a dynamic thick restarted semi refined able algorithm for computing a few selected eigentriplets of large nonsymmetric matrices able method can be used to compute eigentriplets of large scale nonsymmetric matrices however there is no guarantee for the ritz vectors obtained by this method to converge even if the subspace is good enough furthermore for the sake of storage limitation and amount of computations restarting techniques are often needed in order to deal with these problems we propose a dynamic thick restarted semi refined able algorithm sable in which we use semi refined ritz vectors to approximate the desired eigenvectors the relationship between the novel method and the classical one is given theoretical results show that the new method can circumvent the possible danger that exists in the standard one in some degree numerical experiments are made on real world problems and comparisons are drown on the thick restarted able algorithm table and the dynamic thick restarted semi refined able algorithm sable they show that the latter is often more powerful and attractive than its standard counterpart

bounds for the variation of the roots of a polynomial and the eigenvalues of a matrix we derive some new bounds for the distance between the roots of two polynomials in terms of their coefficients and for the distance between the eigenvalues of two matrices in terms of the norm of their difference

formulation and solution of the non linear damped eigenvalue problem for skeletal systems this paper presents and discusses an arnoldi based eigensolution technique for evaluating the complex natural frequencies and mode shapes from frequency dependent quadratic eigenproblems associated with vibration analysis of damped structures the new solution technique is used in conjunction with a mixed finite element modelling procedure which utilizes both the polynomial and frequency dependent displacement fields in formulating the system matrices this modelling provides the ability to represent a frequency dependent damping matrix in vibration analysis of skeletal systems the eigensolution methodology presented here is based upon the ability to evaluate a specific set of parametrized curves for the non linear eigenvalue problem at given values of the parameter numerical examples illustrate that this method used in conjunction with a secant interpolation accurately evaluates the complex natural frequencies and modes of the quadratic non linear eigenproblem and verifies that the new eigensolution technique coupled with the mixed finite element modelling procedure is more accurate than the conventional finite element models

resolvent bounds and spectral variation abstract a new spectral variation bound is given for linear operators acting on normed linear spaces this bound is derived using the cayley hamilton theorem and chebychev polynomials

higher order derivatives and perturbation bounds for determinants the first derivative of the determinant function is given by the well known jacobi s formula we obtain three different expressions for all higher order derivatives norms of these derivatives are then evaluated exactly

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

on the variation of the spectra of matrix pencils several distances between the spectra of two regular matrix pencils are discussed and compared the relations obtained enable us to estimate upper bounds on one distance via known upper bounds on others we also give some perturbation bounds and then show how to apply the relations obtained in this note

unitary invariance and spectral variation abstract we call a norm on operators or matrices weakly unitarily invariant if its value at operator a is not changed by replacing a by u au provided only that u is unitary this class includes such norms as the numerical radius we extend to all such norms an inequality that bounds the spectral variation when a normal operator a is replaced by another normal b in terms of the arclength of any normal path from a to b computed using the norm in question related results treat the local metric geometry of the manifold of normal operators we introduce a representation for weakly unitarily invariant matrix norms in terms of function norms over the unit ball and identify this correspondence explicitly in certain cases

lanczos algorithms and field of value rotations for symmetric matrix pencils abstract iterative algorithms for the eigensolution of symmetric pencils of matrices are considered it is shown that the symmetric lanczos algorithm the nonsymmetric lanczos algorithm and the arnoldi algorithm are closely related in this case the applicability of this class of algorithms to indefinite pencils is discussed a new field of values concept is used to describe the symmetric pencil problem then spectral transformation corresponds to a rotation in the complex plane and the inertia count gives the number of eigenvalues corresponding to points in one of two half planes

an augmented analysis of the perturbed two sided lanczos tridiagonalization process abstract we generalize an augmented rounding error result that was proven for the symmetric lanczos process in c c paige an augmented stability result for the lanczos hermitian matrix tridiagonalization process siam j matrix anal appl 31 2010 2347 2359 to the two sided usually unsymmetric lanczos process for tridiagonalizing a square matrix we extend the analysis to more general perturbations than rounding errors in order to provide tools for the analysis of inexact and related methods the aim is to develop a deeper understanding of the behavior of all these methods our results take the same form as those for the symmetric lanczos process except for the bounds on the backward perturbation terms the generalizations of backward rounding errors for the augmented system in general we cannot derive tight a priori bounds for these terms as was done for the symmetric process but a posteriori bounds are feasible while bounds related to certain properties of matrices would be theoretically desirable

backward errors for eigenproblem of two kinds of structured matrices in this paper we consider backward errors in the eigenproblem of symmetric centrosymmetric and symmetric skew centrosymmetric matrices by making use of the properties of symmetric centrosymmetric and symmetric skew centrosymmetric matrices we derive explicit formulae for the backward errors of approximate eigenpairs

analysis of spectral variation and some inequalities a geometric method based on a decomposition of the space of complex matrices is employed to study the variation of the spectrum of a matrix when adapted to special cases this leads to some classical inequalities as well as some new ones as an example of the latter we show that if u v are unitary matrices and k is a skew hermitian matrix such that uv exp k then for every unitary invariant norm the distance between the eigenvalues of v and those of vis bounded by k this generalises two earlier results which used particular unitary invariant norms

variation of grassman powers and spectra the operator norm of the derivative of the map which takes a finite dimensional linear operator to its kth grassman power the kth compound is evaluated this leads to a bound for the distance between the grassman powers of two operators as an important application a bound for the distance between the eigenvalues of two operators is obtained

variation of induced linear operators abstract let v be an n dimensional inner product space let be an irreducible character of the symmetric group s m and let v be the symmetry class of tensors associated with it let a be a linear operator on v and let k a be the operator it induces on v we obtain an explicit expression for the norm of the derivative of the map a k a in terms of the singular values of a two special cases of this problem antisymmetric and symmetric tensor products have been studied earlier and our results reduce to the earlier ones in these cases

variation of symmetric tensor powers and permanents the norm of the derivative of the symmetric tensor power of an operator is evaluated exactly a bound for the distance between the permanents of two matrices is obtained as a consequence

improving spectral variation bounds with chebyshev polynomials a new spectral variation bound is established for linear operators a and b on finite dimensional vector spaces this bound is v a b c a b 1n a b n 1 n with c 8 this result affirmatively answers a conjecture of friedland

error analysis of the lanczos algorithm for the nonsymmetric eigenvalue problem this paper presents an error analysis of the lanczos algorithm in finite precision arithmetic for solving the standard nonsymmetric eigenvalue problem if no breakdown occurs an analog of paige s theory on the relationship between the loss of orthogonality among the lanezos vectors and the convergence of ritz values in the symmetric lanczos algorithm is discussed the theory developed illustrates that in the nonsymmetric lanczos scheme if ritz values are well conditioned then the loss of biorthogonality among the computed lanczos vectors implies the convergence of a group of ritz triplets in terms of small residuals numerical experimental results confirm this observation

on the variation of the spectra of matrices abstract several distances between the spectra of two matrices are discussed and compared optimal bounds are given which enable us to reduce certain bounds on the eigenvalue variation of matrices by a factor of about two the results of bhatia and mukherjea and of bhatia and friedland on the eigenvalue variation are derived in an elementary way using results of henrici on the spectral variation

an implementation of the look ahead lanczos algorithm for non hermitian matrices the nonsymmetric lanczos method can be used to compute eigenvalues of large sparse non hermitian matrices or to solve large sparse non hermitian linear systems however the original lanczos algorithm is susceptible to possible breakdowns and potential instabilities an implementation of a look ahead version of the lanczos algorithm is presented that except for the very special situation of an incurable breakdown overcomes these problems by skipping over those steps in which a breakdown or near breakdown would occur in the standard process the proposed algorithm can handle look ahead steps of any length and requires the same number of matrix vector products and inner products as the standard lanczos process without look ahead

a look ahead lanczos algorithm for unsymmetric matrices abstract the two sided lanczos algorithm is known to suffer from serious breakdowns these occur when the associated moment matrix does not permit triangular factorization we modify the algorithm slightly so that it corresponds to using a 2 x 2 pivot in triangular factorization whenever a 1 x 1 pivot would be dangerous the incidence of breakdown is greatly reduced the price paid is that the tridiagonal matrix produced by the algorithm now has bumps whenever a 2 x 2 pivot is used author

an optimal bound for the spectral variation of two matrices abstract we establish a bound for the spectral variation of two complex n n matrices a b in terms of a b and a b here denotes the spectral norm it is always better than a bound previously given by bhatia and friedland and it is optimal we describe the set of pairs a b for which the bound is attained

rational krylov sequence methods for eigenvalue computation abstract algorithms to solve large sparse eigenvalue problems are considered a new class of algorithms which is based on rational functions of the matrix is described the lanczos method the arnoldi method the spectral transformation lanczos method and rayleigh quotient iteration all are special cases but there are also new algorithms which correspond to rational functions with several poles in the simplest case a basis of a rational krylov subspace is found in which the matrix eigenvalue problem is formulated as a linear matrix pencil with a pair of hessenberg matrices

derivatives for antisymmetric tensor powers and perturbation bounds abstract in an earlier paper r bhatia t jain higher order derivatives and perturbation bounds for determinants linear algebra appl 431 2009 2102 2108 we gave formulas for derivatives of all orders for the map that takes a matrix to its determinant in this paper we continue that work and find expressions for the derivatives of all orders for the antisymmetric tensor powers and for the coefficients of the characteristic polynomial we then evaluate norms of these derivatives and use them to obtain perturbation bounds

qmr based projection techniques for the solution of non hermitian systems with multiple right hand sides in this work we consider the simultaneous solution of large linear systems of the form ax j b j j 1 k where a is sparse and non hermitian we describe single seed and block seed projection approaches to these multiple right hand side problems that are based on the qmr and block qmr algorithms respectively we use block qmr to solve the block seed system and generate the relevant biorthogonal subspaces approximate solutions to the nonseed systems are simultaneously generated by minimizing their appropriately projected block residuals after the initial block seed has converged the process is repeated by choosing a new block seed from among the remaining nonconverged systems and using the previously generated approximate solutions as initial guesses for the new seed and nonseed systems we give theory for the single seed case that helps explain the convergence behavior under certain conditions implementation details for both the single seed and block seed algorithms are discussed and advantages of the block seed algorithm in cache based serial and parallel environments are noted the computational savings of our methods over using qmr to solve each system individually are illustrated in two examples