10,109 research outputs found
Rayleigh-Ritz majorization error bounds of the mixed type
The absolute change in the Rayleigh quotient (RQ) for a Hermitian matrix with
respect to vectors is bounded in terms of the norms of the residual vectors and
the angle between vectors in [\doi{10.1137/120884468}]. We substitute
multidimensional subspaces for the vectors and derive new bounds of absolute
changes of eigenvalues of the matrix RQ in terms of singular values of residual
matrices and principal angles between subspaces, using majorization. We show
how our results relate to bounds for eigenvalues after discarding off-diagonal
blocks or additive perturbations.Comment: 20 pages, 1 figure. Accepted to SIAM Journal on Matrix Analysis and
Application
Angles between subspaces and their tangents
Principal angles between subspaces (PABS) (also called canonical angles)
serve as a classical tool in mathematics, statistics, and applications, e.g.,
data mining. Traditionally, PABS are introduced via their cosines. The cosines
and sines of PABS are commonly defined using the singular value decomposition.
We utilize the same idea for the tangents, i.e., explicitly construct matrices,
such that their singular values are equal to the tangents of PABS, using
several approaches: orthonormal and non-orthonormal bases for subspaces, as
well as projectors. Such a construction has applications, e.g., in analysis of
convergence of subspace iterations for eigenvalue problems.Comment: 15 pages, 1 figure, 2 tables. Accepted to Journal of Numerical
Mathematic
Explicit model predictive control accuracy analysis
Model Predictive Control (MPC) can efficiently control constrained systems in
real-time applications. MPC feedback law for a linear system with linear
inequality constraints can be explicitly computed off-line, which results in an
off-line partition of the state space into non-overlapped convex regions, with
affine control laws associated to each region of the partition. An actual
implementation of this explicit MPC in low cost micro-controllers requires the
data to be "quantized", i.e. represented with a small number of memory bits. An
aggressive quantization decreases the number of bits and the controller
manufacturing costs, and may increase the speed of the controller, but reduces
accuracy of the control input computation. We derive upper bounds for the
absolute error in the control depending on the number of quantization bits and
system parameters. The bounds can be used to determine how many quantization
bits are needed in order to guarantee a specific level of accuracy in the
control input.Comment: 6 pages, 7 figures. Accepted to IEEE CDC 201
Efficient Cluster Algorithm for Spin Glasses in Any Space Dimension
Spin systems with frustration and disorder are notoriously difficult to study
both analytically and numerically. While the simulation of ferromagnetic
statistical mechanical models benefits greatly from cluster algorithms, these
accelerated dynamics methods remain elusive for generic spin-glass-like
systems. Here we present a cluster algorithm for Ising spin glasses that works
in any space dimension and speeds up thermalization by at least one order of
magnitude at temperatures where thermalization is typically difficult. Our
isoenergetic cluster moves are based on the Houdayer cluster algorithm for
two-dimensional spin glasses and lead to a speedup over conventional
state-of-the-art methods that increases with the system size. We illustrate the
benefits of the isoenergetic cluster moves in two and three space dimensions,
as well as the nonplanar chimera topology found in the D-Wave Inc.~quantum
annealing machine.Comment: 5 pages, 4 figure
A Structural Small Open-Economy Model for Canada
The authors develop a small open-economy dynamic stochastic general-equilibrium (DSGE) model in an attempt to understand the dynamic relationships in Canadian macroeconomic data. The model differs from most recent DSGE models in two key ways. First, for prices and wages, the authors use the time-dependent staggered contracting model of Dotsey, King, and Wolman (1999) and Wolman (1999), rather than the Calvo (1983) specification. Second, to model investment, the authors adopt Edge's (2000a, b) framework of time-to-build with ex-post inflexibilities. The model's parameters are chosen to minimize the distance between the structural model's impulse responses to interest rate, demand (consumption), and exchange rate shocks and those from an estimated vector autoregression (VAR). The majority of the model's theoretical impulse responses fall within the 5 and 95 per cent confidence intervals generated by the VAR.Business fluctuations and cycles; Economic models; Inflation and prices
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