This paper proposes the matrix-weighted consensus algorithm, which is a generalization of the consensus algorithm. Given a networked dynamical system where the interconnections between agents are weighted by nonnegati...
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This paper proposes the matrix-weighted consensus algorithm, which is a generalization of the consensus algorithm. Given a networked dynamical system where the interconnections between agents are weighted by nonnegative definite matrices, it is shown that consensus and clustering phenomena naturally exist. We examine algebraic and algebraic graph conditions for achieving a consensus. (C) 2017 Elsevier Ltd. All rights reserved.
The paper revisits the classical problem of evaluating f (A) for a real function f and a matrix A with real spectrum. The evaluation is based on expanding f in Chebyshev polynomials, and the focus of the paper is to s...
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The paper revisits the classical problem of evaluating f (A) for a real function f and a matrix A with real spectrum. The evaluation is based on expanding f in Chebyshev polynomials, and the focus of the paper is to study the convergence rates of these expansions. In particular, we derive bounds on the convergence rates which reveal the relation between the smoothness of f and the diagonalizability of the matrix A. We present several numerical examples to illustrate our analysis. (C) 2018 Elsevier Inc. All rights reserved.
Two-dimensional time-fractional diffusion equations with given initial condition and homogeneous Dirichlet boundary conditions in a bounded domain are considered. A semidiscrete approximation scheme based on the pseud...
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Two-dimensional time-fractional diffusion equations with given initial condition and homogeneous Dirichlet boundary conditions in a bounded domain are considered. A semidiscrete approximation scheme based on the pseudospectral method to the time-fractional diffusion equation leads to a system of ordinary fractional differential equations. To preserve the high accuracy of the spectral approximation, an approach based on the evaluation of the Mittag-Leffler function on matrix arguments is used for the integration along the time variable. Some examples along with numerical experiments illustrate the effectiveness of the proposed approach. Copyright (c) 2016 John Wiley & Sons, Ltd.
For a simple graph G = (V, E) with eigenvalues of the adjacency matrix lambda(1) >= lambda(2) >= . . . >= lambda(n), the energy of the graph is defined by E (G) =Sigma(n)(j=1) vertical bar lambda(j)vertical b...
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For a simple graph G = (V, E) with eigenvalues of the adjacency matrix lambda(1) >= lambda(2) >= . . . >= lambda(n), the energy of the graph is defined by E (G) =Sigma(n)(j=1) vertical bar lambda(j)vertical bar Myriads of papers have been published in the mathematical and chemistry literature about properties of this graph invariant due to its connection with the energy of (bipartite) conjugated molecules. However, a structural interpretation of this concept in terms of the contributions of even and odd walks, and consequently on the contribution of subgraphs, is not yet known. Here, we find such an interpretation and prove that the (adjacency) energy of any graph (bipartite or not) is a weighted sum of the traces of even powers of the adjacency matrix. We then use such result to find bounds for the energy in terms of subgraphs contributing to it. The new bound are studied for some specific simple graphs, such as cycles and fullerenes. We observe that including contributions from subgraphs of sizes not bigger than 6 improves some of the best known bounds for energy, and more importantly gives insights about the contributions of specific subgraphs to the energy of these graphs. (C) 2017 Elsevier B.V. All rights reserved.
We consider the approximate computation of spectral projectors for symmetric banded matrices. While this problem has received considerable attention, especially in the context of linear scaling electronic structure me...
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We consider the approximate computation of spectral projectors for symmetric banded matrices. While this problem has received considerable attention, especially in the context of linear scaling electronic structure methods, the presence of small relative spectral gaps challenges existing methods based on approximate sparsity. In this work, we show how a data-sparse approximation based on hierarchical matrices can be used to overcome this problem. We prove a priori bounds on the approximation error and propose a fast algorithm based on the QR-based dynamically weighted Halley (QDWH) algorithm, along the lines of works by Nakatsukasa and colleagues. Numerical experiments demonstrate that the performance of our algorithm is robust with respect to the spectral gap. A preliminary MATLAB implementation becomes faster than eig already for matrix sizes of a few thousand.
We suggest a rational Krylov subspace approximation for products of matrix functions and a vector appearing in exponential integrators. We consider matrices with a field-of-values in a sector lying in the left complex...
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We suggest a rational Krylov subspace approximation for products of matrix functions and a vector appearing in exponential integrators. We consider matrices with a field-of-values in a sector lying in the left complex half-plane. The choice of die poles for our method is suggested by a fixed rational approximation based on contour integration along a hyperbola around the sector. Compared to the fixed approximation, our rational Krylov subspace method exhibits an accelerated and more stable convergence of order O (e(-Cn)). (C) 2016 Elsevier B.V. All rights reserved.
This paper provides a new numerical strategy for solving fractional-in-space reaction-diffusion equations on bounded domains under homogeneous Dirichlet boundary conditions. Using the matrix transfer technique the fra...
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This paper provides a new numerical strategy for solving fractional-in-space reaction-diffusion equations on bounded domains under homogeneous Dirichlet boundary conditions. Using the matrix transfer technique the fractional Laplacian operator is replaced by a matrix which, in general, is dense. The approach here presented is based on the approximation of this matrix by the product of two suitable banded matrices. This leads to a semilinear initial value problem in which the matrices involved are sparse. Numerical results are presented to verify the effectiveness of the proposed solution strategy.
The success probability in an ancilla-based circuit generally decreases exponentially in the number of qubits consisted in the ancilla. Although the probability can be amplified through the amplitude amplification pro...
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The success probability in an ancilla-based circuit generally decreases exponentially in the number of qubits consisted in the ancilla. Although the probability can be amplified through the amplitude amplification process, the input dependence of the amplitude amplification makes difficult to sequentially combine two or more ancilla-ased circuits. A new version of the amplitude amplification known as the oblivious amplitude amplification runs independently of the input to the system register. This allows us to sequentially combine two or more ancilla-based circuits. However, this type of the amplification only works when the considered system is unitary or non-nitary but somehow close to a unitary. In this paper, we present a general framework to simulate non-unitary processes on ancilla-based quantum circuits in which the success probability is maximized by using the oblivious amplitude amplification. In particular, we show how to extend a non-unitary matrix to an almost unitary matrix. We then employ the extended matrix by using an ancilla-based circuit design along with the oblivious amplitude amplification. Measuring the distance of the produced matrix to the closest unitary matrix, a lower bound for the fidelity of the final state obtained from the oblivious amplitude amplification process is presented. Numerical simulations for random matrices of different sizes show that independent of the system size, the final amplified probabilities are generally around 0.75 and the fidelity of the final state is mostly high and around 0.95. Furthermore, we discuss the complexity analysis and show that combining two such ancilla-based circuits, a matrix product can be imple-mented. This may lead us to efficiently implement matrix functions represented as infinite matrix products on quantum computers.
Algorithms and implementations for computing the sign function of a triangular matrix are fundamental building blocks for computing the sign of arbitrary square real or complex matrices. We present novel recursive and...
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Algorithms and implementations for computing the sign function of a triangular matrix are fundamental building blocks for computing the sign of arbitrary square real or complex matrices. We present novel recursive and cache-efficient algorithms that are based on Higham's stabilized specialization of Parlett's substitution algorithm for computing the sign of a triangular matrix. We show that the new recursive algorithms are asymptotically optimal in terms of the number of cache misses that they generate. One algorithm that we present performs more arithmetic than the nonrecursive version, but this allows it to benefit from calling highly optimized matrix multiplication routines;the other performs the same number of operations as the nonrecursive version, suing custom computational kernels instead. We present implementations of both, as well as a cache-efficient implementation of a block version of Parlett's algorithm. Our experiments demonstrate that the blocked and recursive versions are much faster than the previous algorithms and that the inertia strongly influences their relative performance, as predicted by our analysis.
A variety of block Krylov subspace methods have been successfully developed for linear systems and matrix equations. The application of block Krylov methods to compute matrix functions is, however, less established, d...
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A variety of block Krylov subspace methods have been successfully developed for linear systems and matrix equations. The application of block Krylov methods to compute matrix functions is, however, less established, despite the growing prevalence of matrix functions in scientific computing. Of particular importance is the evaluation of a matrix function on not just one but multiple vectors. The main contribution of this paper is a class of efficient block Krylov subspace methods tailored precisely to this task. With the full orthogonalization method (FOM) for linear systems forming the backbone of our theory, the resulting methods are referred to as B(FOM)(2): block FOM for functions of matrices. Many other important results are obtained in the process of developing these new methods. matrix-valued inner products are used to construct a general framework for block Krylov subspaces that encompasses already established results in the literature. Convergence bounds for B(FOM)(2) are proven for Stieltjes functions applied to a class of matrices which are self-adjoint and positive definite with respect to the matrix-valued inner product. A detailed algorithm for B(FOM)(2) with restarts is developed, whose efficiency is based on a recursive expression for the error, which is also used to update the solution. Numerical experiments demonstrate the power and versatility of this new class of methods for a variety of matrix-valued inner products, functions, and matrices.
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