1 Introduction.- 2 Definitions and First Properties of Computation.- 3 Computation over a Ring.- 4 Decision Problems and Complexity over a Ring.- 5 The Class NP and NP-Complete Problems.- 6 Integer Machines.- 7 Algebraic Settings for the Problem "P ? NP?".- 8 Newton's Method.- 9 Fundamental Theorem of Algebra: Complexity Aspects.- 10 Bezout's Theorem.- 11 Condition Numbers and the Loss of Precision of Linear Equations.- 12 The Condition Number for Nonlinear Problems.- 13 The Condition Number in ?(H(d).- 14 Complexity and the Condition Number.- 15 Linear Programming.- 16 Deterministic Lower Bounds.- 17 Probabilistic Machines.- 18 Parallel Computations.- 19 Some Separations of Complexity Classes.- 20 Weak Machines.- 21 Additive Machines.- 22 Nonuniform Complexity Classes.- 23 Descriptive Complexity.- References.

Complexity and Real Computation

In this paper I present a history of tractability of continuous problems, which has its beginning in the successful numerical tests for highdimensional integration of finance problems. Tractability results will be illustrated for two multivariate problems, integration and linear tensor products problems, in the worst case setting. My talk at FoCM'08 in Hong Kong and this paper are based on the book Tractability of Multivariate Problems , written jointly with Erich Novak. The first volume of our book has been recently published by the European Mathematical Society. Introduction Many people have recently become interested in studying the tractability of continuous problems. This area of research addresses the computational complexity of multivariate problems defined on spaces of functions of d variables, with d that can be in the hundreds or thousands; in fact, d can even be arbitrarily large. Such problems occur in numerous applications including physics, chemistry, finance, economics, and the computational sciences. As with all problems arising in information-based complexity, we want to solve multivariate problems to within ∈, using algorithms that use finitely many functions values or values of some linear functionals. Let n (∈, d ) be the minimal number of function values or linear functionals that is needed to compute the solution of the d -variate problem to within ∈. For many multivariate problems defined over standard spaces of functions n (∈, d ) is exponentially large in d .

Tractability of Multivariate Problems

https://web.maths.unsw.edu.au/~josefdick/preprints/Novak_Wozniakowski_book_1.pdf

Tractability of Multivariate Problems: Volume I: Linear Information

http://comet.lehman.cuny.edu/vpan/pdf/article.pdf

Univariate polynomials: nearly optimal algorithms for numerical factorization and root-finding

A Tutorial on Computable Analysis

An Efficient Algorithm for the Complex Roots Problem

An O(n^1+epsilon log b) Algorithm for the Complex Roots Problem

Given a univariate polynomial f(z of degree n with complex coefficients, whose real an 2 imaginary parts can be expressed as a ratio of two integers less than 2m in magnitude, the root problem is to find all the roots of f(z) up to specified precision 2-p. Assuming the arithmetic model for computation, we provide, for any E > 0, an algorithm which has complexity O(n'+'logb), where b = m + p. This improves on the previous best known algorithm for the problem which has Complexity O(n2 log b). We claim it then follows from the fact t hat we can bound the precision required in all the arithmetic computations, that the complexity of our algorithm in the Boolean model of computation is 0 (n2+'(n + b) log' blog log b) . The details of the proof of this claim will appear in a subsequent version of this paper. We introduce some techniques [splitting sets and binary search for 'centered'points) which hinge on a recent result of Coppersmith and Neff [CN 941, along with an unbalanced divide and conquer strategy.

http://ieeexplore.ieee.org/iel2/2955/8384/00365737.pdf

C. Andrew Neff

Papers

An Efficient Algorithm for the Complex Roots Problem

An O(n^1+epsilon log b) Algorithm for the Complex Roots Problem

An O(nls'log b) Algorithm for the Complex Roots Problem