An improved computational technique for perturbations of the generalized symmetric linear algebraic eigenvalue problem

A note on the homotopy method for linear algebraic eigenvalue problems

Convergence analysis of Lanczos-type methods for the linear response eigenvalue problem

The pth moment Lyapunov exponent of an n-dimensional linear stochastic system is the principal eigenvalue of a second-order partial differential eigenvalue problem, which can be established using the theory of stochastic dynamical system. An analytical-numerical approach for the determination of the pth moment Lyapunov exponents, for all values of p, is presented. The approach is illustrated through a two-dimensional system under bounded noise or real noise parametric excitation. Series expansions of the eigenfunctions using orthogonal functions are employed to transform the partial differential eigenvalue problems to linear algebraic eigenvalue problems, which are then solved numerically. The numerical values obtained are compared with approximate analytical results with weak noise amplitudes.

Because of its vital role of the trust-region subproblem (TRS) in various applications, for example, in optimization and in ill-posed problems, there are several factorization-free algorithms for solving the large-scale sparse TRS. The truncated Lanczos approach proposed by N. I. M. Gould, S. Lucidi, M. Roma, and P. L. Toint [SIAM J. Optim., 1999, 9: 504–525] is a natural extension of the classical Lanczos method for the symmetric linear system and eigenvalue problem and, indeed follows the classical Rayleigh-Ritz procedure for eigenvalue computations. It consists of 1) projecting the original TRS to the Krylov subspaces to yield smaller size TRS’s and then 2) solving the resulted TRS’s to get the approximates of the original TRS. This paper presents a posterior error bounds for both the global optimal value and the optimal solution between the original TRS and their projected counterparts. Our error bounds mainly rely on the factors from the Lanczos process as well as the data of the original TRS and, could be helpful in designing certain stopping criteria for the truncated Lanczos approach.

In this paper, we introduce a geometric model for linear symmetric eigenvalue problem, which is motivated by the fact that any eigenvalue of a symmetric positive definite matrix A is the reciprocal of the square length of an axis of the ellipsoid xTAx = 1. Hence, to find the largest eigenvalue is equivalent to calculate the shortest axis of the corresponding ellipsoid. Two sequential subspace projection algorithms based on this idea are proposed, and we establish the global convergence and local linear convergence rate of our proposed algorithms. Numerical experiments demonstrate that our algorithm outperforms the MATLAB built-in solver "EIGS" which calls the famous package "ARPACK".

Firstly we study necessary and sufficient conditions for the constant row sums symmetric inverse eigenvalue problem to have a solution and sufficient conditions for the symmetric stochastic inverse eigenvalue problem to have a solution. Then we introduce the concept of general solutions for the symmetric stochastic inverse eigenvalue problem and the concept of totally general solutions for the $3\times3$ symmetric stochastic inverse eigenvalue problem. Finally we study the necessary and sufficient conditions for the symmetric stochastic inverse eigenvalue problems of order 3 to have general solutions, and the necessary and sufficient conditions for the symmetric stochastic inverse eigenvalue problems of order 3 to have a totally general solution.

In the field of operator algebra, the concept of antieigenvalue theory was first introduced by Karl Gustafson with a special emphasis on accretive operators. In a wide range of scientific applications, antieigenvalue naturally occurs. Due to the inclusion of nonlinear Euler equations in the antieigenvalue theory, computing antieigenvalues is a difficult task as compared to that of computing eigenvalues of the operator. In the current paper, we consider linear two-parameter eigenvalue problems (LTEP) and will discuss the abstract algebraic setting of the problem as proposed by Atkinson. We analyze the generalized antieigenvalue pair and their corresponding generalized antieigenvectors for LTEP using the consequences of the Cauchy Schwarz inequality. Some generalized antieigenvalue bounds will also be derived. Generalized antieigenvalues and their corresponding generalized antieigenvectors will be calculated solving their relevant optimization problem. For numerical computations, three examples will be provided. Real symmetric matrices are used in the first case, while real diagonal matrices are used in the second case and finally arbitrary matrices are considered.

Nonlinear flexural response of a slender cantilever beam of constant thickness and linearly-varying width to a primary resonance excitation

The sequence of mini-max subproblems procedures for the solution of symmetric eigenvalue problems

A simple and novel finite element (FE) formulation is proposed to study the thermal post-buckling of composite and FGM columns with axially immovable ends and operating in severe thermal environment. A linear eigenvalue analysis gives the critical buckling temperature but practically the buckled columns can withstand additional thermal load beyond critical temperature, which can be obtained using von-Karman geometric nonlinearity, applicable for moderately large deflections. In the present study, the solution of the non-linear post-buckling problem is obtained by treating it as a linear eigenvalue problem using the concept of effective stiffness. Here, the total degrees of freedom (dof) of the discretized column are reduced and the post-buckling load is obtained without the need for iterative analysis. Comparison of the numerical results obtained from this FE formulation is in very good agreement with those obtained from the earlier FE formulations.

Since its inception by Richard Feynman in 1982, quantum computing has provided an intriguing opportunity to advance computational capabilities over classical computing. Classical computers use bits to process information in terms of zeros and ones. Quantum computers use the complex world of quantum mechanics to carry out calculations using qubits (the quantum analog of a classical bit). The qubit can be in a superposition of the zero and one state simultaneously unlike a classical bit. The true power of quantum computing comes from the complexity of entanglement between many qubits. When entanglement is realized, quantum algorithms for problems such as factoring numbers and solving linear algebra problems show exponential speed-up relative to any known classical algorithm. Linear algebra problems are of particular interest in engineering application for solving problems that use finite element and finite difference methods. Here, we explore quantum linear algebra problems where we design and implement a quantum circuit that can be tested on IBM’s quantum computing hardware. A set of quantum gates are assimilated into a circuit and implemented on the IBM Q system to demonstrate its algorithm capabilities and its measurement methodology.

We investigate the complexity of enumerative approximation of two elementary problems in linear algebra, computing the rank and the determinant of a matrix. In particular, we show that if there exists an enumerator that, given a matrix, outputs a list of constantly many numbers, one of which is guaranteed to be the rank of the matrix, then it can be determined in AC0 (with oracle access to the enumerator) which of these numbers is the rank. Thus, for example, if the enumerator is an FL function, then the problem of computing the rank is in FL. The result holds for matrices over any commutative ring whose size grows at most polynomially with the size of the matrix. The existence of such an enumerator also implies a slightly stronger collapse of the exact counting logspace hierarchy. For the determinant function we establish the following two results: (1) If the determinant is poly-enumerable in logspace, then it can be computed exactly in FL. (2) For any prime p, if computing the determinant modulo p is (p-1)-enumerable in FL, then computing the determinant modulo p can be done in FL. This gives a new perspective on the approximability of many elementary linear algebra problems equivalent to computing the rank or the determinant.KeywordsCommutative RingEquivalence GraphOracle AccessInteger MatriceCircuit ClassThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

One of the important problems in linear algebra is the determination of eigenvalues and eigenvectors of a matrix. Eigenvalue problems in differential and integral equations can also be approximated by algebraic eigenvalue problems. In spite of the simplicity of its formulation, many algorithms are required to deal efficiently with a wide spectrum of problems, which are encountered in practice. The variety in problems arises because of the variety in type of matrix and the varying requirements from the solution. The classification of matrices into different types has been described in Section 3.1 and will not be repeated here. The varying requirements from the solution arise because we may need all eigenvalues and eigenvectors or only some of them. Alternately, we may be interested in only eigenvalues. In some problems, we may be interested in only the eigenvalue with the largest magnitude, or the one with the largest real part, while in other problems we may require all eigenvalues in a given region of the complex plane. In general, there is some correlation between the type of matrix and the amount of information required from the solution. For example, with large matrices which usually arise from discretisation of differential or integral equations, we are normally interested in only a few eigenvalues and possibly the eigenvectors. On the other hand, for small matrices we normally want all eigenvalues and some or all eigenvectors.

This article is concerned with research on parallel pro cessing for chemically reactive supersonic flows. Effi ciency of autotasking and microtasking is examined for the problems of linear algebra and two-dimensional Euler equations with the finite element method. For microtasking, satisfactory improvements in speedup (2.63 for 4 CPUs and 1.67 for 2 CPUs) are achieved, with the load balance and the amount of parallelism being the dominant factors for the speedup. Autotask ing has advantages in providing automatic paralleliza tion, but it has been shown that in the solution of finite element Euler equations based on conventional algo rithms, parallelization by means of user's intervention is required for significant speedups (3.35 for 4 CPUs and 1.79 for 2 CPUs). A much better performance may be achieved with a combination of autotasking and microtasking. These results are verified from various example problems of linear algebra and supersonic diffuser flow field calculations with hydrogen-oxygen chemical reactions.

Standard approaches to measuring the condition of various problems in numerical linear algebra compress all sensitivity information into a single condition number. Thus, a loss of information can occur in situations in which this standard condition number does not accurately reflect the actual sensitivity of a solution or particular entries of a solution. We describe a new method that overcomes these and other common deficiencies. The new procedure measures the effects on the solution of small random changes in the input data. And, by properly scaling the results, obtains condition estimates for each entry of a computed solution. This approach, which is referred to as small-sample statistical condition estimation (SCE), applies to both linear and nonlinear problems. The method has a rigorous statistical theory available for the probability of accuracy of the condition estimates. The theory of this approach is described along with several illustrative examples related to solving linear equations, linear least squares problems, and eigenvalue/eigenvector problems. Applications to problems of specific interest to control engineers are also discussed, including the solution of matrix Lyapunov, Sylvester, and Riccati equations.