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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Journal ArticleDOI
A Unifying Local Convergence Result for Newton's Method in Riemannian Manifolds
TL;DR: A unified result is proved for the existence and local uniqueness of the solution, and for the local convergence of a Riemannian version of Newton's method, based on Kantorovich's majorant principle.
Journal ArticleDOI
Logical Approaches to Computational Barriers
Journal ArticleDOI
Fast Linear Homotopy to Find Approximate Zeros of Polynomial Systems
Carlos Beltrán,Luis Miguel Pardo +1 more
TL;DR: A new complexity bound, polynomial on the average, is proved for the problem of finding an approximate zero of systems of polynometric equations, and it is proved that this last task can be done in running time which is linear in the Bézout number of the system andPolynomial in the size of the input.
Journal ArticleDOI
Complexity estimates depending on condition and round-off error
Felipe Cucker,Steve Smale +1 more
TL;DR: 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.
Book ChapterDOI
A Survey on Continuous Time Computations
TL;DR: An overview of theories of continuous time computation is provided, which allow us to understand both the hardness of questions related to continuous time dynamical systems and the computational power of continuously time analog models.
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.