a lanczos type method for multiple starting vectors given a square matrix and single right and left starting vectors the classical nonsymmetric lanczos process generates two sequences of bior thogonal basis vectors for the right and left krylov subspaces induced by the given matrix and vectors in this paper we propose a lanczos type algorithm that extends the classical lanczos process for single starting vectors to mul tiple starting vectors given a square matrix and two blocks of right and left starting vectors the algorithm generates two sequences of biorthogonal basis vectors for the right and left block krylov subspaces induced by the given data the algorithm can handle the most general case of right and left start ing blocks of arbitrary sizes while all previously proposed extensions of the lanczos process are restricted to right and left starting blocks of identical sizes other features of our algorithm include a built in deation procedure to detect and delete linearly dependent vectors in the block krylov sequences and the option to employ look ahead to remedy the potential breakdowns that may occur in nonsymmetric lanczos type methods

a krylov subspace method for large scale second order cone linear complementarity problem in this paper we first show that the second order cone linear complementarity problem soclcp can be solved by finding a positive zero s in mathbb r of a particular rational function h s and we then propose a krylov subspace method to reduce h s to h ell s as in the model reduction the zero s of h s can be accurately approximated by that of h ell s 0 which itself can be cast as a small eigenvalue problem the new method is made possible by a complete characterization of the curve of h s and it has several advantages over the bisection newton sf bn iteration recently proposed by l h zhang and w h yang math comp 83 2013 pp 1701 1720 and shown to be very efficient for small to medium size problems the method is tested and compared against the sf bn iteration and two other state of the art packages sf sdpt3 and sf sedumi our numerical results show that the method is very efficient for both small to medium dense problems and large scale ones

eigensolution of nonclassically damped structures by complex subspace iteration a nonclassically damped structure includes the consistent damping matrix all structural matrices are symmetric the stiffness matrix has to be positive definite this leads to a quadratic eigenproblem with complex eigenvalues and eigenvectors the common solution technique is the transformation into a linear equation system in state space form which doubles the order of the system and to solve it e g by the lanczos method the presented approach solves the quadratic eigenproblem directly i e the order is not increased and the sparseness or bandedness of the system matrices is retained the solution of the eigenproblem is obtained by the principle of subspace iteration in comparison to undamped modes the complex subspace iteration operates on a modal subspace of double size the iteration sequence is based on a filter principle this enables to utilise pre factorised system equations and starting information of eigenvectors from a previous solution even when the structural matrices are updated shift techniques can be applied which makes the algorithm useful for numerically tough problems including a large number of overdamped eigensolutions

on the partial realization problem abstract in this paper we take a unified approach to the partial realization problem in which we seek to incorporate ideas from numerical linear algebra most of which were originally developed in other contexts we approach the partial realization problem from several different angles and explore the connections to such topics as factoriza tion of hankel matrices block tridiagonalization generalizations of the lanczos process for biorthogonalization the euclidean algorithm and the principal part con tinued fractions of arne magnus the pade table and the berlekamp masseyalgo rithm in this way we are able to clarify some previous results by rissanen kalman and others and place them in a broader context this leads to several results and concepts which we think are new our analysis is restricted to the scalar case but some definitions and formulations have been rigged to facilitate an extension to the matrix case

qmr a quasi minimal residual method for non hermitian linear systems the biconjugate gradient bcg method is the natural generalization of the classical conjugate gradient algorithm for hermitian positive definite matrices to general non hermitian linear systems unfortunately the original bcg algorithm is susceptible to possible breakdowns and numerical instabilities in this paper we present a novel bcg like approach the quasi minimal residual qmr method which overcomes the problems of bcg an implementation of qmr based on a look ahead version of the nonsymmetric lanczos algorithm is proposed it is shown how bcg iterates can be recovered stably from the qmr process some further properties of the qmr approach are given and an error bound is presented finally numerical experiments are reported

solving linear equations over gf 2 block lanczos algorithm abstract we propose a method of solving large sparse systems of linear equations over gf 2 the field with two elements we use the lanczos algorithm modified in two ways a lookahead lanczos algorithm is needed for the problem of dividing by an inner product whose value happens to be 0 a block version of the algorithm allows us to perform 32 matrix vector operations for the cost of one the resulting algorithm is competitive with structured gaussian elimination in terms of time and has much lower space requirements it may be useful in the last stage of integer factorization

low rank modification of the unsymmetric lanczos algorithm the unsymmetric lanczos algorithm is an important method for eigenvalue estimation and for solving linear equations unfortunately the algorithm may break down without providing useful information this is referred to as a serious breakdown in the literature here we introduce a low rank modification of the original matrix a in the case of a serious breakdown this modification can be used to cure a serious breakdown as long as we have orthogonality of the already computed lanczos vectors we can switch to a new rank 1 modified matrix a a ab t such that the lanczos algorithm has no serious breakdown in this step when applied on a the already computed variables in the lanczos algorithm for a and a coincide using a lanczos based iterative solver e g bcg or qmr with start vectors x 0 0 and 1 f we have a 1 f a 1 f and thus by continuing the lanczos algorithm with a we automatically get the desired solution a 1 f also if the lanczos vectors have lost their orthogonality we show theoretically and by numerical examples that the modified lanczos method has the same convergence behavior as the lanczos method without breakdown thus in the case of a serious breakdown we only have to compute the new rank 1 modified matrix a and step further in the original algorithm now using a

reduced order modeling for time dependent optimization problems with initial value controls this paper presents a new reduced order model rom hessian approximation for linear quadratic optimal control problems where the optimal control is the initial value such problems arise in parameter identification and data assimilation where the parameters to be identified appear in the initial data and also as subproblems in multiple shooting formulations of more general optimal control problems the new rom hessians can provide a substantially better approximation than the underlying basic rom approximation and thus can substantially reduce the computing time needed to solve these optimal control problems the computation of a hessian vector product requires the solution of the linearized state equation with initial value given by the vector to which the hessian is applied followed by the solution of the second order adjoint equation projection based roms of these two linear differential equations are used to generate the hessian approximation while the objective function and gradient are compute

iterative solution of linear systems in the 20th century this paper sketches the main research developments in the area of iterative methods for solving linear systems during the 20th century although iterative methods for solving linear systems nd their origin in the early 19th century work by gauss the eld has seen an explosion of activity spurred by demand due to extraordinary technological advances in engineering and sciences the past ve decades have been particularly rich in new developments ending with the availability of large toolbox of specialized algorithms for solving the very large problems which arise in scientic and industrial computational models as in any other scientic area research in iterative methods has been a journey characterized by a chain of contributions building on each other it is the aim of this paper not only to sketch the most signicant of these contributions during the past century but also to relate them to one another c 2000 elsevier science b v all rights reserved

symmetric sor method for augmented systems in this paper we develop symmetric successive overrelaxation ssor method to find solution of augmented systems these systems have appeared in many different applications of scientific computing e g the finite element approximation to solve the navier stokes equation and the constrained optimization the convergence of ssor method are studied finally we apply the new method to solve augmented systems the iteration number of this method to receive the solution is very better than sor like method

quasi kernel polynomials and their use in non hermitian matrix iterations abstract some of the most efficient iterative algorithms for large sparse hermitian matrix computations are based on orthogonal or kernel polynomials for the case of non hermitian matrices methods based on orthogonal or kernel polynomials are less satisfactory in that the resulting algorithms involve long recurrences consequently it is usually too expensive to run the full algorithms and restarts are necessary a typical example is the generalized minimal residual method gmres for solving non hermitian linear systems where work and storage per iteration grow linearly with the iteration number recently two quasi minimal residual methods qmr for solving non hermitian linear systems have been proposed which unlike gmres are based on short recurrences and hence can be used as true iterative schemes without restarts in this paper the concept of quasi kernel polynomials is introduced some general theory for quasi kernel polynomials is developed such as recurrence relations and a characterization of roots of quasi kernel polynomials as generalized eigenvalues it is pointed out that the qmr approaches are based on two particular instances of quasi kernel polynomials also the use of quasi kernel polynomials for approximating eigenvalues or pseudospectra of large sparse non hermitian matrices is briefly discussed

saving flops in lu based shift and invert strategy the shift and invert method is very efficient in eigenvalue computations in particular when interior eigenvalues are sought this method involves solving linear systems of the form email protected z b the shift s is variable hence when a direct method is used to solve the linear system the lu factorization of email protected needs to be computed for every shift change we present two strategies that reduce the number of floating point operations performed in the lu factorization when the shift changes both methods perform first a preprocessing step that aims at eliminating parts of the matrix that are not affected by the diagonal change this leads to about 43 and 50 flops savings respectively for the dense matrices

complex shift and invert strategies for real matrices abstract when using an iterative method for solving a generalized nonsymmetric eigenvalue problem of the form fu mu where f and m are real banded matrices it is often desirable to work with the shifted and inverted operator b f m 1 m in order to enhance the eigenvalue separation and improve efficiency unfortunately the shift is generally complex and so is the matrix b the question then is whether it is possible to avoid complex arithmetic while preserving any advantages of bandedness of the pair f m for the classical problem where m i and f is banded complex arithmetic can be avoided by using double shifts i e by working with the real matrix bb whose bandwidth is double that of f this satisfactory solution extends to the case where m is diagonal as well in the generalized case the answer to the above question is negative in the sense that complex arithmetic can be avoided only at the expense of losing the advantage of bandedness one solution is to factor the shifted matrix f m in complex arithmetic but employ real arithmetic subsequently in the iterative procedure this paper examines several approaches and discusses their respective merits under various circumstances

a comparative study of iterative solutions to linear systems arising in quantum mechanics this study is mainly focused on iterative solutions with simple diagonal preconditioning to two complex valued nonsymmetric systems of linear equations arising from a computational chemistry model problem proposed by sherry li of nersc numerical experiments show the feasibility of iterative methods to some extent when applied to the problems and reveal the competitiveness of our recently proposed lanczos biconjugate a orthonormalization methods to other classic and popular iterative methods by the way experiment results also indicate that application specific preconditioners may be mandatory and required for accelerating convergence

a look ahead algorithm for the solution of general hankel systems the solution of systems of linear equations with hankel coefficient matrices can be computed with onlyo n 2 arithmetic operations as compared too n 3 operations for the general cases however the classical hankel solvers require the nonsingularity of all leading principal submatrices of the hankel matrix the known extensions of these algorithms to general hankel systems can handle only exactly singular submatrices but not ill conditioned ones and hence they are numerically unstable in this paper a stable procedure for solving general nonsingular hankel systems is presented using a look ahead technique to skip over singular or ill conditioned submatrices the proposed approach is based on a look ahead variant of the nonsymmetric lanczos process that was recently developed by freund gutknecht and nachtigal we first derive a somewhat more general formulation of this look ahead lanczos algorithm in terms of formally orthogonal polynomials which then yields the look ahead hankel solver as a special case we prove some general properties of the resulting look ahead algorithm for formally orthogonal polynomials these results are then utilized in the implementation of the hankel solver we report some numerical experiments for hankel systems with ill conditioned submatrices

a study of symbolic processing and computational aspects in helicopter dynamics even research models of helicopter dynamics often lead to a large number of equations of motion with periodic coefficients and floquet theory is a widely used mathematical tool for dynamic analysis presently three approaches are used in generating the equations of motion these are 1 general purpose symbolic processors such as reduce and macsyma 2 a special purpose symbolic processor dehim dynamic equations for helicopter interpretive models and 3 completely numerical approaches in this paper comparative aspects of the first two purely algebraic approaches are studied by applying reduce and dehim to the same set of problems these problems range from a linear model with one degree of freedom to a mildly non linear multi bladed rotor model with several degrees of freedom further computational issues in applying floquet theory are also studied which refer to 1 the equilibrium solution for periodic forced response together with the transition matrix for perturbations about that response and 2 a small number of eigenvalues and eigenvectors of the unsymmetric transition matrix the study showed the following 1 compared to reduce dehim is far more portable and economical but it is also less user friendly particularly during learning phases 2 the problems of finding the periodic response and eigenvalues are well conditioned

chebyshev acceleration techniques for solving nonsymmetric eigenvalue problems abstract the present paper deals with the problem of computing a few of the eigenvalues with largest or smallest real parts of a large sparse nonsymmetric matrix we present a general acceleration technique based on chebyshev polynomials and discuss its practical application to arnoldi s method and the subspace iteration method the resulting algorithms are compared with the classical ones in a few experiments which exhibit a sharp superiority of the arnoldi chebyshev approach

an analysis of the composite step biconjugate gradient method the composite step biconjugate gradient method csbcg is a simple modification of the standard biconjugate gradient algorithm bcg which smooths the sometimes erratic convergence of bcg by computing only a subset of the iterates we show that 2 2 composite steps can cure breakdowns in the biconjugate gradient method caused by near singularity of principal submatrices of the tridiagonal matrix generated by the underlying lanczos process we also prove a best approximation result for the method some numerical illustrations showing the effect of roundoff error are given

large scale complex eigenvalue problems abstract the nonsymmetric eigenvalue problem ax bx is discussed with special emphasis on linear algebra theory on algorithms implemented for solving large scale systems and on interpreting complex spectra obtained in applications from physics and engineering present day supercomputers make the required matrix computations feasible especially in view of the increase in memory

the symplectic eigenvalue problem the butterfly form the sr algorithm and the lanczos method abstract we discuss some aspects of the recently proposed symplectic butterfly form which is a condensed form for symplectic matrices any 2n 2n symplectic matrix can be reduced to this condensed form which contains 8n 4 nonzero entries and is determined by 4n 1 parameters the symplectic eigenvalue problem can be solved using the sr algorithm based on this condensed form the sr algorithm preserves this form and can be modified to work only with the 4n 1 parameters instead of the 4n2 matrix elements the reduction of symplectic matrices to symplectic butterfly form has a close analogy to the reduction of arbitrary matrices to hessenberg form a lanczos like algorithm for reducing a symplectic matrix to butterfly form is also presented

a new implementation of the cmrh method for solving dense linear systems the cmrh method h sadok methodes de projections pour les systemes lineaires et non lineaires habilitation thesis university of lille1 lille france 1994 h sadok cmrh a new method for solving nonsymmetric linear systems based on the hessenberg reduction algorithm numer algorithms 20 1999 303 321 is an algorithm for solving nonsymmetric linear systems in which the arnoldi component of gmres is replaced by the hessenberg process which generates krylov basis vectors which are orthogonal to standard unit basis vectors rather than mutually orthogonal the iterate is formed from these vectors by solving a small least squares problem involving a hessenberg matrix like gmres this method requires one matrix vector product per iteration however it can be implemented to require half as much arithmetic work and less storage moreover numerical experiments show that this method performs accurately and reduces the residual about as fast as gmres with this new implementation we show that the cmrh method is the only method with long term recurrence which requires not storing at the same time the entire krylov vectors basis and the original matrix as in the gmres algorithm a comparison with gaussian elimination is provided

a new iterative method for solving complex symmetric linear systems based on implementation of the quasi minimal residual qmr and biconjugate a orthogonal residual bicor method a new krylov subspace method is presented for solving complex symmetric linear systems the new method can be combined with arbitrary symmetric preconditioners the preconditioned modified hermitian and skew hermitian splitting pmhss preconditioner is used to accelerate the convergence rate of this method numerical experiments indicate that the proposed method and its preconditioned version are efficient and robust in comparison with other krylov subspace methods

the methods of vorobyev and lanczos abstract this paper deals with the method of moments of vorobyev which is a particular case of galerkin s method this method is generalized in different ways and some of its properties are studied in detail as an application it is showed that the lanczos method for solving systems of linear equations can be derived from it and that some generalizations could be obtained

a krylov subspace projection method for simultaneous solution of helmholtz problems at multiple frequencies abstract a krylov subspace projection method which provides simultaneous solutions of the helmholtz equation at multiple frequencies in one solution step is presented the projector is obtained with an unsymmetric block lanczos algorithm applied to a transfer function derived from a finite element discretization this approach is equivalent to a matrix valued pade approximation of the transfer function the proposed method is an extension of the formulation presented in j comput acoust 8 2000 223 to unsymmetric systems and allows the treatment of a much wider range of practical problems including near field and fluid structure interaction computations

rational interpolation via orthogonal polynomials abstract relations between rational interpolants and hankel matrices are investigated a modification of a jacobi like algorithm for rational interpolation proposed in 1 is developed the proposed algorithm eliminates all the auxiliary assumptions required to implement the algorithm of 1 in this way it can be used to construct all existing rational interpolants with the cost of o n 2 arithmetic operations where n 1 is the number of data points

qmr smoothing for lanczos type product methods based on three term rrecurrences for the solution of large sparse non hermitian linear systems lanczos type product methods that are based on the lanczos three term recurrence are derived in a systematic way these methods either square the lanczos process or combine it with a local minimization of the residual for them a quasi minimal residual qmr smoothing is proposed that can also be implemented by short term recurrences the practical performance of these methods and the qmr smoothing is demonstrated in a number of numerical examples

ml n bicgstabt a ml n bicgstab variant with a transpose the 1980 idr method wesseling and sonneveld 1980 12 plays an important role in the history of krylov subspace methods it started the research of transpose free krylov subspace methods the ml n bicgstab method yeung 2012 is one of such methods in this paper we present a new ml n bicgstab variant that involves a transpose in its implementation comparison of this new algorithm with the existing ml n bicgstab algorithms and some other krylov subspace algorithms will be presented

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

an implementation of the qmr method based on coupled two term recurrences recently we proposed a new krylov subspace iteration the quasi minimal residual algorithm qmr 1 for solving general nonsingular non hermitian systems of linear equations ax b 1

methods for overcoming breakdown problems in the unsymmetric lanczos reduction method the unsymmetric lanczos reduction ulr method is developed to solve the finite element based solution to the contaminant transport problem the method sometimes suffers from breakdown when at some step division by a pivot which is zero or near zero causes numerical instability in this paper the maximum pivot new start vector method is developed to overcome such breakdowns by constructing a new starting vector with the possible maximum pivot some cases of instability cannot be remedied by this approach pathological breakdowns and the switch method is developed to complete the solution by changing the algorithm to an arnoldi reduction approach investigation of some two dimensional examples and field problems illustrates the efficiency of the methods and substantial time savings over other existing solution methods 1998 john wiley sons ltd

a breakdown free variation of the nonsymmetric lanczos algorithms the nonsymmetric lanczos tridiagonalization algorithm is essentially the gram schmidt biorthogonalization method for generating biorthogonal bases of a pair of krylov subspaces it suffers from breakdown and instability when a pivot at some step is zero or nearly zero which is often the result of mismatch of the two krylov subspaces in this paper we propose to modify one of the two krylov subspaces by introducing a new start vector when a pivot is small this new start vector generates another krylov subspace which we add to the old one in an appropriate way so that the gram schmidt method for the modified subspaces yields a recurrence similar to the lanczos algorithm our method enforces the pivots to be above a certain threshold and can handle both exact breakdown and near breakdown in particular we recover lookahead lanczos algorithms and arnoldi s algorithm as two special cases we also discuss theoretical and practical issues concerning the new start procedure and present a convergence analysis as well as some numerical examples

computational enhancement of an unsymmetric block lanczos algorithm 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

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 rayleigh chebyshev procedure for finding the smallest eigenvalues and associated eigenvectors of large sparse hermitian matrices a procedure is presented for finding a number of the smallest eigenvalues and their associated eigenvectors of large sparse hermitian matrices the procedure a modification of an inverse subspace iteration procedure uses adaptively determined chebyshev polynomials to approximate the required application of the inverse operator on the subspace the method is robust converges with acceptable rapidity and can easily handle operators with eigenvalues of multiplicity greater than one numerical results are shown that demonstrate the utility of the procedure

new look ahead lanczos type algorithms for linear systems a breakdown due to a division by zero can arise in the algorithms for implementing lanczos method because of the non existence of some formal orthogonal polynomials or because the recurrence relationship used is not appropriate such a breakdown can be avoided by jumping over the polynomials involved this strategy was already used in some algorithms such as the mrz and its variants in this paper we propose new implementations of the recurrence relations of these algorithms which only need the storage of a fixed number of vectors independent of the length of the jump these new algorithms are based on horner s rule and on a different way for computing the coefficients of the recurrence relationships moreover these new algorithms seem to be more stable than the old ones and they provide better numerical results numerical examples and comparisons with other algorithms will be given

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

a review of formal orthogonality in lanczos based methods krylov subspace methods and their variants are presently the favorite iterative methods for solving a system of linear equations although it is a purely linear algebra problem it can be tackled by the theory of formal orthogonal polynomials this theory helps to understand the origin of the algorithms for the implementation of krylov subspace methods and moreover the use of formal orthogonal polynomials brings a major simplification in the treatment of some numerical problems related to these algorithms this paper reviews this approach in the case of lanczos method and its variants the novelty being the introduction of a preconditioner

adaptive frequency windowing for multifrequency solutions in structural acoustics based on the matrix pade via lanczos algorithm 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

computation of matrix valued formally orthogonal polynomials and applications we present a computational procedure for generating formally orthogonal polynomials associated with a given bilinear hankel form with rectangular matrix valued moments our approach covers the most general case of moments of any size and is not restricted to square moments moreover our algorithm has a built in deflation procedure to handle linearly dependent or almost linearly dependent columns and rows of the block hankel matrix associated with the bilinear form possible singular or close to singular leading principal submatrices of the deflated block hankel matrix are avoided by means of look ahead techniques applications of the computational procedure to eigenvalue computations reduced order modeling the solution of multiple linear systems and the fast solution of block hankel systems are also briefly described

an efficient nonsymmetric lanczos method on parallel vector computers abstract in this paper we introduce the s step biorthogonal lanczos method for finding a few eigenvalues of a large sparse nonsymmetric matrix and we prove that the s step method generates reduction matrices which are similar to reduction matrices generated by the standard method we prove that the breakdown conditions of the s step method are less stringent than the standard one one iteration of the s step biorthogonal lanczos algorithm corresponds to s iterations of the standard biorthogonal lanczos algorithm and the s step method has improved data locality and minimized global communication and superior parallel properties to the standard one on parallel machines chronopoulos and gear 1989 and kim and chronopoulos 1991 we implement the s step biorthogonal lanczos method on the cray 2 super computer and discuss the breakdown conditions and demonstrate the superior performance of the s step method to the standard one

semi convergence analysis of gmssor methods for singular saddle point problems recently some authors darvishi and hessari 2011 zhang et al 2012 discussed the generalized modified symmetric successive over relaxation gmssor method for saddle point problems in this paper we further study the gmssor method for solving singular saddle point problems we prove the semi convergence of the gmssor method without the assumption that the 1 1 block sub matrix a should be symmetric and analyze the spectral property of the corresponding preconditioned matrix numerical experiments are given to illustrate the efficiency of gmssor with appropriate parameters

restarting the nonsymmetric lanczos algorithm for eigenvalues and linear equations including multiple right hand sides a restarted nonsymmetric lanczos algorithm is given for computing eigenvalues and both right and left eigenvectors the restarting limits the storage so that finding eigenvectors is practical restarting also makes it possible to deal with roundoff error in new ways we give a scheme for avoiding near breakdown and discuss maintaining biorthogonality a system of linear equations can be solved simultaneously with the eigenvalue computations deflation from the presence of the eigenvectors allows the linear equations to generally have good convergence in spite of the restarting the right and left eigenvectors generated while solving the linear equations can be used to help solve systems with multiple right hand sides

adaptive control of a class of slowly time varying plants abstract an adaptive control system of the type considered earlier by the author which is designed to be stable under the assumption that the unknown plant parameters are constant is shown to maintain stability when the plant parameters are slowly time varying

galerkin eigenvector approximations how close are galerkin eigenvectors to the best approximation available out of the trial subspace under a variety of conditions the galerkin method gives an approximate eigenvector that approaches asymptotically the projection of the exact eigenvector onto the trial subspace and this occurs more rapidly than the underlying rate of convergence of the approximate eigenvectors both orthogonal galerkin and petrov galerkin methods are considered here with a special emphasis on nonselfadjoint problems thus extending earlier studies by chatelin babuska and osborn and knyazev consequences for the numerical treatment of elliptic pdes discretized either with finite element methods or with spectral methods are discussed new lower bounds to the sep of a pair of operators are developed as well

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

the lanczos algorithm and hankel matrix factorization abstract we explore the connections between the lanczos algorithm for matrix tridiagonalization and two fast algorithms for hankel matrix factorization we show how the asymmetric lanczos process is related to the berlekamp massey algorithm and how the symmetrized lanczos process is related to the phillips algorithm we also discuss conditions under which the analysis of phillips applies

a breakdown free lanczos type algorithm for solving linear systems lanczos type algorithms for solving systems of linear equations have their foundations in the theory of formal orthogonal polynomials and the method of moments which leads to a determinantal formula for their iterates the various lanczos type algorithms mainly differ by the way of computing the coefficients entering into the recurrence formulae if the denominator in the formula for one of these coefficients is zero then a breakdown occurs in the algorithm and it must be stopped such a breakdown is in fact due to the non existence of some orthogonal polynomial in this paper we show how to jump over such a singularity by computing the next existing orthogonal polynomial by the block bordering method the resulting algorithm called mrz is equivalent to the nongeneric biodir algorithm which is a look ahead lanczos type algorithm but our derivation is much simpler

a convergence analysis for nonsymmetric lanczos algorithms a convergence analysis for the nonsymmetric lanczos algorithm is presented by using a tridiagonal structure of the algorithm some identities concerning ritz values and ritz vectors are established and used to derive approximation bounds in particular the analysis implies the classical results for the symmetric lanczos algorithm

the lanczos biorthogonalization algorithm and other oblique projection methods for solving large unsymmetric systems many powerful methods for solving systems of equations can be regarded as projection methods most of the projection methods known for solving linear systems are orthogonal projection methods but little attention has been given to the class of nonorthogonal or oblique projection methods which is particularly attractive for large nonsymmetric systems the purpose of the paper is to present some methods in the general setting of oblique projection methods and to give some theoretical results some experiments comparing the various algorithms are reported

on the squared unsymmetric lanczos method the biorthogonal lanczos and the biconjugate gradient methods have been proposed as iterative methods to approximate the solution of nonsymmetric and indefinite linear systems sonneveld 1989 obtained the conjugate gradient squared by squaring the matrix polynomials of the biconjugate gradient method here we square the unsymmetric or biorthogonal lanczos method for computing the eigenvalues of nonsymmetric matrices three forms of restarted squared lanczos methods for solving unsymmetric linear systems of equations were derived numerical experiments with unsymmetric in definite linear systems of equations comparing these methods to a restarted orthogonal krylov subspace iterative method showed that the new methods are competitive and they require that a fixed small number of direction vectors be stored in the main memory

an implicitly restarted symplectic lanczos method for the hamiltonian eigenvalue problem abstract an implicitly restarted symplectic lanczos method for the hamiltonian eigenvalue problem is presented the lanczos vectors are constructed to form a symplectic basis the inherent numerical difficulties of the symplectic lanczos method are addressed by inexpensive implicit restarts the method is used to compute eigenvalues eigenvectors and invariant subspaces of large and sparse hamiltonian matrices and low rank approximations to the solution of continuous time algebraic riccati equations with large and sparse coefficient matrices

numerical solutions for large sparse quadratic eigenvalue problems abstract we study the quadratic eigenvalue problem a b 2c x 0 where a b and c are symmetric real n n matrices and a c are positive definite we propose an efficient numerical algorithm to compute a few of the smallest positive eigenvalues of the problem and their associated eigenvectors the new algorithm includes two parts the first part gives iterative methods which can be used to compute the smallest positive eigenvalue we develop a globally linearly convergent basic iteration and two locally quadratically convergent iterations the second part uses the nonequivalence deflation technique this technique allows us to transform the original problem to a new problem with different a b and c the new problem has the same eigenvalues as the old problem except that the smallest positive eigenvalue of the old problem is replaced by zero therefore the second smallest positive eigenvalue of the old problem becomes the smallest positive one for the new problem then the above proposed iterative methods can be applied again to find the second smallest positive eigenvalue proceeding in this way we can find out the 3rd 4th smallest positive eigenvalues our algorithm utilizes the symmetry and positivity of the given matrices and avoids computing the undesired complex conjugate eigenvalues under some mild conditions is efficient and reliable the above process can also be used to find a few of the largest negative eigenvalues

a transpose free quasi minimal residual algorithm for non hermitian linear systems the biconjugate gradient method bcg for solving general non hermitian linear systems ax b and its transpose free variant the conjugate gradients squared algorithm cgs both typically exhib

stable row recurrences for the pade table and generically superfast lookahead solvers for non hermitian toeplitz systems abstract we present general recurrences for the pade table that allow us to skip ill conditioned pade approximants while we proceed along a row of the table in conjunction with a certain inversion formula for toeplitz matrices these recurrences form the basis for fast algorithms for solving non hermitian toeplitz systems under the assumption that the lookahead step size i e the number of successive skipped approximants remains bounded we give both o n 2 and o n log 2 n algorithms which are presumably weakly stable with little additional work still in o n 2 operations we can also obtain a decomposition of the toeplitz matrix t according to tr ld where r is upper triangular l is unit lower triangular and d is block diagonal the relation to continued fractions is also discussed

a transpose free quasi minimal residual variant of the cors method for solving non hermitian linear systems recently the conjugate a orthogonal residual squared cors method is often competitive and superior to other krylov subspace methods for solving complex non hermitian linear systems from many realistic problems however like the conjugate gradient squared cgs method the cors method often exhibits irregular convergence behavior with wild oscillations in the residual norm to try and remedy this problem we develop a transpose free quasi minimal residual tfqmr variant of the cors method which leads to smooth convergence curves with a convergence rate similar to the cors method we also give convergence results for this new method finally numerical examples are reported to illustrate the effectiveness of the proposed method

breakdowns in the implementation of the lanczos method for solving linear systems abstract the lanczos method for solving systems of linear equations is based on formal orthogonal polynomials its implementation is realized via some recurrence relationships between polynomials of a family of orthogonal polynomials or between those of two adjacent families of orthogonal polynomials a division by zero can occur in such recurrence relations thus causing a breakdown in the algorithm which has to be stopped in this paper two types of breakdowns are discussed the true breakdowns which are due to the nonexistence of some polynomials and the ghost breakdowns which are due to the recurrence relationship used among all the recurrence relationships which can be used and all the algorithms for implementing the lanczos method which came out from them the only reliable algorithm is lanczos orthodir which can only suffer from true breakdowns it is shown how to avoid true breakdowns in this algorithm other algorithms are also discussed and the case of near breakdown is treated the same treatment applies to other methods related to lanczos

lanczos type variants of the cocr method for complex nonsymmetric linear systems motivated by the celebrated extending applications of the well established complex biconjugate gradient cbicg method to deal with large three dimensional electromagnetic scattering problems by pocock and walker m d pocock s p walker the complex bi conjugate gradient solver applied to large electromagnetic scattering problems computational costs and cost scalings ieee trans antennas propagat 45 1997 140 146 three lanczos type variants of the recent conjugate a orthogonal conjugate residual cocr method of sogabe and zhang t sogabe s l zhang a cocr method for solving complex symmetric linear systems j comput appl math 199 2007 297 303 are explored for the solution of complex nonsymmetric linear systems the first two can be respectively considered as mathematically equivalent but numerically improved popularizing versions of the bicr and crs methods for complex systems presented in sogabe s ph d dissertation and the last one is somewhat new and is a stabilized and more smoothly converging variant of the first two in some circumstances the presented algorithms are with the hope of obtaining smoother and hopefully faster convergence behavior in comparison with the cbicg method as well as its two corresponding variants this motivation is demonstrated by numerical experiments performed on some selective matrices borrowed from the university of florida sparse matrix collection by davis

a novel nonsymmetric k lanczos algorithm for the generalized nonsymmetric k eigenvalue problems in this article we present a novel algorithm named nonsymmetric k lanczos algorithm for computing a few extreme eigenvalues of the generalized eigenvalue problem mx al where the matrices m and l have the so called k structures we demonstrate a k tridiagonalization procedure preserves the k structures an error bound for the extreme k ritz value obtained from this new algorithm is presented when compared with the class nonsymmetric lanczos approach this method has the same order of computational complexity and can be viewed as a special 2 x 2 block nonsymmetric lanczos algorithm numerical experiments with randomly generated k matrices show that our algorithm converges faster and more accurate than the nonsymmetric lanczos algorithm 0 ekevier science inc 1997

the lanczos algorithm applied to unsymmetric generalized eigenvalue problem 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

matrix pade via lanczos solutions for vibrations of fluid structure interaction for multiple frequency full field solutions of the boundary value problem describing small fluid structure interaction vibration superimposed on a nominal state with prestress we propose an efficient reduced order method by constructing the full field matrix pade approximant of its finite element matrix function exploiting the matrix pade via lanczos connection the pade coefficients are computed in a stable and efficient way via an unsymmetric banded lanczos process the full field pade type approximant is the result of one sided projection onto krylov subspace we established its order of accuracy which is not maximal the superiority of this method in terms of various problem dimensions and parameters is established by complexity analysis via flop counts numerical examples obtained by using a model problem verified the accuracy of this full field matrix pade approximant copyright 2010 john wiley sons ltd

a restarted induced dimension reduction method to approximate eigenpairs of large unsymmetric matrices this work presents a new algorithm to compute eigenpairs of large unsymmetric matrices using the induced dimension reduction method idr s which was originally proposed for solving systems of linear equations we obtain a hessenberg decomposition from which we approximate the eigenvalues and eigenvectors of a matrix this decomposition has two main advantages first idr s is a short recurrence method which is attractive for large scale computations second the idr s polynomial used to create this hessenberg decomposition is also used as a filter to discard the unwanted eigenvalues additionally we incorporate the implicitly restarting technique proposed by d c sorensen in order to approximate specific portions of the spectrum and improve the convergence

the second order biorthogonalization procedure and its application to quadratic eigenvalue problems given a pair of matrices and starting vectors we present a procedure to generate the biorthonormal basis of the second order right and left krylov subspaces the application is to solve the large scale quadratic eigenvalue problems via oblique projection technique this method can take full advantage of the sparseness of large scale system as well as the superior convergence behavior of krylov subspace based methods by implicit linearization which makes the solution acceptable in terms of both cost and time

an explicit formula for lanczos polynomials abstract the lanczos algorithm for tridiagonalizing a given matrix a generates a sequence of approximating matrices a p that can naturally be obtained as restrictions to subspaces the eigenvalues of these approximating matrices are well known to be good approximations to the extreme eigenvalues of a in this paper we produce explicit formulas for the characteristics polynomials of the a p in terms of the eigenvalues of a these formulas can be used to explain heuristically why these approximations are often quite good at present we have no concrete analytic argument that explains the quality of the approximation the main result draws on the formal relationship between the lanczos algorithm and pade approximations to the moment generating function of a this result is one of the few analytic results available for the unsymmetric lanczos algorithm