1,872 research outputs found
Hitting Time of Quantum Walks with Perturbation
The hitting time is the required minimum time for a Markov chain-based walk
(classical or quantum) to reach a target state in the state space. We
investigate the effect of the perturbation on the hitting time of a quantum
walk. We obtain an upper bound for the perturbed quantum walk hitting time by
applying Szegedy's work and the perturbation bounds with Weyl's perturbation
theorem on classical matrix. Based on the definition of quantum hitting time
given in MNRS algorithm, we further compute the delayed perturbed hitting time
(DPHT) and delayed perturbed quantum hitting time (DPQHT). We show that the
upper bound for DPQHT is actually greater than the difference between the
square root of the upper bound for a perturbed random walk and the square root
of the lower bound for a random walk.Comment: 9 page
On the Convergence of Ritz Pairs and Refined Ritz Vectors for Quadratic Eigenvalue Problems
For a given subspace, the Rayleigh-Ritz method projects the large quadratic
eigenvalue problem (QEP) onto it and produces a small sized dense QEP. Similar
to the Rayleigh-Ritz method for the linear eigenvalue problem, the
Rayleigh-Ritz method defines the Ritz values and the Ritz vectors of the QEP
with respect to the projection subspace. We analyze the convergence of the
method when the angle between the subspace and the desired eigenvector
converges to zero. We prove that there is a Ritz value that converges to the
desired eigenvalue unconditionally but the Ritz vector converges conditionally
and may fail to converge. To remedy the drawback of possible non-convergence of
the Ritz vector, we propose a refined Ritz vector that is mathematically
different from the Ritz vector and is proved to converge unconditionally. We
construct examples to illustrate our theory.Comment: 20 page
Improved Accuracy and Parallelism for MRRR-based Eigensolvers -- A Mixed Precision Approach
The real symmetric tridiagonal eigenproblem is of outstanding importance in
numerical computations; it arises frequently as part of eigensolvers for
standard and generalized dense Hermitian eigenproblems that are based on a
reduction to tridiagonal form. For its solution, the algorithm of Multiple
Relatively Robust Representations (MRRR) is among the fastest methods. Although
fast, the solvers based on MRRR do not deliver the same accuracy as competing
methods like Divide & Conquer or the QR algorithm. In this paper, we
demonstrate that the use of mixed precisions leads to improved accuracy of
MRRR-based eigensolvers with limited or no performance penalty. As a result, we
obtain eigensolvers that are not only equally or more accurate than the best
available methods, but also -in most circumstances- faster and more scalable
than the competition
The eigenvalues and eigenvectors of finite, low rank perturbations of large random matrices
We consider the eigenvalues and eigenvectors of finite, low rank
perturbations of random matrices. Specifically, we prove almost sure
convergence of the extreme eigenvalues and appropriate projections of the
corresponding eigenvectors of the perturbed matrix for additive and
multiplicative perturbation models. The limiting non-random value is shown to
depend explicitly on the limiting eigenvalue distribution of the unperturbed
random matrix and the assumed perturbation model via integral transforms that
correspond to very well known objects in free probability theory that linearize
non-commutative free additive and multiplicative convolution. Furthermore, we
uncover a phase transition phenomenon whereby the large matrix limit of the
extreme eigenvalues of the perturbed matrix differs from that of the original
matrix if and only if the eigenvalues of the perturbing matrix are above a
certain critical threshold. Square root decay of the eigenvalue density at the
edge is sufficient to ensure that this threshold is finite. This critical
threshold is intimately related to the same aforementioned integral transforms
and our proof techniques bring this connection and the origin of the phase
transition into focus. Consequently, our results extend the class of `spiked'
random matrix models about which such predictions (called the BBP phase
transition) can be made well beyond the Wigner, Wishart and Jacobi random
ensembles found in the literature. We examine the impact of this eigenvalue
phase transition on the associated eigenvectors and observe an analogous phase
transition in the eigenvectors. Various extensions of our results to the
problem of non-extreme eigenvalues are discussed.Comment: 27 pages, 1 figure. The paragraph devoted to rectangular matrices has
been suppressed in this version (it will appear independently in a
forthcoming paper
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