a filtered lanczos procedure for extreme and interior eigenvalue problems when combined with krylov projection methods polynomial filtering can provide a powerful method for extracting extreme or interior eigenvalues of large sparse matrices this general approach can be quite efficient in the situation when a large number of eigenvalues is sought however its competitiveness depends critically on a good implementation this paper presents a technique based on such a combination to compute a group of extreme or interior eigenvalues of a real symmetric or complex hermitian matrix the technique harnesses the effectiveness of the lanczos algorithm with partial reorthogonalization and the power of polynomial filtering numerical experiments indicate that the method can be far superior to competing algorithms when a large number of eigenvalues and eigenvectors is to be computed

an improved lanczos algorithm for solving ill conditioned linear equations abstract in the existing lanczos algorithms for solving systems of linear equations the estimate for the residual is effective for well conditioned systems however in actual implementation on a computer we find that the estimate is no longer reliable for ill conditioned cases we first analyze in theory this observation then develop an improved lanczos algorithm numerical examples are also given to demonstrate the effectiveness of the present algorithm

some perspectives on the eigenvalue problem this expository paper explores the relationships among a number of algorithms for solving eigenvalue problems including the power method subspace iteration the qr algorithm and the arnoldi and symmetric lanczos algorithms the symmetric lanczos algorithm is shown to be identical to the three term recursion stieltjes procedure for computing orthogonal polynomials with respect to a measure on the real line the connection between measures on the line and symmetric tridiagonal jacobi matrices is investigated if such a matrix is transformed by a step of the qr algorithm there is a corresponding transformation in the measure the tridiagonal matrices are also exploited for the construction of gaussian quadrature formulas for measures on the line the developments on the real line are replicated with suitable modifications on the unit circle via lanczos like procedures for unitary operators the best known procedure of this type is the recursion of szego for computing orthogonal polynomials on the

a conjugate gradient algorithm for the treatment of multiple incident electromagnetic fields an iterative method based on the conjugate gradient cg algorithm is developed for the efficient treatment of equations involving multiple excitations examples show that significant time savings can be obtained as compared to treating each excitation individually with the conjugate gradient algorithm however these savings are not obtained without the drawback of increased memory requirements to store the additional excitations residuals and solutions the efficiency of this algorithm tends to increase as additional excitations are added

multidimensional esprit for damped and undamped signals algorithm computations and perturbation analysis in this paper we present and analyze the performance of multidimensional esprit n d esprit method for estimating parameters of n d superimposed damped and or undamped exponentials n d esprit algorithm is based on low rank decomposition of multilevel hankel matrices formed by the n d data in order to reduce the computational complexity for large signals we propose a fast n d esprit using truncated singular value decomposition svd then through a first order perturbation analysis we derive simple expressions of the variance of the estimates in n d multiple tones case these expressions do not involve the factors of the svd we also derive closed form expressions of the variances of the complex modes frequencies and damping factors estimates in the n d single tone case computer results are presented to show effectiveness of the fast version of n d esprit and verify theoretical expressions

high performance domainwise parallel direct solver for large scale structural analysis large scale structural analysis using a finite element method requires a high performance and efficient parallel algorithm that is scalable in terms of both performance and storage most of the research for large scale parallel structural analysis has focused on iterative solution methods because they are much easier to parallelize in contrast direct solution methods have generally been considered in adequate for large scale finite element computations because of many difficulties and disadvantages for carrying out large scale problems however direct solution methods are still generally preferred due to their ease of use and the numerical robustness that guarantees the solution to be obtained within an estimated time without failure therefore for the general application of large scale structural analysis to a wide range of problems an efficient parallel direct solution method that has scalability comparable to that of iterative methods is proposed a new parallel direct solver for large scale finite element analysis the domainwise multifrontal solver is proposed by realizing domainwise parallelism for a direct solver by the use of the proposed solver with our own structural analysis code good scalability can be shown as a direct method and can solve the largest problem ever known to be solved by direct solvers

beyond automated multilevel substructuring domain decomposition with rational filtering this paper proposes a rational filtering domain decomposition technique for the solution of large and sparse symmetric generalized eigenvalue problems the proposed technique is purely algebraic an

analysis of the symmetric lanczos algorithm with reorthogonalization methods we present an error analysis of the symmetric lanczos algorithm in finite precision arithmetic the loss of orthogonality among the computed lanczos vectors is explained with the help of a recurrence formula a backward error analysis shows that semiorthogonality among the lanczos vectors is enough to guarantee the accuracy of the computed quantities up to machine precision the results of this analysis are then extended to the more general case of the lanczos algorithm with a semiorthogonalization strategy based on the recurrence formula a new reorthogonalization method called partial reorthogonalization is introduced we show that both partial reorthogonalization and selective orthogonalization as introduced by parlett and scott 15 are semiorthogonalization strategies finally we discuss the application of our results to the solution of linear systems of equations and to the eigenvalue problem

application of the lanczos algorithm to the solution of the groundwater flow equation the solution to the finite element matrix differential equations resulting from the discretization of the groundwater flow equation is normally carried out by a finite difference approximation to the time derivative the total computational effort in solving a fluid flow problem is then directly related to the number of unknowns and the number of time steps required to obtain accurate and stable solutions the lanczos algorithm uses an orthogonal matrix transformation to reduce the finite element equations to a much smaller tridiagonal system of first order differential equations this new system can be solved by a standard tridiagonal solution algorithm with very little computational effort a matrix vector multiplication is then used to obtain the original solution at desired time steps the algorithm is used to accurately simulate the drawdown of synthetic two dimensional aquifers including ones with substantial hydraulic conductivity contrasts the method affords an efficient means of solving large problems particularly when time durations are long

a lanczos method for approximating composite functions abstract we seek to approximate a composite function h x g f x with a global polynomial the standard approach chooses points x in the domain of f and computes h x at each point which requires an evaluation of f and an evaluation of g we present a lanczos based procedure that implicitly approximates g with a polynomial of f by constructing a quadrature rule for the density function of f we can approximate h x using many fewer evaluations of g the savings is particularly dramatic when g is much more expensive than f or the dimension of x is large we demonstrate this procedure with two numerical examples i an exponential function composed with a rational function and ii a navier stokes model of fluid flow with a scalar input parameter that depends on multiple physical quantities

deflated and restarted symmetric lanczos methods for eigenvalues and linear equations with multiple right hand sides a deflated restarted lanczos algorithm is given for both solving symmetric linear equations and computing eigenvalues and eigenvectors the restarting limits the storage so that finding eigenvectors is practical meanwhile the deflating from the presence of the eigenvectors allows the linear equations to generally have good convergence in spite of the restarting some reorthogonalization is necessary to control roundoff error and several approaches are discussed the eigenvectors generated while solving the linear equations can be used to help solve systems with multiple right hand sides experiments are given with large matrices from quantum chromodynamics that have many right hand sides

the lanczos algorithm with partial reorthogonalization the lanczos algorithm is becoming accepted as a powerful tool for finding the cigenvalues and for solving linear systems of equations any practical implementation of the algorithm suffers however from roundoff errors which usually cause the lanczos vectors to lose their mutual orthogonality in order to maintain some level of orthogonality full reorthogonalization fro and selective orthogonalization so have been used in the past as a remedy here partial reorthogonalization pro is proposed as a new method for maintain ing semiorthogonality among the lanczos vectors pro is based on a simple recurrence which allows us to monitor the loss of orthogonality among the lanczos vectors directly wvithout computing the inner products based on the information from the recurrence reorthogonalizations occur only when necessary thus substantial savings are made as compared to fro in some numerical examples we apply the lanczos algorithm with pro to the solution of large symmetric systems of linear equations and show that it is a robust and efficient algorithm for maintaining semiorthogonality among the lanczos vectors the results obtained compare favorably with the conjugate gradient method

numerical methods for electronic structure calculations of materials the goal of this article is to give an overview of numerical problems encountered when determining the electronic structure of materials and the rich variety of techniques used to solve these problems the paper is intended for a diverse scientific computing audience for this reason we assume the reader does not have an extensive background in the related physics our overview focuses on the nature of the numerical problems to be solved their origin and the methods used to solve the resulting linear algebra or nonlinear optimization problems it is common knowledge that the behavior of matter at the nanoscale is in principle entirely determined by the schrodinger equation in practice this equation in its original form is not tractable successful but approximate versions of this equation which allow one to study nontrivial systems took about five or six decades to develop in particular the last two decades saw a flurry of activity in developing effective software one of the main practical variants of the schrodinger equation is based on what is referred to as density functional theory dft the combination of dft with pseudopotentials allows one to obtain in an efficient way the ground state configuration for many materials this article will emphasize pseudopotential density functional theory but other techniques will be discussed as well

on the real convergence rate of the conjugate gradient method abstract we present a parametrized class of matrices for which the rate of convergence of the conjugate gradient method varies greatly with the parameter and does not appreciably depend on the algorithm implementation a small change in the eigenvalue distribution can lead to a large change in the sensitivity of cg to rounding errors a theorem is proved which gives a necessary and sufficient condition for ordering exact arithmetic cg processes for systems with different spectra according to the energy norm of the error theorems 4 1 and 4 2 continue paige s and greenbaum s work

new methods for calculations of the lowest eigenvalues of the real symmetric generalized eigenvalue problem a new iterative method based on a newton correction vector for extension of the krylov subspace its diagonal and band versions are proposed for calculation of selected lowest eigenvalues and corresponding eigenvectors of the generalized symmetric eigenvalue problem additionally diagonal and band jacobi davidson methods are introduced test calculations show that the new iterative method usually converges faster than quadratic near a solution the new iterative method along with its band version uses a smaller number of iterative steps to obtain a solution compared to the jacobi davidson band jacobi davidson method and generalized davidson method correspondingly the diagonal version of the new method preserves an advantage over the diagonal jacobi davidson and the davidson method

the use of a refined error bound when updating eigenvalues of tridiagonals abstract the lanczos algorithm is used to compute some eigenvalues of a given symmetric matrix of large order at each step of the lanczos algorithm it is valuable to know which eigenvalues of the associated tridiagonal matrix have stabilized at eigenvalues of the given symmetric matrix we present a robust algorithm which is fast 20 j to 40 j operations at the j th lanczos step uses about 30 words of extra storage and has a fairly short program approxiamately 200 executable statements

low rank matrix approximation using the lanczos bidiagonalization process with applications low rank approximation of large and or sparse matrices is important in many applications and the singular value decomposition svd gives the best low rank approximations with respect to unitarily invariant norms in this paper we show that good low rank approximations can be directly obtained from the lanczos bidiagonalization process applied to the given matrix without computing any svd we also demonstrate that a so called one sided reorthogonalization process can be used to maintain an adequate level of orthogonality among the lanczos vectors and produce accurate low rank approximations this technique reduces the computational cost of the lanczos bidiagonalization process we illustrate the efficiency and applicability of our algorithm using numerical examples from several applications areas

two conjugate gradient type methods for unsymmetric linear equations we propose two new conjugate gradient type methods for the solution of sparse unsymmetric linear systems we present a new tridiagonalization process for unsymmetric matrices that is closely related to the lanczos process we use orthogonal factorizations of the tridiagonal matrix to derive the new algorithms usymlq and usymqr in the same fashion as symmlq and minres c c paige and m a saunders solution of sparse indefinite systems of linear equations siam j numer anal 12 1975 pp 617 629 for symmetric matrices some numerical results for the new methods and comparisons with other methods are presented

lanczos versus subspace iteration for solution of eigenvalue problems 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

automated adaptive multilevel solver abstract this paper presents an automated adaptive multilevel solver for linear or linearized system of equations the multilevel aspect of the solver is aimed at securing an optimal rate of convergence while keeping the size of the coarsest problem sufficiently small to ensure that the direct portion of the solution does not dominate the total computational cost adaptivity in terms of a priori selection of the number of levels one or more and construction of the optimal multilevel preconditioner is the key to the robustness of the method the number of levels is selected on the basis of estimated conditioning sparsity of the factor and available memory the auxiliary coarse models if required are automatically constructed on the basis of spectral characteristics of individual aggregates groups of neighboring elements an obstacle test consisting of twenty industry and model problems was designed to i determine the optimal values of computational parameters and to ii compare the adaptive multilevel scheme with existing state of the art equation solvers including the multifrontal solver 17 with the mmd reordering scheme and the pcg solver with the nearly optimal modified incomplete cholesky factorization preconditioner

deflated block krylov subspace methods for large scale eigenvalue problems we discuss a class of deflated block krylov subspace methods for solving large scale matrix eigenvalue problems the efficiency of an arnoldi type method is examined in computing partial or closely clustered eigenvalues of large matrices as an improvement we also propose a refined variant of the arnoldi type method comparisons show that the refined variant can further improve the arnoldi type method and both methods exhibit very regular convergence behavior

state space h controller design for descriptor systems this paper proposes a new linear matrix inequality lmi method to design state space h controllers for linear time invariant descriptor systems unlike preceding studies where descriptor type controllers are first computed and then numerically transformed to state space controllers the proposed method carries out the transformation analytically in the parameter domain we derive a necessary and sufficient lmi condition for the existence of a state space controller with the same dynamic order of the descriptor system to be controlled which makes the closed loop system regular impulse free stable and guarantees the h norm bound imposed on the closed loop transfer function furthermore we present parameterization of all such state space controllers by variables satisfying the lmi condition and an arbitrary nonsingular matrix the lmis utilized in this paper are strict ones that is those containing no equality while lmis with equality constraints have been extensively used in the analysis and design for descriptor systems the strict lmis play key roles in deriving the results of this paper

a new shift strategy for the implicitly restarted refined harmonic lanczos method abstract the implicitly restarted refined harmonic lanczos method is widely used for computing the interior eigenpairs of large real symmetric matrices in each restart it takes the wanted refined harmonic ritz pairs to be the approximate eigenpairs and takes the refined harmonic shifts to implicitly restart the algorithm in this paper we modify this method by replacing the refined harmonic shifts by new shifts based on a special direct sum decomposition of the projected subspace we build a new subspace in each restart and then take the harmonic ritz values onto it to be the new shifts the new shifts can be obtained from a small generalized eigenvalue problem and the computational cost is negligible numerical experiments show that the new shifts are generally better than the refined harmonic shifts

optimal lower bounds for rank and select indexes we develop a new lower bound technique for data structures we show an optimal omega n lg lg n lg n space lower bounds for storing an index that allows to implement rank and select queries on a bit vector b provided that b is stored explicitly these results improve upon miltersen soda 05 we show omega m t lg t lower bounds for storing rank select index in the case where b has m 1 bits in it e g low 0 th entropy and the algorithm is allowed to probe t bits of b we simplify the select index given in raman et al soda 02 and show how to implement both rank and select queries with an index of size 1 o 1 n lg lg n lg n o n lg n i e we give an explicit constant for storage in the ram model with word size lg n

efficient denoising algorithms for large experimental datasets and their applications in fourier transform ion cyclotron resonance mass spectrometry modern scientific research produces datasets of increasing size and complexity that require dedicated numerical methods to be processed in many cases the analysis of spectroscopic data involves the denoising of raw data before any further processing current efficient denoising algorithms require the singular value decomposition of a matrix with a size that scales up as the square of the data length preventing their use on very large datasets taking advantage of recent progress on random projection and probabilistic algorithms we developed a simple and efficient method for the denoising of very large datasets based on the qr decomposition of a matrix randomly sampled from the data this approach allows a gain of nearly three orders of magnitude in processing time compared with classical singular value decomposition denoising this procedure called urqrd uncoiled random qr denoising strongly reduces the computer memory footprint and allows the denoising algorithm to be applied to virtually unlimited data size the efficiency of these numerical tools is demonstrated on experimental data from high resolution broadband fourier transform ion cyclotron resonance mass spectrometry which has applications in proteomics and metabolomics we show that robust denoising is achieved in 2d spectra whose interpretation is severely impaired by scintillation noise these denoising procedures can be adapted to many other data analysis domains where the size and or the processing time are crucial

resolving seismic anisotropy sparse matrix methods for geophysical inverse problems two techniques for the singular value decomposition svd of large sparse matrices are directly applicable to geophysical inverse problems subspace iteration and the lanczos method both methods require very little in core storage and efficiently compute a set of singular values and singular vectors a comparison of the singular value and vector estimates of these iterative approaches with the results of a conventional in core svd algorithm demonstrates their accuracy hence it is possible to conduct an assessment of inversion results for larger sparse inverse problems such as those arising in seismic tomography as an example we examine the resolution matrix associated with a crosswell seismic inversion of first arrival times for lateral variations in anisotropy the application to a set of first arrival times from a crosswell survey at the grimsel laboratory emphasizes the utility of including anisotropy in a traveltime inversion the isotropic component of the estimated velocity structure appears to be well constrained even when anisotropy is included in the inversion in the case of the grimsel experiment we are able to resolve a fracture zone in agreement with borehole fracture intersections elements of the resolution matrix reveal moderate averaging among anisotropy coefficients as well as between anisotropy coefficients and source receiver static terms the information on anisotropy such as the directions of maximum velocity appears sensitive to lateral variations in velocity and must be interpreted with some caution

adjusting the rayleigh quotient in semiorthogonal lanczos methods in a semiorthogonal lanczos algorithm the orthogonality of the lanczos vectors is allowed to deteriorate to roughly the square root of the rounding unit after which the current vectors are reorthogonalized a theorem of simon linear algebra appl 61 1984 pp 101 132 shows that the rayleigh quotient i e the tridiagonal matrix produced by the lanczos recursion contains fully accurate approximations to the ritz values in spite of the lack of orthogonality unfortunately the same lack of orthogonality can cause the ritz vectors to fail to converge it also makes the classical estimate for the residual norm misleadingly small in this paper we show how to adjust the rayleigh quotient to overcome this problem

on the lanczos method for solving symmetric linear systems with several right hand sides this paper analyzes a few methods based on the lanczos algorithm for solving large sparse symmetric linear systems with several right hand sides the methods examined are suitable for the situation when the right sides are not too different from one another as is often the case in time dependent or parameter dependent problems we propose a theoretical error bound for the approximation obtained from a projection process onto a krylov subspace generated from processing a previous right hand side

projection techniques for iterative solution of ax b with successive right hand sides we present two projection techniques for computing approximate solutions to linear systems of the form ax n b n for a sequence n 1 2 e g such as arises from time discretization of a partial differential equation the inexpensive approximate solutions can be used as initial guesses for iterative solution of the system resulting in significantly reduced computational expense examples of two and three dimensional incompressible navier stokes calculations are presented in which x represents the pressure and a is a discrete poisson operator in flows containing significant dynamic activity these projection techniques lead to as much as a two fold reduction in solution time

computing the generalized singular values vectors of large sparse or structured matrix pairs we present a numerical algorithm for computing a few extreme generalized singular values and corresponding vectors of a sparse or structured matrix pair a b the algorithm is based on the cs decomposition and the lanczos bidiagonalization process at each iteration step of the lanczos process the solution to a linear least squares problem with a rm t b rm t rm t as the coefficient matrix is approximately computed and this consists the only interface of the algorithm with the matrix pair a b numerical results are also given to demonstrate the feasibility and efficiency of the algorithm

a comparison of lanczos and optimization methods in the partial solution of sparse symmetric eigenproblems 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