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
The Hardness of Polynomial Equation Solving
TL;DR: In this article, the authors investigated the intrinsic sequential time complexity of universal elimination procedures for arbitrary continuous data structures encoding input and output objects of elimination theory and admitting the representation of certain limit objects.
Journal ArticleDOI
Complexity of Bezout’s Theorem VI: Geodesics in the Condition (Number) Metric
TL;DR: In this paper, a new complexity measure of a path of (problems, solutions) pairs in terms of the length of the path in the condition metric is introduced. But the complexity measure is not applicable to the case of geodesic paths.
Journal ArticleDOI
How to Compute Spectra with Error Control.
TL;DR: This work affirmatively resolve the question of whether reliable methods for computing the spectra of operators exist, and the algorithms provided are optimal, realizing the boundary of what digital computers can achieve.
Journal ArticleDOI
On the number of real roots of random polynomials
Hoi H. Nguyen,Oanh Nguyen,Van Vu +2 more
TL;DR: For polynomials with Gaussian coefficients, the error term O(1) was shown to be 2 π log n + o(log n) in this paper.
Journal ArticleDOI
High probability analysis of the condition number of sparse polynomial systems
TL;DR: An upper bound on the probability that the condition number of f in a region U is larger than 1/e is presented, thus recovering the classical mixed volume when U = (C*)n.
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.