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We propose a quadratically convergent algorithm for inverse symmetric eigenvalue problems based on matrix equations. The basic idea is seen in a recent study by Ogita and Aishima, while they derive an efficient iterative refinement algorithm for symmetric eigenvalue problems using special matrix equations. In other words, this study is interpreted as a unified view on quadratically convergent algorithms for eigenvalue problems and inverse eigenvalue problems based on matrix equations. To the best of our knowledge, such a unified development of algorithms is provided for the first time. Since the proposed algorithm for the inverse eigenvalue problems can be regarded as the Newton's method for the matrix equations, the quadratic convergence is naturally proved. Our algorithm is interpreted as an improved version of the Cayley transform method for the inverse eigenvalue problems. Although the Cayley transform method is one of the effective iterative methods, the Cayley transform takes O ( n 3 ) arithmetic operations to produce an orthogonal matrix using a skew-symmetric matrix in each iteration. Our algorithm can refine orthogonality without the Cayley transform, which reduces the operations in each iteration. It is worth noting that our approach overcomes the limitation of the Cayley transform method to the inverse standard eigenvalue problems, resulting in an extension to inverse generalized eigenvalue problems.

This paper proposes a method for a fast estimation of the largest eigenvalue of an asymmetric tridiagonal matrix. The proposed method is based on the Power method and the computation of the square of the original matrix. The matrix square is computed through a proposed fast algorithm designed specifically for tridiagonal matrices. Implementations for compressed column (CCS) and compressed row storage (CRS) formats are provided, discussed and compared to a standard scientific library. We investigate the roundoff numerical errors, showing that the proposed method provides errors no greater than the usual Power method. We provide numerical results with simulations in C/C++ implementation in order to demonstrate the effectiveness of the proposed method.

In this paper, some new absolute perturbation bounds of partitioned generalized Hermitian positive definite eigenvalue problem are established by two different ways. Numerical results show that our bounds are sharper than the ones in the literature.

A new real structure-preserving Jacobi algorithm is proposed for solving the eigenvalue problem of quaternion Hermitian matrix. By employing the generalized JRS-symplectic Jacobi rotations, the new quaternion Jacobi algorithm can preserve the symmetry and JRS-symmetry of the real counterpart of quaternion Hermitian matrix. Moreover, the proposed algorithm only includes real operations without dimension-expanding and is generally superior to the state-of-the-art algorithm. Numerical experiments are reported to indicate its efficiency and accuracy.

In the present paper we investigate the spectrum of operator corresponding to eigenvalue problem with parameter dependent boundary condition. Trace formula for that operator is also established.

Hybrid algorithms for solving the partial generalized eigenvalue problem for symmetric positive definite sparse matrices of different structures by hybrid computers with graphic processors are proposed, coefficients for the efficiency of the algorithms are obtained, and approbation of the developed algorithms for test and practical problems is carried out.

In this paper, we consider the partial eigenvalue assignment problem of high order control systems. Based on the orthogonality relations, we propose a new method for solving this problem by which the undesired eigenvalues are moved to desired values and keep the remaining eigenvalues unchanged. Using the inverse of Cauchy matrix, we give the solvable condition and the explicit solutions to this problem. Numerical examples show that our method is effective.

A positive semi-definite eigenvalue problem for second-order self-adjoint elliptic differential operator definedon a bounded domain in the planewith smooth boundary and Dirichlet boundary condition is considered. This problem has a nondecreasing sequence of positive eigenvalues of finite multiplicity with a limit point at infinity. To the sequence of eigenvalues, there corresponds an orthonormal system of eigenfunctions. The original differential eigenvalue problem is approximated by the finite element method with numerical integration and Lagrange curved triangular finite elements of arbitrary order. Error estimates for approximate eigenvalues and eigenfunctions are established.

This paper focuses on C 0IPG adaptive algorithms for the biharmonic eigenvalue problem with the clamped boundary condition. We prove the reliability and efficiency of the a posteriori error indicator of the approximating eigenfunctions and analyze the reliability of the a posteriori error indicator of the approximating eigenvalues. We present two adaptive algorithms, and numerical experiments indicate that both algorithms are efficient.

We apply a commutation formula and the Rayleigh–Ritz inequality to study the ratio of the first two eigenvalues of one-dimensional Schrodinger operators with Dirichlet boundary conditions. The potentials we consider here are allowed to change sign. We give conditions under which the first eigenvalue is positive, and establish some new results on the upper bounds of the eigenvalue ratio.

In this paper, we discuss a new problem of establishing feasibility conditions for regional eigenvalue assignment of positive observers. Previous results on regional eigenvalue assignment for observers of linear time-invariant positive systems are also improved. Unlike observable dynamical systems whose closed-loop eigenvalues can be assigned arbitrarily, eigenvalues of positive observers are indicated unable to be assigned into any arbitrary region. We derive feasibility conditions in the form of constrained convex programming under which the regional eigenvalue assignment is possible. Moreover, we propose a new method for solving the regional eigenvalue problem of positive observers once the feasibility conditions are satisfied. Numerical examples are given to show the efficacy of the proposed method.

We deal with a generalized phase-field-type system that arises as a transformed system of reaction-diffusion equations with a conservation law. We consider the stationary problem which is reduced to a scalar elliptic equation with a nonlocal term, and study the linearized eigenvalue problem. We first prove by the spectral comparison argument that the number of unstable eigenvalues for the problem coincides with the one of the linearized eigenvalue problem for the original system. We next show a limiting behavior of eigenvalues for the scalar problem as the coefficient of the nonlocal term goes to infinity.

In the paper, an improved two-grid scheme based on shifted-inverse power method is proposed to solve the elliptic eigenvalue problems. With this new scheme, the solution of the elliptic eigenvalue problem on a fine grid T h is reduced to the solution of the elliptic eigenvalue problem and the recovered eigenfunction on a much coarser grid T H , and the solution of an elliptic boundary value problem and the recovered solution on the fine grid T h . Theoretical analysis shows that the scheme allows a much coarser mesh to achieve the superconvergence rate. Finally, some numerical experiments are carried out to confirm the theoretical analysis.

Summary This article discusses the use of frequency response surrogates for eigenvalue optimization problems in topology optimization that may be used to avoid solving the eigenvalue problem. The motivation is to avoid complications that arise from multiple eigenvalues and the computational complexity associated with computation of eigenvalues in very large problems. Copyright © 2017 John Wiley & Sons, Ltd.

In this paper, we consider the eigenvalue bounds for real symmetric matrices whose entries are in a given interval. Based on some matrix computation theories, a new algorithm and some theorems are presented which provide ways for solving the left problems in former papers and obtain satisfying results. Furthermore, the ideas are spread to general symmetric interval matrices. Finally, numerical examples are presented which show the validity of the proposed methods.

In this paper, we study nonlinear optimization problems involving eigenvalues of symmetric matrices. One of the difficulties in solving these problems is that the eigenvalue functions are not differentiable when the multiplicity of the function is not one. We apply the $${\mathcal {U}}$$ -Lagrangian theory to analyze the largest eigenvalue function of a convex matrix-valued mapping which extends the corresponding results for linear mapping in the literature. We also provides the formula of first-and second-order derivatives of the $${\mathcal {U}}$$ -Lagrangian under mild assumptions. These theoretical results provide us new second-order information about the largest eigenvalue function along a suitable smooth manifold, and leads to a new algorithmic framework for analyzing the underlying optimization problem.

In this paper, we consider the eigenvalue variation for any perturbation of Hermitian matrices, and we obtain two perturbation bounds. The first bound always improves the existing bound, and the second bound also improves the existing one under a suitable condition. A simple example is given for comparing these bounds.

The research in this paper is motivated by a recent work of I. Barany and J. Solymosi [I. Barany and J. Solymosi. Gershgorin disks for multiple eigenvalues of non-negative matrices. Preprint arXiv no. 1609.07439, 2016.] about the location of eigenvalues of nonnegative matrices with geometric multiplicity higher than one. In particular, an answer to a question posed by Barany and Solymosi, about how the location of the eigenvalues can be improved in terms of their geometric multiplicities is obtained. New inclusion sets for the eigenvalues of a real square matrix, called Ger\v{s}gorin discs of the second type, are introduced. It is proved that under some conditions, an eigenvalue of a real matrix is in a Ger\v{s}gorin disc of the second type. Some relationships between the geometric multiplicities of eigenvalues and these new inclusion sets are established. Some other related results, consequences, and examples are presented. The results presented here apply not only to nonnegative matrices, but extend to all real matrices, and some of them do not depend on the geometric multiplicity.

The paper focuses on the generalization of a nonlinear multi-parameter eigenvalue problem for a system of nonlinear differential equations. The problem is reduced to a system of nonlinear integral equations on a segment. The notion of eigentuple is introduced, the existence of a finite number of isolated eigentuples is proved, and their distribution is described. The corresponding linear multi-parameter eigenvalue problem is studied as well; it is proved that the linear problem has an infinite number of isolated eigentuples. Applications to nonlinear electromagnetic wave propagation theory are demonstrated.

For symmetric eigenvalue problems, we constructed a three-term recurrence polynomial filter by means of Chebyshev polynomials. The new filtering technique does not need to solve linear systems and only needs matrix-vector products. It is a memory conserving filtering technique for its three-term recurrence relation. As an application, we use this filtering strategy to the Davidson method and propose the filtered-Davidson method. Through choosing suitable shifts, this method can gain cubic convergence rate locally. Theory and numerical experiments show the efficiency of the new filtering technique.

给出一类新的非奇异矩阵——p-范数双严格对角占优矩阵(简记为p-范数DSDD矩阵)，并由其得到一个新的矩阵特征值包含区域。文中算例表明在某些情况下本文的矩阵特征值包含区域含于著名的Brauer- Cassini卵形区域之中。 A new class of nonsingular matrices, p-norm double strictly diagonally dominant matrices, (shorthand for p-norm DSDD matrices), is presented, and it is used to get a new eigenvalue inclusion region. A numerical example is given to show that the eigenvalue inclusion in this paper, in some cases, is in the famous Brauer-Cassini oval area.

In this article the computation of the Structured Singular Values (SSV) for the delay eigenvalue problems and polynomial eigenvalue problems is presented and investigated. The comparison of bounds of SSV with the well-known MATLAB routine mussv is investigated.

This paper presents two kinds of symmetric tridiagonal plus paw form (hereafter TPPF) matrices, which are the combination of tridiagonal matrices and bordered diagonal matrices. In particular, we exploit the interlacing properties of their eigenvalues. On this basis, the inverse eigenvalue problems for the two kinds of symmetric TPPF matrices are to construct these matrices from the minimal and the maximal eigenvalues of all their leading principal submatrices respectively. The necessary and sufficient conditions for the solvability of the problems are derived. Finally, numerical algorithms and some examples of the results developed here are given.

Matrix eigenvalue problems play a significant role in many areas of computational science and engineering. In this paper, we propose a fast continuous method for the extreme eigenvalue problem. We first convert such a nonconvex optimization problem into a minimization problem with concave objective function and convex constraints based on the continuous methods developed by Golub and Liao. Then, we propose a continuous method for solving such a minimization problem. To accelerate the convergence of this method, a self-adaptive step length strategy is adopted. Under mild conditions, we prove the global convergence of this method. Some preliminary numerical results are presented to verify the effectiveness of the proposed method eventually.

© 2017 Society for Industrial and Applied Mathematics. We contribute to the perturbation theory of nonlinear eigenvalue problems in three ways. First, we extend the formula for the sensitivity of a simple eigenvalue with respect to a variation of a parameter to the case of multiple nonsemisimple eigenvalues, thereby providing an explicit expression for the leading coefficients of the Puiseux series of the emanating branches of eigenvalues. Second, for a broad class of delay eigenvalue problems, the connection between the finitedimensional nonlinear eigenvalue problem and an associated infinite-dimensional linear eigenvalue problem is emphasized in the developed perturbation theory. Finally, in contrast to existing work on analyzing multiple eigenvalues of delay systems, we develop all theory in a matrix framework, i.e., without reduction of a problem to the analysis of a scalar characteristic quasi-polynomial. ispartof: SIAM Journal on Matrix Analysis and Applications vol:38 issue:2 pages:599-620 status: published

In this paper, the Eigenvalue Complementarity Problem (EiCP) with real symmetric matrices is addressed, which appears in the study of contact problem in mechanics. We discuss a quadratic programming formulation to the problem. The resulting problems are nonlinear programs that can be solved by a line search filter-SQP algorithm.

This article is devoted to both theoretical and numerical studies of eigenvalues of regular fractional $2\alpha $-order Sturm-Liouville problem where $\frac{1}{2}< \alpha \leq 1$. In this paper, we implement the reproducing kernel method RKM) to approximate the eigenvalues. To find the eigenvalues, we force the approximate solution produced by the RKM satisfy the boundary condition at $x=1$. The fractional derivative is described in the Caputo sense. Numerical results demonstrate the accuracy of the present algorithm. In addition, we prove the existence of the eigenfunctions of the proposed problem. Uniformly convergence of the approximate eigenfunctions produced by the RKM to the exact eigenfunctions is proven.

In this paper, we consider the robust partial quadratic eigenvalue assignment problem in vibration by active feedback control. Based on the receptance measurements and the system matrices, we propose an optimization method for the robust and minimum norm partial quadratic eigenvalue assignment problem. We provide a new cost function and the closed-loop eigenvalue sensitivity and the feedback norms can be minimized simultaneously. Our method is also extended to the case of time delay between measurements of state and actuation of control. Numerical tests demonstrate the effectiveness of our method.

This paper discusses the differentiability of a class of functions associated with eigenvalues and eigenvectors of symmetric matrices. Recursive style formulas of partial derivatives for this class of functions are derived and higher order derivatives can be easily obtained from these formulas. Meanwhile, some interesting characteristics of multiple eigenvalues are revealed. Examples involving inverse eigenvalue problems and primary matrix functions are given to illustrate the applications of the results obtained in this paper.

A simple and intuitive matrix iteration method for solving nonlinear eigenvalue problems is described and demonstrated in detail by two problems: (i) the boundary value problem associated with large deflection of a flexible rod, and (ii) the initial value problem associated with normal mode motion of a double pendulum. The two problems are solved by two approaches, the finite difference approach and a continuous realization approach which is similar in spirit to the Rayleigh–Ritz method.

The state-of-the-art algorithms for solving the trust-region subproblem (TRS) are based on an iterative process, involving solutions of many linear systems, eigenvalue problems, subspace optimization, or line search steps. A relatively underappreciated fact, due to Gander, Golub, and von Matt [Linear Algebra Appl., 114 (1989), pp. 815--839], is that TRSs can be solved by one generalized eigenvalue problem, with no outer iterations. In this paper we rediscover this fact and discover its great practicality, which exhibits good performance both in accuracy and efficiency. Moreover, we generalize the approach in various directions, namely by allowing for an ellipsoidal constraint, dealing with the so-called hard case, and obtaining approximate solutions efficiently when high accuracy is unnecessary. We demonstrate that the resulting algorithm is a general-purpose TRS solver, effective both for dense and large-sparse problems, including the so-called hard case. Our algorithm is easy to implement: its essence i...

We study the convergence of an Ulm-like Cayley transform method for solving inverse eigenvalue problems which avoids solving approximate Jacobian equations. Under the nonsingularity assumption of the relative generalized Jacobian matrices at the solution, a convergence analysis covering both the distinct and multiple eigenvalues cases is provided and the quadratical convergence is proved. Moreover, numerical experiments are given in the last section to illustrate our results.

In this paper we present an efficient spectral method based on the Legendre-Galerkin approximation for the transmission eigenvalue problem. A rigorous error analysis is presented by using the minmax principle for the generalized eigenvalue problems associated to a transmission eigenvalue problem. However, this formulation can only compute real eigenvalues. Thus, we also present another formulation based on second order equations and construct an appropriate set of basis functions such that the matrices in the discrete variational form are sparse. For the case of constant medium, we derive the matrix formulations based on the tensor-product for the discrete variational form in two and three-dimensional cases, respectively. In addition, we also establish an optimization scheme based on the Legendre-Galerkin approximation. With this scheme we can estimate the index of refraction of an inhomogeneous medium. We also present ample numerical results to show that our method is very effective and high accurate.

In this paper we examine an eigenvalue optimization problem. Given two nonlinear functions α ( λ ) and β ( λ ) , find a subset D of the unit ball of measure A for which the first Dirichlet eigenvalue of the operator − div ( ( α ( λ ) χ D + β ( λ ) χ D c ) ∇ u ) = λ u is as small as possible. This sort of nonlinear eigenvalue problems arises in the study of some quantum dots taking into account an electron effective mass. We establish the existence of a solution, and we propose a numerical algorithm to obtain an approximate description of the optimizer.

We analyse an eigenvalue problem, involving the anisotropic (p, q)-Laplace operator in a smooth-bounded domain. We show that the set of eigenvalues of the problem is exactly an unbounded open interval.

A method for stability of the descriptor continuous-time linear system is focused. For easily, it is converted to a standard continuous-time linear system by the definition of a derivative state feedback. Then partial eigenvalue assignment is used for obtaining state feedback and showing stability and controllability of the standard system. In partial eigenvalue assignment, just a part of the open loop spectrum of the standard linear systems are reassigned, while leaving the rest of the spectrum invariant and for reassigning, similarity transformation is used. Using partial eigenvalue assignment is easier than using eigenvalue assignment. Because by partial eigenvalue assignment, size of matrices and state and input vectors are decreased and stability is kept, too. Also concluding remarks and an algorithm are proposed to the descriptions be obvious. At the end, convergence of state and input vectors in the descriptor system to balance point (zero) are showed by figures in numerical examples.

In this paper, we offer several lower bounds on eigenvalue summation for the solution of the Lyapunov matrix differential equation applying index matrix eigenvalue inequalities and Holder inequality. Further, we give a numerical example to illustrate the effectiveness of the derived bounds.

We define Atkinson's semi-definite p-Laplacian eigenvalue problems, which include the regular p-Laplacian eigenvalue problems with L1 coefficient functions. Then we show that the Sturm oscillation theorem also holds for this eigenvalue problem.

A variation of Kublanovskaya's nonlinear QR method for solving banded nonlinear eigenvalue problems is presented in this article. The new method is iterative and specifically designed for problems too large to use dense linear algebra techniques. For the unstructurally banded nonlinear eigenvalue problem, a new data structure is used for storing the matrices to keep memory and computational costs low. In addition, an algorithm is presented for computing several nearby nonlinear eigenvalues to already-computed ones. Finally, numerical examples are given to show the efficacy of the new methods, and the source code has been made publicly available.

In this article, convergence of an iterative scheme to approximate the first eigenfunction and related eigenvalue for p-Laplace operator is shown. Moreover, numerical examples are presented that show the efficiency and accuracy of the algorithm.

The transmission eigenvalue problem is an important and challenging topic arising in the inverse scattering theory. In this paper, for the Helmholtz transmission eigenvalue problem, we give a weak formulation which is a nonselfadjoint linear eigenvalue problem. Based on the weak formulation, we first discuss the non-conforming finite element approximation, and prove the error estimates of the discrete eigenvalues obtained by the Adini element, the Morley–Zienkiewicz element, the modified-Zienkiewicz element et al. And we report some numerical examples to validate the efficiency of our approach for solving transmission eigenvalue problem.

The inverse nodal problem for the Sturm–Liouville operator L ⁢ ( q , a ) ${L(q,a)}$ arisen from acoustic scattering problems was considered. The authors showed that q ⁢ ( x ) ${q(x)}$ on the interval [ 0 , 1 ] ${[0,1]}$ can be uniquely determined by the interior twin-dense nodal subset W T ⁢ ( [ 0 , 1 ] ) ${W_{T}([0,1])}$ , or W T ⁢ ( [ a 0 , 1 ] ) ${W_{T}([a_{0},1])}$ , respectively. Using the above results, two uniqueness theorems of the spherically symmetric speed of sound η ⁢ ( r ) ${\eta(r)}$ for acoustic scattering problems are obtained.

In this paper, we present a numerical method for periodic band structure calculation, which is associated with eigenvalue problems for periodic problems, using the boundary element method (BEM). In the BEM, the eigenvalue problems are converted into non-linear eigenvalue problems, which are not tractable with conventional eigensolvers. In the present study, to solve non-linear eigenvalue problems, the block Sakurai–Sugiura (SS) method, which can convert non-linear eigenvalue problems into generalised eigenvalue problems, is utilised. A fast direct solver for the BEM with a fast multipole representation is employed in the algorithm of the block SS method since algebraic equations need to be solved for multiple right-hand sides in the block SS method. We conduct several numerical experiments related to phononic structures to confirm the validity and efficiency of the proposed method. We confirm that the proposed method can calculate the band structure of the phononic structures, and the computational time with the proposed method is less than that with a conventional FEM-based eigensolvers with triangular linear elements even for relatively small problems.

In this paper, a new optimization scheme is proposed for robust eigenvalue placement in high-order descriptor systems in union region based on harmony search algorithm. The specification on the closed-loop eigenvalues is given in terms of robust union region stability constraints. The radius or the position of the subregions can be arbitrarily specified in complex plane for the transient performance request. By using exact eigenvalue placement theory and combining the harmony search algorithm, the robust eigenvalue placement for high-order descriptor linear system in union region is converted into a global dynamical optimization problem. Further, the eigenstructure of the closed-loop system matrix can be optimized with better robustness bound (i.e., the spectral upper bound of the maximum allowable perturbation or uncertainty for the system matrices) by the proposed scheme without imposing restriction on the differentiability of the nonlinear robust measure index function. Consequently, the robust feedback controller can be obtained by dynamically optimizing the eigenvalue and eigenvector pairs of the system. Finally, the simulation results illustrate the effectiveness and superiority of the proposed method.