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Summary This paper tries to accelerate the convergence rate of the general viscous dynamic relaxation method. For this purpose, a new automated procedure for estimating the critical damping factor is developed by employing a simple variant of the Lanczos algorithm, which does not require any re-orthogonalization process. All of the computational operations are performed by simple vector–matrix multiplication without requiring any matrix factorization or inversion. Some numerical examples with geometric nonlinear behavior are analyzed by the proposed algorithm. Results show that the suggested procedure could effectively decrease the total number of convergence iterations compared with the conventional dynamic relaxation algorithms. Copyright © 2017 John Wiley & Sons, Ltd.

Summary Proper range and precision analysis play an important role in the development of fixed-point algorithms for embedded system applications. Numerical linear algebra algorithms used to find singular value decomposition of symmetric matrices are suitable for signal and image-processing applications. These algorithms have not been attempted much in fixed-point arithmetic. The reason is wide dynamic range of data and vulnerability of the algorithms to round-off errors. For any real-time application, the range of the input matrix may change frequently. This poses difficulty for constant and variable fixed-point formats to decide on integer wordlengths during float-to-fixed conversion process because these formats involve determination of integer wordlengths before the compilation of the program. Thus, these formats may not guarantee to avoid overflow for all ranges of input matrices. To circumvent this problem, a novel dynamic fixed-point format has been proposed to compute integer wordlengths adaptively during runtime. Lanczos algorithm with partial orthogonalization, which is a tridiagonalization step in computation of singular value decomposition of symmetric matrices, has been taken up as a case study. The fixed-point Lanczos algorithm is tested for matrices with different dimensions and condition numbers along with image covariance matrix. The accuracy of fixed-point Lanczos algorithm in three different formats has been compared on the basis of signal-to-quantization-noise-ratio, number of accurate fractional bits, orthogonality and factorization errors. Results show that dynamic fixed-point format either outperforms or performs on par with constant and variable formats. Determination of fractional wordlengths requires minimization of hardware cost subject to accuracy constraint. In this context, we propose an analytical framework for deriving mean-square-error or quantization noise power among Lanczos vectors, which can serve as an accuracy constraint for wordlength optimization. Error is found to propagate through different arithmetic operations and finally accumulate in the last Lanczos vector. It is observed that variable and dynamic fixed-point formats produce vectors with lesser round-off error than constant format. All the three fixed-point formats of Lanczos algorithm have been synthesized on Virtex 7 field-programmable gate array using Vivado high-level synthesis design tool. A comparative study of resource usage and power consumption is carried out. Copyright © 2015 John Wiley & Sons, Ltd.

SUMMARY This paper describes the development of explicit time stepping methods for linear PDEs that are specifically designed to cope with the stiffness of the system of ODEs that results from spatial discretization. As stiffness is caused by the contrasting behavior of coupled components of the solution, it is proposed to adopt a componentwise approach in which each coefficient of the solution in an appropriate basis is computed using an individualized approximation of the solution operator. This has been accomplished by Krylov subspace spectral (KSS) methods, which use techniques from ‘matrices, moments and quadrature’ to approximate bilinear forms involving functions of matrices via block Gaussian quadrature rules. These forms correspond to coefficients with respect to the chosen basis of the application of the solution operator of the PDE to the solution at an earlier time. In this paper, it is proposed to substantially enhance the efficiency of KSS methods through the prescription of quadrature nodes on the basis of asymptotic analysis of the recursion coefficients produced by block Lanczos iteration for each Fourier coefficient as a function of frequency. The potential of this idea is illustrated through numerical results obtained from the application of the modified KSS methods to diffusion equations and wave equations. Copyright © 2012 John Wiley & Sons, Ltd.

The specification of a correct background-error covariance matrix is a key issue in data assimilation schemes. The Ensemble Kalman Filter (EnKF) aims at providing simulations of analysis and background errors and then gives a way to determine this background-error covariance matrix. The EnKF can be transposed to variational ensemble assimilation, where a set of perturbed variational analyses are performed. In this case, however, there is an evident important additional cost associated with the use of multiple minimizations. The aim of the paper is to investigate different techniques to reduce the cost of the multiple minimizations that have to be performed. In particular, the use is investigated of a preconditioning technique based on Ritz eigenpairs resulting from a first minimization performed by a combined Lanczos/conjugate-gradient algorithm. The possibility is also studied of improving the starting point of a new perturbed solution, with Lanczos vectors issued from a single prior unperturbed or perturbed minimization. This appears to provide a first significant reduction in the cost of the new minimization. Finally, a new approach is proposed to generalize the previous idea to the use of multiple sets of Lanczos vectors issued from an ensemble of perturbed assimilations. The application of this procedure to a simplified analysis problem shows encouraging results, as it appears to be a possible way for reducing the global cost of an ensemble variational assimilation. Moreover, this seems to provide an efficient strategy for parallelizing such an ensemble variational assimilation but also the deterministic variational assimilation itself. Copyright © 2012 Royal Meteorological Society

This article reviews methods for computing ro-vibrational spectra of van der Waals molecules. Due to the presence of large-amplitude motion, calculations often play an important role in assigning and understanding the spectra of van der Waals molecules. Fortunately, it is possible to make usefully accurate calculations because important parts of the spectrum can be understood by doing calculations that omit the intramolecular coordinates. In this article, we present new ideas for deriving kinetic energy operators and discuss choosing basis functions and doing the matrix–vector products that are required to obtain a spectrum using the Lanczos algorithm. © 2011 John Wiley & Sons, Ltd. WIREs Comput Mol Sci 2011 1 952–963 DOI: 10.1002/wcms.73

Recent advance in high performance computing (HPC) resources has opened the possibility to expand the scope of density functional theory (DFT) simulations toward large and complex molecular systems. This work proposes a numerically robust method that enables scalable diagonalizations of large DFT Hamiltonian matrices, particularly with thousands of computing CPUs (cores) that are usual these days in terms of sizes of HPC resources. The well-known Lanczos method is extensively refactorized to overcome its weakness for evaluation of multiple degenerate eigenpairs that is the substance of DFT simulations, where a multilevel parallelization is adopted for scalable simulations in as many cores as possible. With solid benchmark tests for realistic molecular systems, the fidelity of our method are validated against the locally optimal block preconditioned conjugated gradient (LOBPCG) method that is widely used to simulate electronic structures. Our method may waste computing resources for simulations of molecules whose degeneracy cannot be reasonably estimated. But, compared to LOBPCG method, it is fairly excellent in perspectives of both speed and scalability, and particularly has remarkably less (< 10%) sensitivity of performance to the random nature of initial basis vectors. As a promising candidate for solving electronic structures of highly degenerate systems, the proposed method can make a meaningful contribution to migrating DFT simulations toward extremely large computing environments that normally have more than several tens of thousands of computing cores.

The Lanczos algorithm produces partial eigensolutions at the two extreme ends of the eigenspectrum. The inverse Lanczos algorithm with shift s produces partial eigensolutions nearest to the shift. It generates iteratively one Lanczos vector at a time and the elements of a tridiagonal matrix whose eigenvalues approximate the required eigenvalues. The Lanczos vectors are orthogonal and span the same subspace of the required eigenvectors theoretically. Orthogonality of the Lanczos vectors is lost during iteration owing to the attraction of the round-off errors by some initially found Lanczos vectors, and reorthogonalization is required. Full reorthogonalization by the Gram-Schmidt process is expensive and unstable. An economical and stable method of reorthogonalization by subspace iteration is introduced. The Lanczos vectors without reorthogonalization are taken as initial trial vectors in the subspace method to produce the required eigensolutions. The operation counts of the combined method compare favorably with the Lanczos method with full reorthogonalization for the same accuracy of solutions. Application of the Sturm sequence check is recommended to make sure that no solutions are missed in the required spectrum.

For many problems in structural acoustics, it is desired to obtain solutions at many frequencies over a large range in the frequency domain. A reduced-order multifrequency algorithm based on matrix Pade approximation, using the matrix Pade-via-Lanczos (MPVL) connection, has been previously used to solve both exterior and interior acoustic problems. However, the method is not guaranteed to give the correct solution across the entire frequency region of interest, but only locally around a reference frequency. An adaptive frequency windowing scheme is introduced to address this shortcoming for practical application of this method. The application of this algorithm to tightly coupled problems in interior structural acoustics is presented.

The goal of our paper is to demonstrate the cost-effective use of the Lanczos method for estimating the critical time step in an explicit, transient dynamics code. The Lanczos method can provide a significantly larger estimate for the critical time-step than an element-based method (the typical scheme). However, the Lanczos method represents a more expensive method for calculating a critical time-step than element-based methods. Our paper shows how the additional cost of the Lanczos method can be amortized over a number of time steps and lead to an overall decrease in run-time for an explicit, transient dynamics code. We present an adaptive hybrid scheme that synthesizes the Lanczos-based and element-based estimates and allows us to run near the critical time-step estimate provided by the Lanczos method.

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.

We demonstrate that the combination of simply contracted basis functions and the Lanczos algorithm yields an extremely efficient method for computing vibrational energy levels. We discuss ideas and present some results for HOOH and CH4. The basis functions we use are products of eigenfunctions of reduced-dimension Hamiltonians obtained by freezing coordinates at equilibrium. The basis functions represent the desired wave functions well yet are simple enough that matrix–vector products may be evaluated efficiently. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004

We study the Lanczos method for solving symmetric linear systems with multiple right-hand sides. First, we propose a numerical method of implementing the Lanczos method, which can provide all approximations to the solution vectors of the remaining linear systems. We also seek possible application of this algorithm for solving the linear systems that occur in continuation problems. Sample numerical results are reported. Copyright ? 2002 John Wiley & Sons, Ltd.

An efficient dynamic analysis method using the modified Lanczos co-ordinates is presented. The proposed method is obtained by applying the modified Lanczos algorithm using Lanczos vectors that satisfy the stiffness-orthonormality condition to the conventional Lanczos co-ordinates method. The modified Lanczos co-ordinates method is more efficient than the conventional method in the case of structures under multi-input loads. The effectiveness of the modified Lanczos co-ordinates method is verified by analysing a numerical example. Copyright © 2003 John Wiley & Sons, Ltd.

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.

Applications such as the modal analysis of structures and acoustic cavities require a number of eigenvalues and eigenvectors of large-scale Hermitian eigenvalue problems. The most popular method is probably the spectral transformation Lanczos method. An important disadvantage of this method is that a change of pole requires a complete restart. In this paper, we investigate the use of the rational Krylov method for this application. This method does not require a complete restart after a change of pole. It is shown that the change of pole can be considered as a change of Lanczos basis. The major conclusion of this paper is that the method is numerically stable when the poles are chosen in between clusters of the approximate eigenvalues. Copyright (C) 2001 John Wiley & Sons, Ltd.

In this paper, Lanczos and Arnoldi reduction methods as the special cases of the generalized Hessenberg method are briefly reviewed. Attention is paid to the effect of symmetry of matrices on the behaviour of the reduction schemes, such as serious numerical breakdown. Based on the summation decomposition of matrices, two structures of the upper Hessenberg form of a general unsymmetric matrix and their relationship are revealed, in terms of which, Arnoldi reduction schemes for unsymmetric matrices can be reformulated in two respective forms. The relationship between the reformulated reduction scheme and the current Lanczos schemes for skew and symmetric matrices are also discussed. Copyright © 1999 John Wiley & Sons, Ltd.

This paper describes a reduced finite element formulation for mildly non-linear transient heat transfer analysis problems based on the Lanczos algorithm. In the proposed reduced formulation, all material non-linearities of irradiation boundary elements are included using the pseudo-force method. Numerical time integration of the reduced formulation is conducted by the Galerkin method. The results of numerical examples demonstrate the applicability and the accuracy of the proposed method for transient heat transfer analysis when restricted to mildly non-linear situations. In particular, the example of a bridge under environmental thermal condition shows that the proposed method is about 20 times faster, without loss of solution stability and accuracy, than a direct integration method for the unreduced formulation. Copyright © 1999 John Wiley & Sons, Ltd.

In this paper, a solution method is presented to solve the eigenproblem arising in the dynamic analysis of non-proportional damping systems with symmetric matrices. The method is based on the Lanczos method to generate one pair of Krylov subspaces consisting of trial vectors, which is then used to reduce a large eigenvalue problem into a much smaller one. The method retains the n order quadratic eigenproblem, without employing the method of matrix augmentation traditionally used to cast the problem as a linear eigenproblem of order 2n. In this process, the method preserves the sparseness and symmetry of the system matrices and does not invoke complex arithmetic; thus making it very economical for use in solving large problems. Numerical results are presented to demonstrate the efficiency and accuracy of the proposed method.

For a given real n x n matrix A and initial vectors v(1) and w(1), we examine the sensitivity of the tridiagonal matrix T and the biorthogonal sets of vectors of the Lanczos reduction to small changes in A, v(1) and w(1). We also consider the sensitivity of the developing Krylov subspaces. Copyright (C) 1999 John Wiley & Sons, Ltd.

Two Lanczos subspace propagation techniques are discussed in this work and compared with the Chebyshev method applied to the original Hamiltonian matrix. Both procedures involve the use of a reduced propagator in the Lanczos subspace to calculate the solution to the time-dependent Schrodinger equation but differ in the way the propagator is evaluated. The LSC (Lanczos subspace Chebyshev) expresses the propagator in terms of Chebyshev polynomials that are functions of the tridiagonal Hamiltonian matrix in the Lanczos space. In contrast, the LSV (Lanczos subspace variational) is implemented by solving the eigenproblem in the Lanczos subspace and then performing a variational expansion of the propagator in the M-dimensional eigenvector space. Although the LSV is the same as the reduced propagator scheme proposed by Park and Light, in the present study the LSV is implemented as a one-step long-time propagator. As a numerical example, the interaction of a molecule with a strong laser pulse is investigated. The Hamiltonian is explicitly time dependent in this case, and thus the stationary formalism is employed in this work to solve the time-dependent Schrodinger equation. Application of either the LSC or LSV yields a wave function in the M-dimensional Lanczos subspace. Nonetheless, the transition amplitudes computed from this wave function are in excellent agreement with those calculated by direct application of the Chebyshev method in the original space used to define the Hamiltonian matrix. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 67: 273–285, 1998

A spectral filtering method reported recently [Faraday Disc. 102, 17 (1996)], which utilizes the optimal expansion of the Green operator in a finite Lanczos subspace, is used to construct a new scheme for the calculation of bound vibrational states by filter diagonalization. The method requires storage of only two real vectors in the primary representation, i.e., those required to generate the Lanczos subspace. Calculations for the HO 2 molecule show that the new scheme efficiently generates converged eigenvalues within a nominated energy window which may be scanned through the bound spectrum, and utilizes just a single Lanczos subspace (generated once only). It not only is more efficient than the regular Lanczos algorithm for computing high-lying eigenstates, but also has the interesting property of eliminating duplicated and ghost eigenvalues, which can cause problems in interpretation of the regular Lanczos spectrum.

We study the Lanczos type methods for continuation problems. First we indicate how the symmetric Lanczos method may be used to solve both positive definite and indefinite linear systems. Furthermore, it can be used to monitor the simple bifurcation points on the solution curve of the eigenvalue problems. This includes computing the minimum eigenvalue, the minimum singular value, and the condition number of the partial tridiagonalizations of the coefficient matrices. The Ritz vector thus obtained can be applied to compute the tangent vector at the bifurcation point for branch-switching. Next, we indicate that the block or band Lanczos method can be used to monitor the multiple bifurcations as well as to solve the multiple right hand sides. We also show that the unsymmetric Lanczos method can be exploited to compute the minimum eigenvalue of a nearly symmetric matrix, and therefore to detect the simple bifurcation point as well. Some preconditioning techniques are discussed. Sample numerical results are reported. Our test problems include second order semilinear elliptic eigenvalue problems. © 1997 by John Wiley & Sons, Ltd.

Summary In the present paper, we describe an adaptive modified rational global Lanczos algorithm for model-order reduction problems using multipoint moment matching-based methods. The major problem of these methods is the selection of some interpolation points. We first propose a modified rational global Lanczos process and then we derive Lanczos-like equations for the global case. Next, we propose adaptive techniques for choosing the interpolation points. Second-order dynamical systems are also considered in this paper, and the adaptive modified rational global Lanczos algorithm is applied to an equivalent state space model. Finally, some numerical examples will be given.

The discrete variable representation method is applied to the determination of the rotation-vibration energy levels of the fundamental electronic state of NO2. The Hamiltonian is expressed in Johnson hyperspherical coordinates and developed on a DVR basis for each internal coordinate, while parity-adapted linear combinations of Wigner functions are used to describe the rotational motion. The diagonalization of the Hamiltonian matrix is performed using the Lanczos algorithm for large symmetric and Hermitian matrices. Results for rovibrational states up to J = 11 for the first five vibrational energy levels are presented. © 1997 John Wiley & Sons, Inc.

In the present paper, we analyse the computational performance of the Lanczos method and a recent optimization technique for the calculation of the p (p ≤ 40) leftmost eigenpairs of generalized symmetric eigenproblems arising from the finite element integration of elliptic PDEs. The accelerated conjugate gradient method is used to minimize successive Rayleigh quotients defined in deflated subspaces of decreasing size. The pointwise Lanczos scheme is employed in combination with both the Cholesky factorization of the stiffness matrix and the preconditioned conjugate gradient method for evaluating the recursive Lanczos vectors. The three algorithms are applied to five sample problems of varying size up to almost 5000. The numerical results show that the Lanczos approach with Cholesky triangularization is generally faster (up to a factor of 5) for small to moderately large matrices, while the optimization method is superior for large problems in terms of both storage requirement and CPU time. In the large case, the Lanczos–Cholesky scheme may be very expensive to run even on modern quite powerful computers.

A damped system eigenvalue analysis of acoustical cavities using the boundary element method is presented. The acoustic boundary element eigenproblem formulation found in the literature is extended to include sound absorption in acoustical cavities. A dissipative term is included in the eigenvalue matrix equation to account for boundary absorption. The resulting damped system eigenvalue problem is solved using a new Lanczos subspace algorithm for quadratic eigenproblems. Since the boundary element matrices are unsym-metric, the Lanczos algorithm presented is in its most general form for unsymmetric quadratic eigenprob-lems. Examples are presented to show the application of the method in computing the eigenfrequencies of acoustic cavities with sound absorption.

We present a storage-efficient and robust algorithm for the computation of eigenvectors of large sparse symmetrical matrices using a Lanczos scheme. The algorithm is based upon a linear combination of Lanczos vectors (LCLV) with a variable iteration depth. A simple method is given to determine the iteration depth before the eigenvector computation is performed. Test calculations are reported for tight-binding models of ordered and disordered 2-D systems. The algorithm turns out to be reliable if an eigenvector residual less than 10−4 is required. We report benchmarks for various computers. Possible fields of application are discussed. © 1993 John Wiley & Sons, Inc.

An efficient numerical algorithm is developed to solve the quadratic eigenvalue problems arising in the dynamic analysis of damped structural systems. The algorithm can even be applied to structural systems with non-symmetric matrices. The algorithm is based on the use of Arnoldi's method to generate a Krylov subspace of trial vectors, which is then used to reduce a large eigenvalue problem to a much smaller one. The reduced eigenvalue problem is solved and the solutions are used to construct approximate solutions to the original large system. In the process, the algorithm takes full advantage of the sparseness and symmetry of the system matrices and requires no complex arithmetic, therefore, making it very economical for use in solving large problems. The numerical results from test examples are presented to demonstrate that a large fraction of the approximate solutions calculated are very accurate, indicating that the algorithm is highly effective for extracting a number of vibration modes for a large dynamic system, whether it is lightly or heavily damped.

Generalized conjugate gradient methods are often used to solve large sparse non-symmetric linear systems. Such systems often arise in the numerical solution of convection–diffusion problems. The Lanczos/Orthodir method proposed by Jea and Young suffers the roundoff error problem. In this paper a modified Lanczos/Orthodir method is considered. The modifications are based on Paige's computational procedure and on the adaptive restarting procedure. Numerical results showed that the modified Lanczos/Orthodir method is more stable than the original version. Furthermore, with the adaptive restarting scheme the method performed much better than the Lanczos/Orthomin method.

A three-dimensional potential energy surface of the electronic ground state of ZnH2 () molecule is constructed from more than 7500 ab initio points calculated at the internally contracted multireference configuration interaction with the Davidson correction (icMRCI+Q) level employing large basis sets. The calculated relative energies of various dissociation reactions are in good agreement with the previous theoretical/experimental values. Low-lying vibrational energy levels of ZnH2, ZnD2, and HZnD are calculated on the three-dimensional potential energy surface using the Lanczos algorithm, and found to be in good agreement with the available experimental band origins and the previous theoretical values. © 2009 Wiley Periodicals, Inc. J Comput Chem, 2010

Application of the two-sided Lanczos recursion to the unsymmetric generalized eigenvalue problem is presented. The system matrices are real and unsymmetric. Therefore, the recursions are performed in real arithmetic and complex arithmetic is employed in the QR algorithm used to extract the eigenvalues of the transformed tridiagonal matrix. The biorthonormal transformation of the unsymmetric generalized eigenvalue problem is considered in detail with appropriate proofs presented in Appendices. Issues relating to the computer implementation of the unsymmetric generalized eigenvalue problem are discussed. The example problems solved demonstrate the working of the algorithm in extracting the complex and/or real eignevalues of an unsymmetric system of matrices. Also, the algorithm is applied to extract a few of the eigenvalues of a large fluid-structure interaction problem, and the results are compared with the eigenfrequencies extracted by an unsymmetric subspace iteration procedure presented in the literature.

A three-dimensional global potential energy surface for the electronic ground state of HXeBr molecule is constructed from more than 4200 ab initio points. These points are generated using an internally contracted multi-reference configuration interaction method with the Davidson correction (icMRCI + Q) and large basis sets. The stabilities and dissociation barriers are identified from the potential energy surfaces. The three-body dissociation channel is found to be the dominate dissociation channel for HXeBr. Based on the obtained potentials, low-lying vibrational energy levels of HXeBr calculated using the Lanczos algorithm is found to be in good agreement with the available experimental band origins.

An unsymmetric block Lanczos algorithm has been employed for the dynamic analysis of a large system which has arbitrary damping and/or repeated (or closely spaced) eigenvalues. In the algorithm development, the right and left Lanczos vectors are all theoretically biorthogonal to each other. However, these vectors may lose the biorthogonality owing to cancellation and roundoff errors. For the unsymmetric case there can be a breakdown, even without numerical errors. This paper describes computational techniques which have led to a robust unsymmetric block Lanczos algorithm.

Real and complex Ritz vector bases for dynamic analysis of large linear systems with non-proportional damping are presented and compared. Both vector bases are generated utilizing load dependent vector algorithms that employ recurrence equations analogous to the Lanczos algorithm. The choice of static response to fixed spatial loading distribution, as a starting vector in recurrence equations, is motivated by the static correction concept. Different phases of dynamic response analysis are compared with respect to computational efficiency and accuracy. It is concluded that the real vector basis approach is approximately eight times more efficient than the complex vector basis approach. The complex vector basis has some advantages with respect to accuracy, if the excitation is of piecewise linear form, since the exact solution can be utilized. In addition, it is demonstrated that both Ritz vector bases, real and complex, possess superior accuracy over the adequate eigenvector bases.

The Lanczos algorithm has proved to be a powerful solution method not only for finding the eigenvalues but for solving linear systems of equations. In this work a new implementation of the algorithm is presented for solving linear systems of equations with a sequence of right-hand sides. The versions of the method proposed in the past treat the right-hand side vectors successively by keeping the tridiagonal matrix and the orthonormal basis in fast or secondary storage. The new technique handles all approximations to the solution vectors simultaneously without the necessity for keeping the tridiagonal matrix or the orthonormal basis in fast or secondary storage. Thus, when the first solution vector has converged to a required accuracy good approximations to the remaining solution vectors have simultaneously been obtained. It then takes fewer iterations to reach the final accuracy by working separately on each of the remaining vectors.

When dealing with quantum many-body systems one is faced with problems growing exponentially in the number of particles to be considered. To overcome this curse of dimensionality one has to consider representation formats which scale only polynomially. Physicists developed concepts like matrix product states (MPS) to represent states of interest and formulated algorithms such as the density matrix renormalization group (DMRG) to find such states. We consider the standard Lanczos algorithm and formulate it for vectors given in the MPS format. It turns out that a restarted version which includes a projection onto the MPS manifold gives the same approximation quality as the well-established DMRG method. Moreover, this variant is more flexible and provides more information about the spectrum. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)

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 the largest sparse real and symmetric indefinite matrices of the Anderson model of localization. We compare the Lanczos algorithm in the 1987 implementation by Cullum and Willoughby with the shift-and-invert techniques in the implicitly restarted Lanczos method and in the Jacobi-Davidson method. Our preconditioning approaches for the shift-and-invert symmetric indefinite linear system are based on maximum weighted matchings and algebraic multilevel incomplete $LDL^T$ 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. We demonstrate the effectiveness and the numerical accuracy of these algorithms. Our numerical examples reveal that recent algebraic multilevel preconditioning solvers can accelerative the computation of a large-scale eigenvalue problem corresponding to the Anderson model of localization by several orders of magnitude., Comment: 9 .eps figures, submitted to SIAM J. Sci. Comp., SIAM LaTeX styles (included), high quality figures can be obtained from http://eprints.csc.warwick.ac.uk/archive/00000110/

This paper is concerned with the development of an efficient eigenproblem solution algorithm and an associated computer program for the economical solution of the free vibration problem of complex practical spinning structural systems. Thus, a detailed description of a newly developed block Lanczos procedure is presented in this paper that employs only real numbers in all relevant computations and also fully exploits sparsity of associated matrices. The procedure is capable of computing multiple roots and proves to be most efficient compared to other existing similar techniques.

In this paper the Choleski factorization of the mass or the stiffness matrix and its use in reduction of the general free vibration equation to a standard eigenproblem prior to the application of the Lanczos method is described, and their effects on the necessary computing time in determination of the frequencies are discussed.

In this paper we consider solution of the eigen problem in structural analysis using a recent version of the Lanczos method and the subspace method. The two methods are applied to examples and we conclude that the Lanczos method has advantages which are too good to overlook.

In this paper Kron's primitive composite system matrix is shown to be reducible to a symmetric indefinite matrix. This matrix, although never formed explicitly, can then be transformed to a tridiagonal matrix of reduced order by the Lanczos algorithm. Eigenvalue solutions of this partially tridiagonalized matrix give very good approximations to the eigenvalues of the composite system. The method has multiplication counts which are over 90 per cent lower than when the (condensed) Kron matrix is solved by the Newton-Raphson iteration applied to the Kron scalar equation. Numerical examples illustrating the Kron-Lanczos method when solving for the natural frequencies of frames with large numbers of degrees of freedom are presented.

In this paper we extend the Lanczos algorithm for the dynamic analysis of structures7 to systems with general matrix coefficients. The equations of dynamic equilibrium are first transformed to a system of first order differential equations. Then the unsymmetric Lanczos method is used to generate two sets of vectors. These vectors are used in a method of weighted residuals to reduce the equations of motion to a small unsymmetric tridiagonal system. The algorithm is further simplified for systems of equations with symmetric matrices. By appropriate choice of the starting vectors we obtain an implementation of the Lanczos method that is remarkably close to that in Reference 7, but generalized to the case with indefinite matrix coefficients. This simplification eliminates one of the sets of vectors generated by the unsymmetric Lanczos method and results in a symmetric tridiagonal, but indefinite, system. We identify the difficulties that may arise when this implementation is applied to problems with symmetric indefinite matrices such as vibration of structures with velocity feedback control forces which lead to symmetric damping matrices. This approach is used to evaluate the vibration response of a damped beam problem and a space mast structure with symmetric damping matrix arising from velocity feedback control forces. In both problems, accurate solutions were obtained with as few as 20 Lanczos vectors.

In this work, a modal solution method for transient heat conduction utilizing a co-ordinate transformation matrix generated by the Lanczos algorithm is presented. The special characteristics of this co-ordinate transformation are also discussed. The comparisons made in the NASA Insulation Test problem analysis shown that this approach is more cost-effective than direct solutions, without loss of accuracy.

A procedure for deriving the Lanczos vectors is explained and their use in structural dynamics analysis as an alternative to modal co-ordinates is discussed. The vectors are obtained by an inverse iteration procedure in which orthogonality is imposed between the vectors resulting from successive iteration cycles. Using these Lanczos vectors the equations of motion are transformed to tridiagonal form, which provides for a very efficient time-stepping solution. The effectiveness of the method is demonstrated by a numerical example.

A method for reducing the order of a dynamical model of a large structure with arbitrary damping is developed analytically and demonstrated. A Lanczos algorithm is described which can reduce square unsymmetric system matrices to block-tridiagonal form, and a procedure for defining the reduced-order model from the right and left Lanczos vectors is outlined. Results for sample problems involving the 8-DOF FEM model of a beam-rotor assembly subjected to random and stepped external forces are presented in extensive graphs and briefly characterized.