexity Theory
We begin by establishing the smoothness and irreducibility of certain algebraic varieties. Whereas these facts must be standard to algebraic geometers, they do not seem readily available.
Computational complexity: on the geometry of polynomials and a theory of cost. I
On montre que le nombre d'iterations requises pour qu'un algorithme rapide trouve un zero d'un polynome complexe de degre d, est lineaire en d, pourvu qu'un petit ensemble arbitraire de problemes
The issues are illustrated, the issues are posed, and the perspective toward resolution is posed.
Convergence and Complexity of Newton Iteration for Operator Equations
An optimal convergence condition for Newton iteration in a Banach space is established and which stronger condition must be imposed to also assure good complexity.
Complexity of Bezout's theorem IV: probability of success; extensions
We estimate the probability that a given number of projective Newton steps applied to a linear homotopy of a system of n homogeneous polynomial equations in $n + 1$ complex variables of fixed degrees
Condition number analysis for sparse polynomial systems
A new invariant, the sparse condition number, is introduced and it is shown that a sparse polynomial system analysis in terms of this invariant is easier to solve than a non sparse one.