Matrix Exponential Research Articles

Computing high dimensional multiple integrals involving matrix exponentials

This paper deals with the numerical computation of the high dimensional multiple integrals involving matrix exponentials that can be rewritten as the product of a matrix exponential times vector. To this end, in addition to the conventional iterative methods for computing the action of the matrix exponential on a vector, iterative methods for the action of the phi-function over a vector are also considered. This is illustrated with a Krylov–Padé approximation to the product of phi-function times vector, which reveals potential for computing a variety of high dimensional multiple integrals that arise in several areas of applied mathematics, model identification, control engineering and numerical methods.

A q-analogue of the bipartite distance matrix of a nonsingular tree

All graphs considered here are simple and finite. Let G be a labeled, connected, bipartite graph and let (L,R) be a vertex bipartition of G. The bipartite adjacency matrix A of G is the submatrix of its adjacency matrix determined by the rows corresponding to L and the columns corresponding to R, see [6]. In a similar way, one can consider the bipartite distance matrix of G. It is the submatrix of the usual distance matrix D of G determined by the rows corresponding to L and the columns corresponding to R. Let G=T be a tree with a unique perfect matching. Very recently, in [2], it was shown that this matrix possesses many interesting properties and has a very deep combinatorial relation with the tree structure. One of those results, is the elegant description of the determinant of the bipartite distance matrix of T by a combinatorial sum over the collection of all alternating paths. Another result is reminiscent of the well known result of Graham and Pollack, which tells that the determinant of the usual distance matrix a tree of order 2p is a multiple of 22p−2. It was shown that the determinant of the bipartite distance matrix of a tree of order 2p with a unique perfect matching is a multiple of 2p−1.There are many studies of distance-like matrices available in the literature. In this article, we consider two of them. The first one is the q-distance matrix which is a generalization of the usual distance matrix. We show that the determinant of the q-bipartite distance matrix of a tree with a unique perfect matching can still be expressed as a combinatorial sum over the set of alternating paths with respect to a suitable sequence. The result of [2] follows as a corollary when q=1.The second distance-like matrix we study is the exponential distance matrix, which has a very simple expression for the determinant. We show that an extremely similar result hold here too. This is unexpected, as our matrix is of half the order.Recall that, the determinant of the usual distance matrix of a tree is just dependent on the number of vertices and is independent of the tree structure. In [2], it was shown that the determinant of the bipartite distance matrix of a tree with a unique perfect matching was dependent on the tree structure and on some smaller classes it was independent of the tree structure. We prove similar results for the q-bipartite distance matrix.However, as another surprise, we show that the determinant of the exponential bipartite distance matrix of a tree with a unique perfect matching is independent of the tree structure.

Experimental Realization of a Quantum Refrigerator Driven by Indefinite Causal Orders.

Indefinite causal order (ICO) is playing a key role in recent quantum technologies. Here, we experimentally study quantum thermodynamics driven by ICO on nuclear spins using the nuclear magnetic resonance system. We realize the ICO of two thermalizing channels to exhibit how the mechanism works, and show that the working substance can be cooled or heated albeit it undergoes thermal contacts with reservoirs of the same temperature. Moreover, we construct a single cycle of the ICO refrigerator based on the Maxwell's demon mechanism, and evaluate its performance by measuring the work consumption and the heat energy extracted from the low-temperature reservoir. Unlike classical refrigerators in which the coefficient of performance (COP) is perversely higher the closer the temperature of the high-temperature and low-temperature reservoirs are to each other, the ICO refrigerator's COP is always bounded to small values due to the nonunit success probability in projecting the ancillary qubit to the preferable subspace. To enhance the COP, we propose and experimentally demonstrate a general framework based on the density matrix exponentiation (DME) approach, as an extension to the ICO refrigeration. The COP is observed to be enhanced by more than 3 times with the DME approach. Our Letter demonstrates a new way for nonclassical heat exchange, and paves the way towards construction of quantum refrigerators on a quantum system.

Equations and Character Sums with Matrix Powers, Kloosterman Sums over Small Subgroups, and Quantum Ergodicity

Abstract We obtain a nontrivial bound on the number of solutions to the equation $$ \begin{align*} &\sum_{i=1}^{\nu} A^{x_i} = \sum_{i=\nu+1}^{2\nu} A^{x_i}, \qquad 1 \leqslant x_i \leqslant \tau, \end{align*}$$with a fixed $n\times n$ matrix $A$ over a finite field ${{\mathbb {F}}}_q$ of $q$ elements of multiplicative order $\tau $. We apply our result to obtain a new bound for additive character sums with a matrix exponential function, nontrivial beyond the square-root threshold. For $n=2$, this equation has been considered by Kurlberg and Rudnick (for $\nu =2$) and Bourgain (for large $\nu $) in their study of quantum ergodicity for linear maps over residue rings. We use a new approach to improve their results and also obtain a bound on Kloosterman sums over small subgroups, of size below the square-root threshold.

Quantum principal component analysis based on the dynamic selection of eigenstates

Quantum principal component analysis is a dimensionality reduction method to select the significant features of a dataset. A classical method finds the solution in polynomial time, but when the dimension of feature space scales exponentially, it is inefficient to compute the matrix exponentiation of the covariance matrix. The quantum method uses density matrix exponentiation to find principal components with exponential speedup. We enhance the existing algorithm that applies amplitude amplification using range-based static selection of eigenstates on the output of phase estimation. So, we propose an equivalent quantum method with the same complexity using a dynamic selection of eigenstates. Our algorithm can efficiently find phases of equally likely eigenvalues based on the similarity scores. It obtains principal components associated with highly probable larger eigenvalues. We analyze these methods on various factors to justify the resulting complexity of a proposed method as effective in quantum counterparts.

Lexicons of Key Terms in Scholarly Texts and Their Disciplinary Differences: From Quantum Semantics Construction to Relative-Entropy-Based Comparisons.

Complex networks are often used to analyze written text and reports by rendering texts in the form of a semantic network, forming a lexicon of words or key terms. Many existing methods to construct lexicons are based on counting word co-occurrences, having the advantage of simplicity and ease of applicability. Here, we use a quantum semantics approach to generalize such methods, allowing us to model the entanglement of terms and words. We show how quantum semantics can be applied to reveal disciplinary differences in the use of key terms by analyzing 12 scholarly texts that represent the different positions of various disciplinary schools (of conceptual change research) on the same topic (conceptual change). In addition, attention is paid to how closely the lexicons corresponding to different positions can be brought into agreement by suitable tuning of the entanglement factors. In comparing the lexicons, we invoke complex network-based analysis based on exponential matrix transformation and use information theoretic relative entropy (Jensen–Shannon divergence) as the operationalization of differences between lexicons. The results suggest that quantum semantics is a viable way to model the disciplinary differences of lexicons and how they can be tuned for a better agreement.

Acoustic waves in functionally graded rods: polynomial inhomogeneity

The harmonic acoustic waves in a semi-infinite functionally graded (FG) 1D rod with a longitudinal arbitrary inhomogeneity are analyzed by a combined method based on the modified Cauchy formalism and the exponential matrix method. The closed form dispersion equations for harmonic waves are constructed, yielding the implicit dispersion relations for acoustic waves in FG rods. For longitudinal inhomogeneity of polynomial type the corresponding dispersion relations are constructed explicitly.

Viscoelastic and viscoacoustic modelling using the Lie product formula

ABSTRACTWe present efficient and accurate modelling of seismic wavefields in anelastic media. We use a first‐order viscoelastic wave equation based on the generalized Robertsson's formulation. In our work, viscoacoustic and viscoelastic wave equations use the standard linear solid mechanism. To numerically solve the first‐order wave equation, we employed a scheme derived from the Lie product formula, where the time evolution operator of the analytic solution is written as a product of exponential matrices, and each exponential matrix term is approximated by the Taylor series expansion. The accuracy of the proposed scheme is evaluated by comparison with the analytical solution for a homogeneous medium. We also present simulations of some geological models with different structural complexities, whose results confirm the accuracy of the proposed scheme and illustrate the attenuation effect on the seismic energy during its propagation in the medium. Our results demonstrate that the numerical scheme employed can be used to extrapolate wavefields stably for even larger time steps than those usually used by traditional finite‐difference schemes.

On linear algebra of one type of symmetric matrices with harmonic Fibonacci entries

This paper focuses on a specially constructed matrix whose entries are harmonic Fibonacci numbers and considers its Hadamard exponential matrix. A lot of admiring algebraic properties are presented for both of them. Some of them are determinant, inverse in usual and in the Hadamard sense, permanents, some norms, etc. Additionally, a MATLAB-R2016a code is given to facilitate the calculations and to further enrich the content.

The Legendre Polynomial Axial Expansion Method

This work presents a new formulation of the axial expansion transport method explicitly using Legendre polynomials for arbitrarily high-order expansions. This new formulation also features an alternative method of axial leakage calculation to allow for nonextruded flat source region meshes. This alternative axial leakage is introduced alongside a balance equation requirement to ensure that neutron balance is preserved in the coarse mesh for a given axial leakage formulation, which allows for effective coarse mesh finite difference acceleration. A matrix exponential table method is derived to allow for fast computations of arbitrarily high-order matrix exponentials for this work and precludes the need for further research into matrix exponential calculations for this method. Numerical results are presented that demonstrate the stability of the axial expansion method in systems with voidlike regions, showcase the speedup from matrix exponential tables, and investigate the axial convergence of the method in terms of both expansion order and mesh size.

Benchmark Buckling Solutions of Truncated Conical Shells by Multiplicative Perturbation With Precise Matrix Exponential Computation

Abstract The multiplicative perturbation method with precise matrix exponential computation is developed for the buckling analysis of axially compressed truncated conical shells (TCSs) that are commonly encountered in engineering. To overcome the limitation of conventional methods in terms of assuming solution forms, the multiplicative perturbation method is introduced to tackle the governing partial differential equations (PDEs) with variable coefficients. Specifically, the governing equation in matrix form for a buckled TCS is first formulated in the state space. The multiplicative perturbation method is then employed to convert the matrix differential equation with variable coefficients into the state transition equations with constant coefficients, in which the arisen matrix exponential is computed by the precise integral method. Finally, the state transition equations and the boundary conditions are integrated into an entire matrix equation, whose solution provides the buckling loads and buckling modes of the TCS. The convergence study and comprehensive numerical and graphic results are presented. Given the new solutions, the effects of some crucial size parameters as well as boundary conditions on the critical buckling loads are quantitatively studied. Due to the merits on solving PDEs with variable coefficients, the developed method may be extended to more intractable plate and shell problems.

3D anisotropic TEM modeling with loop source using model reduction method

Abstract For model reduction techniques, there have been relatively few studies performed regarding the forward modeling of anisotropic media in comparison to transient electromagnetic (TEM) forward modeling of isotropic media. The transient electromagnetic method (TEM) responses after the current has been turned off can be represented as a homogeneous ordinary differential equation (ODE) with an initial value, and the ODE can be solved using a matrix exponential function. However, the order of the matrix exponential function is large and solving it directly is challenging, thus this study employs the Shift-and-Invert (SAI-Krylov) subspace algorithm. The SAI-Krylov subspace technique is classified as a single-pole approach compared to the multi-pole rational Krylov subspace approach. It only takes one LU factorization of the coefficient matrix, along with hundreds of backward substitutions. The research in this paper shows that the anisotropic medium has little effect on the optimal shift ${\gamma _{opt}}$ and subspace order m. Furthermore, as compared to the mimetic finite volume method (SAI-MFV) of the SAI-Krylov subspace technique, the method proposed in this paper (SAI-FEM) can further improve the computing efficiency by roughly 13%. In contrast to the standard implicit time step iterative technique, the SAI-FEM method does not require discretization in time, and the TEM response at any moment within the off-time period can be easily computed. Next, the accuracy of the SAI-FEM algorithm was verified by 1D solutions for an anisotropic layer model and a 3D anisotropic model. Finally, the electromagnetic characteristics of the anisotropic anomalous body of the center loop device and separated device of the airborne transient electromagnetic method were analyzed, and it was found that horizontal conductivity has a considerable influence on the TEM response of the anisotropic medium.

Computing the Riemannian Logarithm on the Stiefel Manifold: Metrics, Methods, and Performance

We address the problem of computing Riemannian normal coordinates on the real, compact Stiefel manifold of orthonormal frames. The Riemannian normal coordinates are based on the so-called Riemannian exponential and the associated Riemannian logarithm map and enable one to transfer almost any computational procedure to the realm of the Stiefel manifold. To compute the Riemannian logarithm is to solve the (local) geodesic endpoint problem. Instead of restricting the consideration to geodesics with respect to a single selected metric, we consider a family of Riemannian metrics introduced by Hüper, Markina, and Silva Leite that includes the Euclidean and the canonical metric as prominent examples. As main contributions, we provide (1) a unified, structured, reduced formula for the Stiefel geodesics. The formula is unified in the sense that it works for the full family of metrics under consideration. It is structured in the sense that it relies on matrix exponentials of skew-symmetric matrices exclusively. It is reduced in relation to the dimension of the matrices of which matrix exponentials have to be calculated. We provide (2) a unified method to tackle the geodesic endpoint problem numerically, and (3) we improve the existing Riemannian log algorithm under the canonical metric in terms of the computational efficiency. The findings are illustrated by means of numerical examples, where the novel algorithms prove to be the most efficient methods known to this date.

Application of MGA and EGA algorithms on large-scale linear systems of ordinary differential equations

In this paper, we propose two new approaches to solve the large linear system of ordinary differential equations (LSODE), which are based on the projection technique on the extended global Krylov (EGKS) or global Krylov (GKS) subspaces if A is not invertible. The first approach (EGA-expm or MGA-expm if A is not invertible) is based on the projection of matrix exponential function (expm), the second approach (EGA-ROS2, EGA-BDF2 or MGA-ROS2, MGA-BDF2 if A is not invertible) is based on a projection of the initial problem onto an EGKS (or GKS if A is not invertible) to get a new LSODE of small-size then solve it with the 2-stage Rosenbrock method (ROS2) or the backward differentiation formula (BDF2) method and the 2-stage obtained solution is used to have a low-rank (LR) approximate solution of the initial problem. Moreover, we will present the expression of the residual and the error norms. Finally, some attractive and interesting examples of specific problems are presented to confirm the strength of the new approaches.

Near-linear convergence of the Random Osborne algorithm for Matrix Balancing

We revisit Matrix Balancing, a pre-conditioning task used ubiquitously for computing eigenvalues and matrix exponentials. Since 1960, Osborne’s algorithm has been the practitioners’ algorithm of choice, and is now implemented in most numerical software packages. However, the theoretical properties of Osborne’s algorithm are not well understood. Here, we show that a simple random variant of Osborne’s algorithm converges in near-linear time in the input sparsity. Specifically, it balances K in {mathbb {R}}_{ge 0}^{n times n} after O(m varepsilon ^{-2} log kappa ) arithmetic operations in expectation and with high probability, where m is the number of nonzeros in K, varepsilon is the ell _1 accuracy, and kappa = sum _{ij} K_{ij} / ( min _{ij : K_{ij} ne 0} K_{ij}) measures the conditioning of K. Previous work had established near-linear runtimes either only for ell _2 accuracy (a weaker criterion which is less relevant for applications), or through an entirely different algorithm based on (currently) impractical Laplacian solvers. We further show that if the graph with adjacency matrix K is moderately connected—e.g., if K has at least one positive row/column pair—then Osborne’s algorithm initially converges exponentially fast, yielding an improved runtime O(m varepsilon ^{-1} log kappa ). We also address numerical precision issues by showing that these runtime bounds still hold when using O(log (nkappa /varepsilon ))-bit numbers. Our results are established through an intuitive potential argument that leverages a convex optimization perspective of Osborne’s algorithm, and relates the per-iteration progress to the current imbalance as measured in Hellinger distance. Unlike previous analyses, we critically exploit log-convexity of the potential. Notably, our analysis extends to other variants of Osborne’s algorithm: along the way, we also establish significantly improved runtime bounds for cyclic, greedy, and parallelized variants of Osborne’s algorithm.

Guided waves in stratified media with equal acoustic impedances

Dispersion of guided waves propagating in layered media containing layers with equal acoustic impedances is analyzed by Cauchy sextic formalism coupled with the exponential fundamental matrix method. The cases of layers with equal acoustic impedances apparently have not been studied yet. However, dispersion in such a medium exhibits several peculiarities that may be essential for developing the acoustic non-destructive testing methods.

Exploring a new discrete delayed Mittag–Leffler matrix function to investigate finite‐time stability of Riemann–Liouville fractional‐order delay difference systems

In this paper, firstly, a new discrete delayed Mittag–Leffler matrix function is introduced, which generalizes the existing discrete delayed exponential matrix function. Secondly, based on it, the explicit formula of the solution of homogeneous Riemann–Liouville (RL) fractional‐order delay difference system is obtained. Thirdly, the explicit formulas of the solutions of nonhomogeneous RL fractional‐order delay difference systems are also derived in terms of the superposition principle and the new discrete delayed Mittag–Leffler matrix function. Furthermore, a finite‐time stability (FTS) criterion of nonhomogeneous RL fractional‐order delay difference system is given using the properties of the new discrete delayed Mittag–Leffler matrix function. Finally, three examples are presented to demonstrate the effectiveness of the proposed theorems.

Fundamental modes of guided waves in stratified plates: appearing transverse quasi-resonances

It has long been known that the transverse resonances may occur at propagating higher modes of guided waves in homogeneous plates. These resonances are associated with the appearing zero group velocities (ZGV) at which the flux of energy becomes orthogonal to the direction of propagation. The current research focuses on the transverse resonances found in propagating fundamental symmetric modes (S 0) in some multilayered plates. The observed phenomenon may clarify the interlayer damages and delamination at the guided wave excitation and propagation in heterogeneous structures. Cauchy sextic formalism, coupled with the exponential fundamental matrix method, is used for analyzing the transverse resonances.

A Novel Security Framework for Medical Data in IoT Ecosystems

The Internet of Things (IoT) is integrated with the healthcare system to create an intelligent hospital ecosystem. Protecting the privacy and security of medical data is a central problem in such an ecosystem. This article proposes a novel yet simple security framework to ensure the confidentiality of medical data during the communication between IoT hops. The proposed framework utilizes the concept of encryption to ensure the security. It essentially involves inducing confusion among pixels by employing image pixel scrambling achieved by spiralizing the medical data matrix, followed by a diffusion process using matrix exponentials and pseudorandom number generation by sequential chaotic maps. A reliable decryption scheme is finally designed to obtain the original data from the encrypted data. Experimental results and analyzes demonstrate the efficiency and robustness of the proposed framework and, thus, making it more suitable for the IoT ecosystem.

Improved Fourier Modal Method Analyzing 2-D Ultrathin Periodic Structures Without Solving Eigenvalues

Intensive computation for solving eigenvalues of large eigenmatrix restricted the development of Fourier modal method (FMM) in analyzing periodic structures, especially 2-D ultrathin periodic structures. To alleviate the problem of intensive computation in the traditional FMM algorithm when analyzing 2-D ultrathin periodic structures, an efficient FMM algorithm without solving eigenvalues is proposed in this letter. Based on the first-order Taylor Expansion, the exponential term matrix in the traditional algorithm can be replaced by two simple diagonal matrices, which avoids the large dimensional matrix inversion process, and the newly formed execution equation is only related to the eigenmatrix. Two numerical examples show that the results of the traditional FMM algorithm, improved FMM algorithm and CST simulation, are highly consistent. Compared with the traditional algorithm, the improved algorithm can greatly alleviate the computing-intensive problems, and the improved algorithm can significantly reduce the CPU time by more than 90% and reduce computation memory by about 70%.