Open AccessBook
Complexity and Real Computation
TLDR
This chapter discusses decision problems and Complexity over a Ring and the Fundamental Theorem of Algebra: Complexity Aspects.Abstract:
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.read more
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TL;DR: This book provides a solid fundament for studying various aspects of computability and complexity in analysis and is written in a style suitable for graduate-level and senior students in computer science and mathematics.
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Mathematical problems for the next century
TL;DR: Arnabels invitation is inspired in part by Hilbert's list of 1900 (see e.g. [Browder, 1976]) and I have used that list to help design this essay.
References
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Journal ArticleDOI
Sequential and parallel complexity of approximate evaluation of polynomial zeros
TL;DR: These new sequential and parallel algorithms establish new record upper bounds on both arithmetic and Boolean complexity of approximating to complex polynomial zeros and reach the sequential Boolean time bounds O(bn3 log (bn)log log(bn)) for approximation to all the zeros.
Journal ArticleDOI
Constructive Aspects of the Fundamental Theorem of Algebra
B. Dejon,Peter Henrici +1 more
Journal ArticleDOI
On the efficiency of Newton's method in approximating all zeros of a system of complex polynomials
TL;DR: It is shown that, probabilistically, each zero of f is successfully approximated within a determined number of steps.
Journal ArticleDOI
A survey on real structural complexity theory
Klaus Meer,Christian Michaux +1 more
TL;DR: This tutorial paper overviews research being done in the eld of structural complexity and recursion theory over the real numbers and other domains following the approach by Blum, Shub, and Smale.