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Eigenvalue Problem (eigenvalue + problem)
Kinds of Eigenvalue Problem Selected AbstractsGlobal optimization for robust control synthesis based on the Matrix Product Eigenvalue ProblemINTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL, Issue 9 2001Yuji Yamada Abstract In this paper, we propose a new formulation for a class of optimization problems which occur in general robust control synthesis, called the Matrix Product Eigenvalue Problem (MPEP): Minimize the maximum eigenvalue of the product of two block-diagonal positive-definite symmetric matrices under convex constraints. This optimization class falls between methods of guaranteed low complexity such as the linear matrix inequality (LMI) optimization and methods known to be NP-hard such as the bilinear matrix inequality (BMI) formulation, while still addressing most robust control synthesis problems involving BMIs encountered in applications. The objective of this paper is to provide an algorithm to find a global solution within any specified tolerance , for the MPEP. We show that a finite number of LMI problems suffice to find the global solution and analyse its computational complexity in terms of the iteration number. We prove that the worst-case iteration number grows no faster than a polynomial of the inverse of the tolerance given a fixed size of the block-diagonal matrices in the eigenvalue condition. Copyright 2001 © John Wiley & Sons, Ltd. [source] Molecular axes and planes as an Eigenvalue problemJOURNAL OF APPLIED CRYSTALLOGRAPHY, Issue 2 2008Detlef-M. A statistical approach is used to define the long molecular axis and the main molecular plane. While the concept of molecular axis and plane is immediately useful for simple rod-like or planar molecules, it can be easily generalized to arbitrary molecules as well as molecular subgroups. From the molecular axes and planes, the orientation of the molecule with respect to the lattice as well as intermolecular orientations can be characterized. The method is applied for a comparison of pentacene and para -quinquephenyl crystal structures. [source] A digital simulation of the vibration of a two-mass two-spring systemCOMPUTER APPLICATIONS IN ENGINEERING EDUCATION, Issue 3 2010Wei-Pin Lee Abstract In this study, we developed a computer program to simulate the vibration of a two-mass two-spring system by using Visual BASIC. Users can enter data for the two-mass two-spring system. The software will derive the eigenvalue problem from the input data. Then the software solves the eigenvalue problem and illustrates the results numerically and graphically on the screen. In addition, the program uses animation to demonstrate the motions of the two masses. The displacements, velocities, and accelerations of the two bodies can be shown if the corresponding checkboxes are selected. This program can be used in teaching courses, such as Linear Algebra, Advanced Engineering Mathematics, Vibrations, and Dynamics. Use of the software may help students to understand the applications of eigenvalue problems and related topics such as modes of vibration, natural frequencies, and systems of differential equations. © 2009 Wiley Periodicals, Inc. Comput Appl Eng Educ 18: 563,573, 2010; View this article online at wileyonlinelibrary.com; DOI 10.1002/cae.20241 [source] Response of unbounded soil in scaled boundary finite-element methodEARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICS, Issue 1 2002John P. Wolf Abstract The scaled boundary finite-element method is a powerful semi-analytical computational procedure to calculate the dynamic stiffness of the unbounded soil at the structure,soil interface. This permits the analysis of dynamic soil,structure interaction using the substructure method. The response in the neighbouring soil can also be determined analytically. The method is extended to calculate numerically the response throughout the unbounded soil including the far field. The three-dimensional vector-wave equation of elasto-dynamics is addressed. The radiation condition at infinity is satisfied exactly. By solving an eigenvalue problem, the high-frequency limit of the dynamic stiffness is constructed to be positive definite. However, a direct determination using impedances is also possible. Solving two first-order ordinary differential equations numerically permits the radiation condition and the boundary condition of the structure,soil interface to be satisfied sequentially, leading to the displacements in the unbounded soil. A generalization to viscoelastic material using the correspondence principle is straightforward. Alternatively, the displacements can also be calculated analytically in the far field. Good agreement of displacements along the free surface and below a prism foundation embedded in a half-space with the results of the boundary-element method is observed. Copyright © 2001 John Wiley & Sons, Ltd. [source] Localized spectral analysis on the sphereGEOPHYSICAL JOURNAL INTERNATIONAL, Issue 3 2005Mark A. Wieczorek SUMMARY It is often advantageous to investigate the relationship between two geophysical data sets in the spectral domain by calculating admittance and coherence functions. While there exist powerful Cartesian windowing techniques to estimate spatially localized (cross-)spectral properties, the inherent sphericity of planetary bodies sometimes necessitates an approach based in spherical coordinates. Direct localized spectral estimates on the sphere can be obtained by tapering, or multiplying the data by a suitable windowing function, and expanding the resultant field in spherical harmonics. The localization of a window in space and its spectral bandlimitation jointly determine the quality of the spatiospectral estimation. Two kinds of axisymmetric windows are here constructed that are ideally suited to this purpose: bandlimited functions that maximize their spatial energy within a cap of angular radius ,0, and spacelimited functions that maximize their spectral power within a spherical harmonic bandwidth L. Both concentration criteria yield an eigenvalue problem that is solved by an orthogonal family of data tapers, and the properties of these windows depend almost entirely upon the space,bandwidth product N0= (L+ 1) ,0/,. The first N0, 1 windows are near perfectly concentrated, and the best-concentrated window approaches a lower bound imposed by a spherical uncertainty principle. In order to make robust localized estimates of the admittance and coherence spectra between two fields on the sphere, we propose a method analogous to Cartesian multitaper spectral analysis that uses our optimally concentrated data tapers. We show that the expectation of localized (cross-)power spectra calculated using our data tapers is nearly unbiased for stochastic processes when the input spectrum is white and when averages are made over all possible realizations of the random variables. In physical situations, only one realization of such a process will be available, but in this case, a weighted average of the spectra obtained using multiple data tapers well approximates the expected spectrum. While developed primarily to solve problems in planetary science, our method has applications in all areas of science that investigate spatiospectral relationships between data fields defined on a sphere. [source] Computing eigenvalue bounds of structures with uncertain-but-non-random parameters by a method based on perturbation theoryINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 11 2007Huinan Leng Abstract In this paper, an eigenvalue problem that involves uncertain-but-non-random parameters is discussed. A new method is developed to evaluate the reliable upper and lower bounds on frequencies of structures for these problems. In this method the matrix in the deviation amplitude interval is considered to be a perturbation around the nominal value of the interval matrix, and the upper and lower bounds to the maximum and minimum eigenvalues of this perturbation matrix are computed, respectively. Then based on the matrix perturbation theory, the eigenvalue bounds of the original interval eigenvalue problem can be obtained. Finally, two numerical examples are provided and the results show that the proposed method is reliable and efficient. Copyright © 2006 John Wiley & Sons, Ltd. [source] Parallel computation of arbitrarily shaped waveguide modes using BI-RME and Lanczos methodsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 4 2007A. M. Vidal Abstract This paper is devoted to the parallelization of a new method for solving large, structured eigenvalue problems, which appear in the electromagnetic modal analysis of arbitrarily shaped waveguides, typically present in many modern passive devices. This new method, based on the boundary integral-resonant mode expansion (BI-RME) technique and in the Lanczos method (for solution of the eigenvalue problem), was recently proposed by the authors, showing important advantages in terms of CPU time and memory over previously used solutions. As it will be fully described in this paper, the parallel version of such a new method allows further important savings in the overall CPU computation time. Comparative benchmarks and scalability issues related to the implemented parallel algorithm are discussed. Copyright © 2006 John Wiley & Sons, Ltd. [source] Iterative modal perturbation and reanalysis of eigenvalue problemINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 4 2003X. L. Liu Abstract This paper presents an examination of the methods for the iterative modal perturbation and the application of these methods to the reanalysis of the eigenvalue problem. The iteration is based on the first-order modal perturbation. In two examples, it is shown that the iterative analysis has the advantage of accuracy over the addition of higher-order perturbations and it is an appropriate approach for the reanalysis of the eigenvalue problem in terms of accuracy and computational efficiency. Copyright © 2003 John Wiley & Sons, Ltd. [source] Computation of a few smallest eigenvalues of elliptic operators using fast elliptic solversINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 8 2001Janne Martikainen Abstract The computation of a few smallest eigenvalues of generalized algebraic eigenvalue problems is studied. The considered problems are obtained by discretizing self-adjoint second-order elliptic partial differential eigenvalue problems in two- or three-dimensional domains. The standard Lanczos algorithm with the complete orthogonalization is used to compute some eigenvalues of the inverted eigenvalue problem. Under suitable assumptions, the number of Lanczos iterations is shown to be independent of the problem size. The arising linear problems are solved using some standard fast elliptic solver. Numerical experiments demonstrate that the inverted problem is much easier to solve with the Lanczos algorithm that the original problem. In these experiments, the underlying Poisson and elasticity problems are solved using a standard multigrid method. Copyright © 2001 John Wiley & Sons, Ltd. [source] Wavelet Galerkin method in multi-scale homogenization of heterogeneous mediaINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 3 2006Shafigh Mehraeen Abstract The hierarchical properties of scaling functions and wavelets can be utilized as effective means for multi-scale homogenization of heterogeneous materials under Galerkin framework. It is shown in this work, however, when the scaling functions are used as the shape functions in the multi-scale wavelet Galerkin approximation, the linear dependency in the scaling functions renders improper zero energy modes in the discrete differential operator (stiffness matrix) if integration by parts is invoked in the Galerkin weak form. An effort is made to obtain the analytical expression of the improper zero energy modes in the wavelet Galerkin differential operator, and the improper nullity of the discrete differential operator is then removed by an eigenvalue shifting approach. A unique property of multi-scale wavelet Galerkin approximation is that the discrete differential operator at any scale can be effectively obtained. This property is particularly useful in problems where the multi-scale solution cannot be obtained simply by a wavelet projection of the finest scale solution without utilizing the multi-scale discrete differential operator, for example, the multi-scale analysis of an eigenvalue problem with oscillating coefficients. Copyright © 2005 John Wiley & Sons, Ltd. [source] A comparison of eigensolvers for large-scale 3D modal analysis using AMG-preconditioned iterative methodsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 2 2005Peter Arbenz Abstract The goal of our paper is to compare a number of algorithms for computing a large number of eigenvectors of the generalized symmetric eigenvalue problem arising from a modal analysis of elastic structures. The shift-invert Lanczos algorithm has emerged as the workhorse for the solution of this generalized eigenvalue problem; however, a sparse direct factorization is required for the resulting set of linear equations. Instead, our paper considers the use of preconditioned iterative methods. We present a brief review of available preconditioned eigensolvers followed by a numerical comparison on three problems using a scalable algebraic multigrid (AMG) preconditioner. Copyright © 2005 John Wiley & Sons, Ltd. [source] On singularities in the solution of three-dimensional Stokes flow and incompressible elasticity problems with cornersINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 4 2004A. Dimitrov Abstract In this paper, a numerical procedure is presented for the computation of corner singularities in the solution of three-dimensional Stokes flow and incompressible elasticity problems near corners of various shape. For obtaining the order and mode of singularity, a neighbourhood of the singular point is considered with only local boundary conditions. The weak formulation of this problem is approximated using a mixed u, p Galerkin,Petrov finite element method. Additionally, a separation of variables is used to reduce the dimension of the original problem. As a result, the quadratic eigenvalue problem (P+,Q+,2R)d=0 is obtained, where the saddle-point-type matrices P, Q, R are defined explicitly. For a numerical solution of the algebraic eigenvalue problem an iterative technique based on the Arnoldi method in combination with an Uzawa-like scheme is used. This technique needs only one direct matrix factorization as well as few matrix,vector products for finding all eigenvalues in the interval ,,(,) , (,0.5, 1.0), as well as the corresponding eigenvectors. Some benchmark tests show that this technique is robust and very accurate. Problems from practical importance are also analysed, for instance the surface-breaking crack in an incompressible elastic material and the three-dimensional viscous flow of a Newtonian fluid past a trihedral corner. Copyright © 2004 John Wiley & Sons, Ltd. [source] Efficient computation of order and mode of corner singularities in 3D-elasticityINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 8 2001A. Dimitrov Abstract A general numerical procedure is presented for the efficient computation of corner singularities, which appear in the case of non-smooth domains in three-dimensional linear elasticity. For obtaining the order and mode of singularity, a neighbourhood of the singular point is considered with only local boundary conditions. The weak formulation of the problem is approximated by a Galerkin,Petrov finite element method. A quadratic eigenvalue problem (P+,Q+,2R) u=0 is obtained, with explicitly analytically defined matrices P,Q,R. Moreover, the three matrices are found to have optimal structure, so that P,R are symmetric and Q is skew symmetric, which can serve as an advantage in the following solution process. On this foundation a powerful iterative solution technique based on the Arnoldi method is submitted. For not too large systems this technique needs only one direct factorization of the banded matrix P for finding all eigenvalues in the interval ,e(,),(,0.5,1.0) (no eigenpairs can be ,lost') as well as the corresponding eigenvectors, which is a great improvement in comparison with the normally used determinant method. For large systems a variant of the algorithm with an incomplete factorization of P is implemented to avoid the appearance of too much fill-in. To illustrate the effectiveness of the present method several new numerical results are presented. In general, they show the dependence of the singular exponent on different geometrical parameters and the material properties. Copyright © 2001 John Wiley & Sons, Ltd. [source] Linear stability analysis of flow in a periodically grooved channelINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 6 2003T. Adachi1 Abstract We have conducted the linear stability analysis of flow in a channel with periodically grooved parts by using the spectral element method. The channel is composed of parallel plates with rectangular grooves on one side in a streamwise direction. The flow field is assumed to be two-dimensional and fully developed. At a relatively small Reynolds number, the flow is in a steady-state, whereas a self-sustained oscillatory flow occurs at a critical Reynolds number as a result of Hopf bifurcation due to an oscillatory instability mode. In order to evaluate the critical Reynolds number, the linear stability theory is applied to the complex laminar flow in the periodically grooved channel by constituting the generalized eigenvalue problem of matrix form using a penalty-function method. The critical Reynolds number can be determined by the sign of a linear growth rate of the eigenvalues. It is found that the bifurcation occurs due to the oscillatory instability mode which has a period two times as long as the channel period. Copyright © 2003 John Wiley & Sons, Ltd. [source] Frequency/time-domain modelling of 3D waveguide structures by a BI-RME approachINTERNATIONAL JOURNAL OF NUMERICAL MODELLING: ELECTRONIC NETWORKS, DEVICES AND FIELDS, Issue 1 2002P. Arcioni This paper presents a full wave method for the determination of the mathematical model of a 3D waveguide structure in the form of the pole expansion in the s -plane of its generalized admittance matrix. The method is based on a boundary integral-resonant mode expansion approach. By the introduction of appropriate state-variables, the method leads to the pole expansion by solving a linear generalized eigenvalue problem, like in the well-known techniques used up to now in frequency/time domain modelling based on finite difference or finite element methods. With respect to these methods we have the advantage of a significant reduction in both memory allocation and computing time. Two examples show the accuracy of the results and the efficiency of the method. Copyright © 2002 John Wiley & Sons, Ltd. [source] Comment on the connected-moments polynomial approachINTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY, Issue 7 2008M. G. Marmorino Abstract Bartashevich has recently proposed two new methods for approximating eigenvalues of a Hamiltonian. The first method uses Hamiltonian moments generated from a trial function and his second method is a generalization of local energy methods. We show that the first method is equivalent to a variational one, a matrix eigenvalue problem using a Lanzcos subspace. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2008 [source] Exponentially accurate quasimodes for the time-independent Born,Oppenheimer approximation on a one-dimensional molecular systemINTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY, Issue 5 2005George A. Hagedorn Abstract We consider the eigenvalue problem for a one-dimensional molecular-type quantum Hamiltonian that has the form where h(y) is an analytic family of self-adjoint operators that has a discrete, nondegenerate electronic level ,(y) for y in some open subset of ,. Near a local minimum of the electronic level ,(y) that is not at a level crossing, we construct quasimodes that are exponentially accurate in the square of the Born,Oppenheimer parameter , by optimal truncation of the Rayleigh,Schrödinger series. That is, we construct an energy E, and a wave function ,,, such that the L2 -norm of ,, is ,,(1) and the L2 -norm of (H(,) , E,),, is bounded by , exp(,,/,2) with , > 0. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005 [source] On the asymptotic behaviour of the discrete spectrum in buckling problems for thin platesMATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 7 2006Monique Dauge Abstract We consider the buckling problem for a family of thin plates with thickness parameter ,. This involves finding the least positive multiple , of the load that makes the plate buckle, a value that can be expressed in terms of an eigenvalue problem involving a non-compact operator. We show that under certain assumptions on the load, we have , = ,,(,2). This guarantees that provided the plate is thin enough, this minimum value can be numerically approximated without the spectral pollution that is possible due to the presence of the non-compact operator. We provide numerical computations illustrating some of our theoretical results. Copyright © 2005 John Wiley & Sons, Ltd. [source] The destabilizing effect of boundary slip on Bénard convectionMATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 7 2006Mark Webber Abstract We investigate the influence of slip boundary conditions on the onset of Bénard convection in an infinite fluid layer. It is shown that the critical Rayleigh number is a decreasing function of the slip length, and therefore boundary slip is seen to have a destabilizing effect. Chebyshev-tau and compound matrix formulations for solving the eigenvalue problem are presented. Copyright © 2005 John Wiley & Sons, Ltd. [source] Sufficient conditions of non-uniqueness for the Coulomb friction problemMATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 1 2004Riad Hassani Abstract We consider the Signorini problem with Coulomb friction in elasticity. Sufficient conditions of non-uniqueness are obtained for the continuous model. These conditions are linked to the existence of real eigenvalues of an operator in a Hilbert space. We prove that, under appropriate conditions, real eigenvalues exist for a non-local Coulomb friction model. Finite element approximation of the eigenvalue problem is considered and numerical experiments are performed. Copyright © 2003 John Wiley & Sons, Ltd. [source] One-dimensional approximations of the eigenvalue problem of curved rodsMATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 12 2001Josip Tamba In this work we analyse the asymptotic behaviour of eigenvalues and eigenfunctions of the linearized elasticity eigenvalue problem of curved rod-like bodies with respect to the small thickness , of the rod. We show that the eigenfunctions and scaled eigenvalues converge, as , tends to zero, toward eigenpairs of the eigenvalue problem associated to the one-dimensional curved rod model which is posed on the middle curve of the rod. Because of the auxiliary function appearing in the model, describing the rotation angle of the cross-sections, the limit eigenvalue problem is non-classical. This problem is transformed into a classical eigenvalue problem with eigenfunctions being inextensible displacements, but the corresponding linear operator is not a differential operator. Copyright © 2001 John Wiley & Sons, Ltd. [source] The Krein,von Neumann extension and its connection to an abstract buckling problemMATHEMATISCHE NACHRICHTEN, Issue 2 2010Mark S. Ashbaugh Abstract We prove the unitary equivalence of the inverse of the Krein,von Neumann extension (on the orthogonal complement of its kernel) of a densely defined, closed, strictly positive operator, S , ,IH for some , > 0 in a Hilbert space H to an abstract buckling problem operator. In the concrete case where in L2(,; dnx) for , , ,n an open, bounded (and sufficiently regular) domain, this recovers, as a particular case of a general result due to G. Grubb, that the eigenvalue problem for the Krein Laplacian SK (i.e., the Krein,von Neumann extension of S), SKv = ,v, , , 0, is in one-to-one correspondence with the problem of the buckling of a clamped plate, (-,)2u = , (-,)u in ,, , , 0, u , H02(,), where u and v are related via the pair of formulas u = SF -1 (-,)v, v = , -1(-,)u, with SF the Friedrichs extension of S. This establishes the Krein extension as a natural object in elasticity theory (in analogy to the Friedrichs extension, which found natural applications in quantum mechanics, elasticity, etc.) (© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] Numerical methods for palindromic eigenvalue problems: Computing the anti-triangular Schur formNUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 1 2009D. Steven Mackey Abstract We present structure-preserving numerical methods for the eigenvalue problem of complex palindromic pencils. Such problems arise in control theory, as well as from palindromic linearizations of higher degree palindromic matrix polynomials. A key ingredient of these methods is the development of an appropriate condensed form,the anti-triangular Schur form. Ill-conditioned problems with eigenvalues near the unit circle, in particular near ±1, are discussed. We show how a combination of unstructured methods followed by a structured refinement can be used to solve such problems accurately. Copyright © 2008 John Wiley & Sons, Ltd. [source] Harmonic and refined Rayleigh,Ritz for the polynomial eigenvalue problemNUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 1 2008Michiel E. Hochstenbach Abstract After reviewing the harmonic Rayleigh,Ritz approach for the standard and generalized eigenvalue problem, we discuss several extraction processes for subspace methods for the polynomial eigenvalue problem. We generalize the harmonic and refined Rayleigh,Ritz approaches which lead to new approaches to extract promising approximate eigenpairs from a search space. We give theoretical as well as numerical results of the methods. In addition, we study the convergence of the Jacobi,Davidson method for polynomial eigenvalue problems with exact and inexact linear solves and discuss several algorithmic details. Copyright © 2008 John Wiley & Sons, Ltd. [source] Linear system solution by null-space approximation and projection (SNAP)NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 1 2007M. Ili Abstract Solutions of large sparse linear systems of equations are usually obtained iteratively by constructing a smaller dimensional subspace such as a Krylov subspace. The convergence of these methods is sometimes hampered by the presence of small eigenvalues, in which case, some form of deflation can help improve convergence. The method presented in this paper enables the solution to be approximated by focusing the attention directly on the ,small' eigenspace (,singular vector' space). It is based on embedding the solution of the linear system within the eigenvalue problem (singular value problem) in order to facilitate the direct use of methods such as implicitly restarted Arnoldi or Jacobi,Davidson for the linear system solution. The proposed method, called ,solution by null-space approximation and projection' (SNAP), differs from other similar approaches in that it converts the non-homogeneous system into a homogeneous one by constructing an annihilator of the right-hand side. The solution then lies in the null space of the resulting matrix. We examine the construction of a sequence of approximate null spaces using a Jacobi,Davidson style singular value decomposition method, called restarted SNAP-JD, from which an approximate solution can be obtained. Relevant theory is discussed and the method is illustrated by numerical examples where SNAP is compared with both GMRES and GMRES-IR. Copyright © 2006 John Wiley & Sons, Ltd. [source] On large-scale diagonalization techniques for the Anderson model of localizationPROCEEDINGS IN APPLIED MATHEMATICS & MECHANICS, Issue 1 2007Olaf Schenk We propose efficient preconditioning algorithms for an eigenvalue problem arising in quantum physics, namely the computation of a few interior eigenvalues and their associated eigenvectors for large-scale sparse real and symmetric indefinite matrices of the Anderson model of localization. Our preconditioning approaches for the shift-and-invert symmetric indefinite linear system are based on maximum weighted matchings and algebraic multi-level incomplete LDLT factorizations. These techniques can be seen as a complement to the alternative idea of using more complete pivoting techniques for the highly ill-conditioned symmetric indefinite Anderson matrices. Our numerical examples reveal that recent algebraic multi-level preconditioning solvers can accelerate the computation of a large-scale eigenvalue problem corresponding to the Anderson model of localization by several orders of magnitude. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] A Quadratic Eigenproblem in the Analysis of a Time Delay SystemPROCEEDINGS IN APPLIED MATHEMATICS & MECHANICS, Issue 1 2006Elias JarlebringArticle first published online: 4 DEC 200 In this work we solve a quadratic eigenvalue problem occurring in a method to compute the set of delays of a linear time delay system (TDS) such that the system has an imaginary eigenvalue. The computationally dominating part of the method is to find all eigenvalues z of modulus one of the quadratic eigenvalue problem where ,1, ,, ,m ,1 , , are free parameters and u a vectorization of a Hermitian rank one matrix. Because of its origin in the vectorization of a Lyapunov type matrix equation, the quadratic eigenvalue problem is, even for moderate size problems, of very large size. We show one way to treat this problem by exploiting the Lyapunov type structure of the quadratic eigenvalue problem when constructing an iterative solver. More precisely, we show that the shift-invert operation for the companion form of the quadratic eigenvalue problem can be efficiently computed by solving a Sylvester equation. The usefulness of this exploitation is demonstrated with an example. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] Linear waves in a symmetric equatorial channelTHE QUARTERLY JOURNAL OF THE ROYAL METEOROLOGICAL SOCIETY, Issue 624 2007C. Erlick Abstract Using a scaling that allows us to separate the effects of the gravity wave speed from those of boundary location, we reduce the equations for linear waves in a zonal channel on the equatorial beta-plane to a single-parameter eigenvalue problem of the Schrödinger type with parabolic potential. The single parameter can be written , = (,,)2/,1/2, where , = gH(2,R),2, ,, is half the channel width, g is the acceleration due to gravity, H is the typical height of the troposphere or ocean, , is the Earth's rotational frequency, and R is the Earth's radius. The Schrödinger-type equation has exact analytical solutions in the limits , , 0 and , , ,, and one can use these to write an approximate expression for the solution that is accurate everywhere to within 4%. In addition to the simple expression for the eigenvalues, the concise and unified theory also yields explicit expressions for the associated eigenfunctions, which are pure sinusoidal in the , , 0 limit and Gaussian in the , , , limit. Using the same scaling, we derive an eigenvalue formulation for linear waves in an equatorial channel on the sphere with a simple explicit formula for the dispersion relation accurate to O{(,,)2}. From this, we find that the phase velocity of the anti-Kelvin mode on the sphere differs by as much as 10% from , ,1/2. Integrating the linearized shallow-water equations on the sphere, we find that for for larger , and ,,, the phase speeds of all of the negative modes differ substantially from their phase speeds on the beta-plane. Furthermore, the dispersion relations of all of the waves in the equatorial channel on the sphere approach those on the unbounded sphere in a smooth asymptotic fashion, which is not true for the equatorial channel on the beta-plane. Copyright © 2007 Royal Meteorological Society [source] A digital simulation of the vibration of a two-mass two-spring systemCOMPUTER APPLICATIONS IN ENGINEERING EDUCATION, Issue 3 2010Wei-Pin Lee Abstract In this study, we developed a computer program to simulate the vibration of a two-mass two-spring system by using Visual BASIC. Users can enter data for the two-mass two-spring system. The software will derive the eigenvalue problem from the input data. Then the software solves the eigenvalue problem and illustrates the results numerically and graphically on the screen. In addition, the program uses animation to demonstrate the motions of the two masses. The displacements, velocities, and accelerations of the two bodies can be shown if the corresponding checkboxes are selected. This program can be used in teaching courses, such as Linear Algebra, Advanced Engineering Mathematics, Vibrations, and Dynamics. Use of the software may help students to understand the applications of eigenvalue problems and related topics such as modes of vibration, natural frequencies, and systems of differential equations. © 2009 Wiley Periodicals, Inc. Comput Appl Eng Educ 18: 563,573, 2010; View this article online at wileyonlinelibrary.com; DOI 10.1002/cae.20241 [source] New approaches for non-classically damped system eigenanalysisEARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICS, Issue 9 2005Karen Khanlari Abstract This paper presents three new approaches for solving eigenvalue problems of non-classically damped linear dynamics systems with fewer calculations than the conventional state vector approach. In the latter, the second-order differential equation of motion is converted into a first-order system by doubling the size of the matrices. The new approaches simplify the approach and reduce the number of calculations. The mathematical formulations for the proposed approaches are presented and the numerical results compared with the existing method by solving a sample problem with different damping properties. Of the three proposed approaches, the expansion approach was found to be the simplest and fastest to compute. Copyright © 2005 John Wiley & Sons, Ltd. [source] | |||||||||||||||||||||||||||||||