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
Citations
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Computation in Dynamically Bounded Asymmetric Systems
TL;DR: In this paper, the authors analyze the behavior of asymmetrical connected networks of linear threshold neurons, whose positive response is unbounded, and show that, for a wide range of parameters, this asymmetry brings interesting and computationally useful dynamical properties.
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Root refinement for real polynomials using quadratic interval refinement
Michael Kerber,Michael Sagraloff +1 more
TL;DR: A variant of QIR is formulated, denoted AQIR ("Approximate QIR"), that considers only approximations of the polynomial coefficients and chooses a suitable working precision adaptively, and achieves near-optimal bounds that essentially match the best known theoretical bounds on root approximation as obtained by very sophisticated algorithms.
Journal ArticleDOI
Approximate Solutions of Analytic Inequality Systems
TL;DR: It is proved that when a system of equalities and inequalities is satisfied approximately at a certain point, it is also satisfied exactly on a nearby point.
Journal ArticleDOI
Condition Number Bounds for Problems with Integer Coefficients
TL;DR: An explicit a priori bound for the condition number associated to each of the following problems is given: general linear equation solving, least squares, nonsymmetric eigenvalue problems, solving univariate polynomials, and solving systems of multivariate poynomials.
Proceedings ArticleDOI
Using SOS for Optimal Semialgebraic Representation of Sets: Finding Minimal Representations of Limit Cycles, Chaotic Attractors and Unions
Morgan Jones,Matthew M. Peet +1 more
TL;DR: In this article, the authors show that Sum-of-Squares optimization can be used to find optimal semialgebraic representations of sets and provide numerical examples for the Lorenz attractor and the Van der Pol limit cycle.
References
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Book
Principles of Algebraic Geometry
Phillip Griffiths,Joe Harris +1 more
TL;DR: In this paper, a comprehensive, self-contained treatment of complex manifold theory is presented, focusing on results applicable to projective varieties, and including discussion of the theory of Riemann surfaces and algebraic curves, algebraic surfaces and the quadric line complex.
Book
The algebraic eigenvalue problem
TL;DR: Theoretical background Perturbation theory Error analysis Solution of linear algebraic equations Hermitian matrices Reduction of a general matrix to condensed form Eigenvalues of matrices of condensed forms The LR and QR algorithms Iterative methods Bibliography.
Proceedings ArticleDOI
The complexity of theorem-proving procedures
TL;DR: It is shown that any recognition problem solved by a polynomial time-bounded nondeterministic Turing machine can be “reduced” to the problem of determining whether a given propositional formula is a tautology.