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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
Probabilistic analysis of a differential equation for linear programming
TL;DR: In this paper, the authors derived a simple expression for the distribution of the convergence rate in the asymptotic limit of large problem size and showed that it is a scaling function of a single variable.
Journal ArticleDOI
The Complexity of the Nucleolus in Compact Games
TL;DR: The complexity of the nucleolus is characterized by exhibiting an upper bound that holds on any class of compact games, and by showing that this bound is tight even on the (structurally simple) class of graph games.
Journal Article
On numerical stability in large scale linear algebraic computations : Plenary lecture presented at the 75th annual GAMM conference, Dresden/Germany, 22-26 march 2004
Z. Strakos,J. Liesen +1 more
TL;DR: In this article, the authors consider numerical stability of iterative methods in matrix computations and present some examples of rounding error analysis that are fundamental to justify numerically computed results, including the Lanczos method, the conjugate gradient (CG) method and the generalised minimal residual (GMRES) method.
Journal ArticleDOI
How Many Bits Does It Take to Track an Open Quantum System
Raisa Karasik,Howard M. Wiseman +1 more
TL;DR: It is shown that, for any ergodic master equation, one can expect to find an adaptive monitoring scheme on the bath that can confine the system state to jumping between only K states, for some K ≥ (D - 1)(2) + 1.
Posted Content
Solving SDP Completely with an Interior Point Oracle
TL;DR: In this paper, the existence of an oracle which solves any semidefinite programming (SDP) problem satisfying Slater's condition simultaneously at its primal and dual sides is considered. But the problem is not solved, since the oracle might not be able to directly solve general SDPs even after certain regularization schemes are applied.
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.