a global rational arnoldi method for model reduction abstract in this paper we propose two new approaches for model order reduction of large scale multi input multi output mimo linear time invariant dynamical systems lti these methods are based on a generalization of the global arnoldi algorithm which is used to generate projection subspaces an adaptive procedure for the selection of shift parameters is proposed and some simple arnoldi relations are established in order to compute error bounds on the transfer function numerical experiments showing the effectiveness of these approaches are provided

frequency limited balanced truncation with low rank approximations in this article we investigate model order reduction of large scale systems using frequency limited balanced truncation which restricts the well known balanced truncation framework to prescribed frequency regions the main emphasis is put on the efficient numerical realization of this model reduction approach we discuss numerical methods to take care of the involved matrix valued functions the occurring large scale lyapunov equations are solved for low rank approximations for which we also establish results regarding the eigenvalues of their solutions these results and also numerical experiments will show that the eigenvalues of the lyapunov solutions in frequency limited balanced truncation often decay faster than those in standard balanced truncation moreover we show in further numerical examples that frequency limited balanced truncation generates reduced order models which are significantly more accurate in the considered frequency region

homotopy techniques for tensor decomposition and perfect identifiability let t be a general complex tensor of format n1 nd when the fraction i ni 1 i ni 1 is an integer and a natural inequality called balancedness is satisfied it is expected that t has finitely many minimal decomposition as a sum of decomposable tensors we show how homotopy techniques allow us to find all the decompositions of t starting from a given one in particular this gives a guess true with high probability about the total number of these decompositions this guess matches exactly with all cases previously known and predicts several unknown cases some surprising experiments yielded two new cases of generic identifiability formats 3 4 5 and 2 2 2 3 which have a unique decomposition as the sum of 6 and 4 decomposable tensors respectively we conjecture that these two cases together with the classically known matrix pencils are the only cases where generic identifiability holds i e the only identifiable cases building on the computational experiments we also use basic tools from algebraic geometry to prove these two new cases are indeed generically identifiable

parallelization of the rational arnoldi algorithm rational krylov methods are applicable to a wide range of scientific computing problems and the rational arnoldi algorithm is a commonly used procedure for computing an orthonormal basis of a rational krylov space typically the computationally most expensive component of this algorithm is the solution of a large linear system of equations at each iteration we explore the option of solving several linear systems simultaneously thus constructing the rational krylov basis in parallel if this is not done carefully the basis being orthogonalized may become badly conditioned leading to numerical instabilities in the orthogonalization process we introduce the new concept of continuation pairs which gives rise to a near optimal parallelization strategy that allows to control the growth of the condition number of this nonorthogonal basis as a consequence we obtain a significantly more accurate and reliable parallel rational arnoldi algorithm the computational benefits are illustrated using several numerical examples from different application areas

automatic smoothness detection of the resolvent krylov subspace method for the approximation of c_0 semigroups the resolvent krylov subspace method builds approximations to operator functions f a times a vector v for the semigroup and related operator functions this method is proved to possess the favorable property that the convergence is automatically faster when the vector v is smoother the user of the method does not need to know the presented theory and alterations of the method are not necessary in order to adapt to the possibly unknown smoothness of v the findings are illustrated by numerical experiments

a projection method for generalized eigenvalue problems using numerical integration in this paper we propose a method for finding certain eigenvalues of a generalized eigenvalue problem that lie in a given domain of the complex plane the proposed method projects the matrix pencil onto a subspace associated with the eigenvalues that are located in the domain via numerical integration

computing eigenvalues of real symmetric matrices with rational filters in real arithmetic powerful algorithms have recently been proposed for computing eigenvalues of large matrices by methods related to contour integrals best known are the works of sakurai and coauthors and polizzi and coauthors even if the matrices are real symmetric most such methods rely on complex arithmetic leading to expensive linear systems to solve an appealing technique for overcoming this starts from the observation that certain discretized contour integrals are equivalent to rational interpolation problems for which there is no need to leave the real axis investigation shows that using rational interpolation per se suffers from instability however related techniques involving real rational filters can be very effective this article presents a technique of this kind that is related to previous work published in japanese by murakami

a rational function preconditioner for indefinite sparse linear systems this paper introduces a rational function preconditioner for linear systems with indefinite sparse matrices a by resorting to rational functions of a the algorithm decomposes the spectrum of a into two disjoint regions and approximates the restriction of a 1 on these regions separately we show a systematic way to construct these rational functions so that they can be applied stably and inexpensively an attractive feature of the proposed approach is that the construction and application of the preconditioner can exploit two levels of parallelism moreover the proposed preconditioner can be modified at a negligible cost into a preconditioner for a nearby matrix of the form a ci which can be useful in some applications the efficiency and robustness of the proposed preconditioner are demonstrated on a few tests with challenging model problems including problems arising from the helmholtz equation in three dimensions

a contour integral approach to the computation of invariant pairs abstract we study some aspects of the invariant pair problem for matrix polynomials as introduced by betcke and kressner 3 and by beyn and thummler 6 invariant pairs extend the notion of eigenvalue eigenvector pairs providing a counterpart of invariant subspaces for the nonlinear case we compute formulations for the condition numbers and the backward error for invariant pairs and solvents we then adapt the sakurai sugiura moment method 1 to the computation of invariant pairs including some classes of problems that have multiple eigenvalues numerical refinement via a variant of newton s method is also studied furthermore we investigate the relation between the matrix solvent problem and the triangularization of matrix polynomials

nonlinear eigenvalue problems and contour integrals in this paper beyn s algorithm for solving nonlinear eigenvalue problems is given a new interpretation and a variant is designed in which the required information is extracted via the canonical polyadic decomposition of a hankel tensor a numerical example shows that the choice of the filter function is very important particularly with respect to where it is positioned in the complex plane

interpreting idr as a petrov galerkin method the induced dimension reduction idr method of sonneveld and van gijzen siam j sci comput 31 2008 pp 1035 1062 is shown to be a petrov galerkin projection method with a particular choice of left krylov subspaces these left subspaces are rational krylov spaces consequently other methods such as bicgstab and ml s bicgstab which are mathematically equivalent to some versions of idr can also be interpreted as petrov galerkin methods the connection with rational krylov spaces inspired a new version of idr called ritz idr where the poles of the rational function are chosen as certain ritz values experiments are presented illustrating the effectiveness of this new version

an error analysis of two related quadrature methods for computing zeros of analytic functions we present an error analysis for two related quadrature methods the delves lyness method and its modification by kravanja sakurai and van barel for computing all the zeros of an analytic function that lie inside the unit circle we consider the forward as well as the backward approximation error in case the integrals are computed via the trapezoidal rule on the unit circle contrary to the delves lyness method the quadrature error that arises from the zeros located inside the unit circle does not affect the results of the approach of kravanja et al numerical experiments illustrate our main results

the exponentially convergent trapezoidal rule it is well known that the trapezoidal rule converges geometrically when applied to analytic functions on periodic intervals or the real line the mathematics and history of this phenomenon are reviewed and it is shown that far from being a curiosity it is linked with computational methods all across scientific computing including algorithms related to inverse laplace transforms special functions complex analysis rational approximation integral equations and the computation of functions and eigenvalues of matrices and operators

from local to global triangularization abstract it is shown that a semigroup of shatten p class operators is simultaneously triangularizable if each pair of operators in the semigroup is triangularizable several sufficient conditions for triangularizability of semigroups are obtained as corollaries a block triangularization theorem for algebras of compact operators is established consequences of which include a number of necessary and sufficient conditions for triangularization of such algebras

convergence analysis of an extended krylov subspace method for the approximation of operator functions in exponential integrators we analyze the convergence of an extended krylov subspace method for the approximation of operator functions that appear in exponential integrators for operators the size of the polynomial part of the extended krylov subspace is restricted according to the smoothness of the initial data this restriction for the continuous operator has a significant influence on the approximation of matrix functions evaluated for matrices stemming from space discretizations of the continuous operator we prove convergence of the method for the continuous operator and in the discrete case this leads to a convergence independent of the norm of the discretized operator uniformly over all possible grids the analysis is illustrated by numerical experiments

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

computing partial spectra with least squares rational filters we present a method for computing partial spectra of hermitian matrices based on a combination of subspace iteration with rational filtering in contrast with classical rational filters derived from cauchy integrals or from uniform approximations to a step function we adopt a least squares ls viewpoint for designing filters one of the goals of the proposed approach is to build a filter that will lead to linear systems that are easier to solve by iterative methods among the main advantages of the proposed ls filters is their flexibility thus we can place poles in more general locations than with the formulations mentioned above and we can also repeat these poles a few times for better efficiency this leads to a smaller number of required poles than in existing methods as a consequence factorization costs are reduced when direct solvers are used and the scheme is also beneficial for iterative solvers the paper discusses iterative schemes to solve the linear systems resulting from the filtered s

dissecting the feast algorithm for generalized eigenproblems we analyze the feast method for computing selected eigenvalues and eigenvectors of large sparse matrix pencils after establishing the close connection between feast and the well known rayleigh ritz method we identify several critical issues that influence convergence and accuracy of the solver the choice of the starting vector space the stopping criterion how the inner linear systems impact the quality of the solution and the use of feast for computing eigenpairs from multiple intervals we complement the study with numerical examples and hint at possible improvements to overcome the existing problems

rational gauss quadrature the existence of standard gauss quadrature rules with respect to a nonnegative measure d mu with support on the real axis easily can be shown with the aid of orthogonal polynomials with respect to this measure efficient algorithms for computing the nodes and weights of an n point gauss rule use the n times n symmetric tridiagonal matrix determined by the recursion coefficients for the first n orthonormal polynomials many rational functions that are orthogonal with respect to the measure d mu and have real or complex conjugate poles also satisfy a short recursion relations this paper describes how banded matrices determined by the recursion coefficients for these orthonormal rational functions can be used to efficiently compute the nodes and weights of rational gauss quadrature rules

adaptive rational krylov subspaces for large scale dynamical systems the rational krylov space is recognized as a powerful tool within model order reduction techniques for linear dynamical systems however its success has been hindered by the lack of a parameter free procedure which would effectively generate the sequence of shifts used to build the space in this paper we propose an adaptive computation of these shifts the whole procedure only requires us to inject some initial rough estimate of the spectral region of the matrix while further information is automatically generated during the process the approach is a full generalization to the nonsymmetric case of the idea first proposed in druskin et al 2010 18 and it is used for two important problems in control the approximation of the transfer function and the numerical solution of large lyapunov equations the procedure can be naturally extended to other related problems such as the solution of the sylvester equation and parametric or higher order systems several numerical experiments are proposed to assess the quality of the rational projection space over its most natural competitors

an integral method for solving nonlinear eigenvalue problems we propose a numerical method for computing all eigenvalues and the corresponding eigenvectors of a nonlinear holomorphic eigenvalue problem that lie within a given contour in the complex plane the method uses complex integrals of the resolvent operator applied to at least k column vectors where k is the number of eigenvalues inside the contour the theorem of keldysh is employed to show that the original nonlinear eigenvalue problem reduces to a linear eigenvalue problem of dimension k no initial approximations of eigenvalues and eigenvectors are needed the method is particularly suitable for moderately large eigenvalue problems where k is much smaller than the matrix dimension we also give an extension of the method to the case where k is larger than the matrix dimension the quadrature errors caused by the trapezoid sum are discussed for the case of analytic closed contours using well known techniques it is shown that the error decays exponentially with an exponent given by the product of the number of quadrature points and the minimal distance of the eigenvalues to the contour

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

a novel numerical method to determine the algebraic multiplicity of nonlinear eigenvalues we generalize the algebraic multiplicity of the eigenvalues of nonlinear eigenvalue problems neps to the rational form and give the extension of the argument principle in addition we propose a novel numerical method to determine the algebraic multiplicity of the eigenvalues of the neps in a given region by the contour integral method finally some numerical experiments are reported to illustrate the effectiveness of our method

a lanczos model order reduction technique to efficiently simulate electromagnetic wave propagation in dispersive media in this paper we present a krylov subspace model order reduction technique for time and frequency domain electromagnetic wave fields in linear dispersive media starting point is a self consistent first order form of maxwell s equations and the constitutive relation this form is discretized on a standard staggered yee grid while the extension to infinity is modeled via a recently developed global complex scaling method by applying this scaling method the time or frequency domain electromagnetic wave field can be computed via a so called stability corrected wave function since this function cannot be computed directly due to the large order of the discretized maxwell system matrix krylov subspace reduced order models are constructed that approximate this wave function we show that the system matrix exhibits a particular physics based symmetry relation that allows us to efficiently construct the time and frequency domain reduced order models via a lanczos type reduction algorithm the frequency domain models allow for frequency sweeps meaning that a single model provides field approximations for all frequencies of interest and dominant field modes can easily be determined as well numerical experiments for two and three dimensional configurations illustrate the performance of the proposed reduction method

singularities of linear systems and the waring problem the waring problem for homogeneous forms asks for additive decomposition of a form f into powers of linear forms a classical problem is to determine when such a decomposition is unique in this paper we answer this question when the degree of f is greater than the number of variables to do this we translate the algebraic statement into a geometric one concerning the singularities of linear systems of p with assigned singularities

using implicitly filtered rks for generalised eigenvalue problems the rational krylov sequence rks method can be seen as a generalisation of arnoldi s method it projects a matrix pencil onto a smaller subspace this projection results in a small upper hessenberg pencil as for the arnoldi method rks can be restarted implicitly using the qr decomposition of a hessenberg matrix this restart comes with a projection of the subspace using a rational function in this paper it is shown how the restart can be worked out in practice in a second part it is shown when the filtering of the subspace basis can fail and how this failure can be handled by deflating a converged eigenvector from the subspace using a schur decomposition

on investigating gmres convergence using unitary matrices for a given matrix a and right hand side b this paper investigates unitary matrices generating with some right hand sides c the same gmres residual norms as the pair a b we give characterizations of this class of unitary matrices and point out the relationship with krylov subspaces and krylov residual subspaces for the pair a b we investigate the eigenvalues of these unitary matrices in relation to the convergence behavior of gmres for the pair a b and describe the indispensable role of the eigenvector information we conclude with a formula for the gmres residual norms generated by a normal matrix b in terms of its eigenvalues and components of the right hand side c in the eigenvector basis

decomposition of homogeneous polynomials with low rank let f be a homogeneous polynomial of degree d in m 1 variables defined over an algebraically closed field of characteristic 0 and suppose that f belongs to the s th secant variety of the d uple veronese embedding of mathbb p m into mathbb p tiny left begin array c rm m d rm d end array right 1 but that its minimal decomposition as a sum of d th powers of linear forms m1 mr is f m 1 d cdots m r d with r s we show that if s r 2d 1 then such a decomposition of f can be split in two parts one of them is made by linear forms that can be written using only two variables the other part is uniquely determined once one has fixed the first part we also obtain a uniqueness theorem for the minimal decomposition of f if r is at most d and a mild condition is satisfied

an implicit filter for rational krylov using core transformations abstract the rational krylov method is a powerful tool for computing a selected subset of eigenvalues in large scale eigenvalue problems in this paper we study a method to implicitly apply a filter in a rational krylov iteration by directly acting on a qr factorized representation of the hessenberg pair from the rational krylov method this filter is used to restart the iteration which is generally required to limit the orthogonalization and storage costs the contribution in this paper is threefold we reformulate existing procedures in terms of operations on core transformations this has the advantage of improved convergence monitoring secondly we demonstrate that the extended qz method is a special case of this more general method finally numerical experiments show the validity and the increased accuracy of the new approach compared with existing methods

inverse system of a symbolic power i abstract the authors determine macaulay s inverse system to a symbolic power of the graded ideal of functions vanishing at a union of irreducible varieties in projective space

superlinear convergence of the rational arnoldi method for the approximation of matrix functions a superlinear convergence bound for rational arnoldi approximations to functions of matrices is derived this bound generalizes the well known superlinear convergence bound for the conjugate gradient method to more general functions with finite singularities and to rational krylov spaces a constrained equilibrium problem from potential theory is used to characterize a max min quotient of a nodal rational function underlying the rational arnoldi approximation where an additional external field is required for taking into account the poles of the rational krylov space the resulting convergence bound is illustrated at several numerical examples in particular the convergence of the extended krylov method for the matrix square root

a fast algorithm for parabolic pde based inverse problems based on laplace transforms and flexible krylov solvers we consider the problem of estimating parameters in large scale weakly nonlinear inverse problems for which the underlying governing equations is a linear time dependent parabolic partial differential equation a major challenge in solving these inverse problems using newton type methods is the computational cost associated with solving the forward problem and with repeated construction of the jacobian which represents the sensitivity of the measurements to the unknown parameters forming the jacobian can be prohibitively expensive because it requires repeated solutions of the forward and adjoint time dependent parabolic partial differential equations corresponding to multiple sources and receivers we propose an efficient method based on a laplace transform based exponential time integrator combined with a flexible krylov subspace approach to solve the resulting shifted systems of equations efficiently our proposed solver speeds up the computation of the forward and adjoint problems thus yielding significant speedup in total inversion time we consider an application from transient hydraulic tomography tht which is an imaging technique to estimate hydraulic parameters related to the subsurface from pressure measurements obtained by a series of pumping tests the algorithms discussed are applied to a synthetic example taken from tht to demonstrate the resulting computational gains of this proposed method

randomized estimation of spectral densities of large matrices made accurate for a large hermitian matrix a in mathbb c n times n a cn n it is often the case that the only affordable operation is matrix vector multiplication in such case randomized method is a powerful way to estimate the spectral density or density of states of a however randomized methods developed so far for estimating spectral densities only extract information from different random vectors independently and the accuracy is therefore inherently limited to mathcal o 1 sqrt n v o 1 nv where n v nv is the number of random vectors in this paper we demonstrate that the mathcal o 1 sqrt n v o 1 nv barrier can be overcome by taking advantage of the correlated information of random vectors when properly filtered by polynomials of a our method uses the fact that the estimation of the spectral density essentially requires the computation of the trace of a series of matrix functions that are numerically low rank by repeatedly applying a to the same set of random vectors and taking different linear combination of the results we can sweep through the entire spectrum of a by building such low rank decomposition at different parts of the spectrum under some assumptions we demonstrate that a robust and efficient implementation of such spectrum sweeping method can compute the spectral density accurately with mathcal o n 2 o n2 computational cost and mathcal o n o n memory cost numerical results indicate that the new method can significantly outperform existing randomized methods in terms of accuracy as an application we demonstrate a way to accurately compute a trace of a smooth matrix function by carefully balancing the smoothness of the integrand and the regularized density of states using a deconvolution procedure

lyapunov inverse iteration for identifying hopf bifurcations in models of incompressible flow the identification of instability in large scale dynamical systems caused by hopf bifurcation is difficult because of the problem of identifying the rightmost pair of complex eigenvalues of large sparse generalized eigenvalue problems a new method developed in k meerbergen and a spence siam j matrix anal appl 31 2010 pp 1982 1999 avoids this computation instead performing an inverse iteration for a certain set of real eigenvalues that requires the solution of a large scale lyapunov equation at each iteration in this study we refine the lyapunov inverse iteration method to make it more robust and efficient and we examine its performance on challenging test problems arising from fluid dynamics various implementation issues are discussed including the use of inexact inner iterations and the impact of the choice of iterative solution for the lyapunov equations and the effect of eigenvalue distribution on performance numerical experiments demonstrate the robustness of the algorithm

an implicitly restarted krylov subspace method for real symmetric skew symmetric eigenproblems abstract a new implicitly restarted krylov subspace method for real symmetric skew symmetric generalized eigenvalue problems is presented the new method improves and generalizes the shira method of mehrmann and watkins 2001 37 to the case where the skew symmetric matrix is singular it computes a few eigenvalues and eigenvectors of the matrix pencil close to a given target point several applications from control theory are presented and the properties of the new method are illustrated by benchmark examples

recursion relations for the extended krylov subspace method abstract the evaluation of matrix functions of the form f a v where a is a large sparse or structured symmetric matrix f is a nonlinear function and v is a vector is frequently subdivided into two steps first an orthonormal basis of an extended krylov subspace of fairly small dimension is determined and then a projection onto this subspace is evaluated by a method designed for small problems this paper derives short recursion relations for orthonormal bases of extended krylov subspaces of the type k m mi 1 a span a m 1 v a 1 v v a v a mi v m 1 2 3 with i a positive integer and describes applications to the evaluation of matrix functions and the computation of rational gauss quadrature rules

on the computation of a very large number of eigenvalues for selfadjoint elliptic operators by means of multigrid methods recent results in the study of quantum manifestations in classical chaos raise the problem of computing a very large number of eigenvalues of selfadjoint elliptic operators the standard numerical methods for large eigenvalue problems cover the range of applications where a few of the leading eigenvalues are needed they are not appropriate and generally fail to solve problems involving a number of eigenvalues exceeding a few hundreds further the accurate computation of a large number of eigenvalues leads to much larger problem dimension in comparison with the usual case dealing with only a few eigenvalues a new method is presented which combines multigrid techniques with the lanczos process the resulting scheme requires o mn arithmetic operations and o n storage requirement where n is the number of unknowns and m the number of needed eigenvalues the discretization of the considered differential operators is realized by means of p finite elements and is applicable on general geometries numerical experiments validate the proposed approach and demonstrate that it allows to tackle problems considered to be beyond the range of standard iterative methods at least on current workstations the ability to compute more than 9000 eigenvalues of an operator of dimension exceeding 8 million on a pc shows the potential of this method practical applications are found e g in the numerical simulation of quantum billiards

a refined shift and invert arnoldi algorithm for large unsymmetric generalized eigenproblems abstract the shift and invert arnoldi method has been popularly used for computing a number of eigenvalues close to a given shift and or the associated eigenvectors of a large unsymmetric matrix pair but there is no guarantee for the approximate eigenvectors ritz vectors obtained by this method to converge even though the subspace is good enough in order to correct this problem a refined shift and invert arnoldi method is proposed that uses certain refined ritz vectors to approximate the desired eigenvectors the refined ritz vectors can be computed cheaply and reliably by small sized singular value decompositions it is shown that the refined method converges a refined shift and invert arnoldi algorithm is developed and several numerical examples are reported comparisons are drawn on the refined algorithm and the shift and invert arnoldi algorithm indicating that the former is considerably more efficient than the latter

stable polefinding and rational least squares fitting via eigenvalues a common way of finding the poles of a meromorphic function f in a domain where an explicit expression of f is unknown but f can be evaluated at any given z is to interpolate f by a rational function pq such that r i f i at prescribed sample points i i 1l and then find the roots of q this is a two step process and the type of the rational interpolant needs to be specified by the user many other algorithms for polefinding and rational interpolation or least squares fitting have been proposed but their numerical stability has remained largely unexplored in this work we describe an algorithm with the following three features 1 it automatically finds an appropriate type for a rational approximant thereby allowing the user to input just the function f 2 it finds the poles via a generalized eigenvalue problem of matrices constructed directly from the sampled values f i in a one step fashion and 3 it computes rational approximants p q in a numerically stable manner in that p p q q f with small p q at the sample points making it the first rational interpolation or approximation algorithm with guaranteed numerical stability our algorithm executes an implicit change of polynomial basis by the qr factorization and allows for oversampling combined with least squares fitting through experiments we illustrate the resulting accuracy and stability which can significantly outperform existing algorithms

a numerical solution of the constrained weighted energy problem a numerical algorithm is presented to solve the constrained weighted energy problem from potential theory as one of the possible applications of this algorithm we study the convergence properties of the rational lanczos iteration method for the symmetric eigenvalue problem the constrained weighted energy problem characterizes the region containing those eigenvalues that are well approximated by the ritz values the region depends on the distribution of the eigenvalues on the distribution of the poles and on the ratio between the size of the matrix and the number of iterations our algorithm gives the possibility of finding the boundary of this region in an effective way we give numerical examples for different distributions of poles and eigenvalues and compare the results of our algorithm with the convergence behavior of the explicitly performed rational lanczos algorithm

generalized block lanczos methods for large unsymmetric eigenproblems generalized block lanczos methods for large unsymmetric eigenproblems are presented which contain the block arnoldi method and the block arnoldi algorithms are developed the convergence of this class of methods is analyzed when the matrix a is diagonalizable upper bounds for the distances between normalized eigenvectors and a block krylov subspace are derived and a priori theoretical error bounds for ritz elements are established compared with generalized lanczos methods which contain arnoldi s method the convergence analysis shows that the block versions have two advantages first they may be efficient for computing clustered eigenvalues second they are able to solve multiple eigenproblems however a deep analysis exposes that the approximate eigenvectors or ritz vectors obtained by general orthogonal projection methods including generalized block methods may fail to converge theoretically for a general unsymmetric matrix a even if corresponding approximate eigenvalues or ritz values do since the convergence of ritz vectors needs more sufficient conditions which may be impossible to satisfy theoretically than that of ritz values does the issues of how to restart and to solve multiple eigenproblems are addressed and some numerical examples are reported to confirm the theoretical analysis

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

numerical algorithms based on analytic function values at roots of unity let f z be an analytic or meromorphic function in the closed unit disk sampled at the n th roots of unity based on these data how can we approximately evaluate f z or f m z at a point z in the disk how can we calculate the zeros or poles of f in the disk these questions exhibit in the purest form certain algorithmic issues that arise across computational science in areas including integral equations partial differential equations and large scale linear algebra we analyze some of the possibilities and emphasize the distinction between algorithms based on polynomial or rational interpolation and those based on trapezoidal rule approximations of cauchy integrals we then show how these developments apply to the problem of computing the eigenvalues in the disk of a matrix of large dimension finally we highlight the power of rational in comparison with polynomial approximations for some of these problems

k3 surfaces of genus 8 and varieties of sums of powers of cubic fourfolds the main result of this paper is that the variety of presentations of a general cubic form f in 6 variables as a sum of 10 cubes is isomorphic to the fano variety of lines of a cubic 4 fold f in general different from f z f a general k3 surface s of genus 8 determines uniquely a pair of cubic 4folds the apolar cubic f s and the dual pfaffian cubic f s or for simplicity f and f as beauville and donagi have shown the fano variety tf of lines on the cubic f is isomorphic to the hilbert scheme hilb2 s of length two subschemes of s the first main result of this paper is that hilb2 s parametrizes the variety vsp f 10 of presentations of the cubic form f with f z f as a sum of 10 cubes which yields an isomorphism between nff and vsp f 10 furthermore we show that vsp f 10 sets up a 6 10 correspondence between f and ff the main result follows by a deformation argument 1 pfaffian and apolar cubic 4 folds associated to k3 surfaces of genus 8 1 1 let v be a 6 dimensional vector space over c fix a basis e e5 for v then ei a ej for 0 i j aijei aej e a2v a skewsymmetric matrix m g aij with aji aj with pliicker coordinates xij the embedding of the grassmannian g g 2 v in p14 p a2v is then precisely the locus of rank 2 skewsymmetric 6 x 6 matrices 0 x01 x02 x03 x04 x05 x0i 0 x12 x13 x14 x15 m x02 x12 0 x23 x24 x25 x03 x13 x23 0 x34 x35 x04 x14 x24 x34 0 x45 x05 x15 x25 x35 x45 0 since the sum of two rank 2 matrices has rank at most 4 and any rank 4 skewsymmetric matrix is the sum of two rank 2 skewsymmetric matrices the secant variety of g is the cubic hypersurface k defined by the 6 x 6 pfaffian m of the matrix m the dual variety of g in p14 p a2v is a cubic hypersurface k k cf 10 k is the secant variety of g g v 2 the grassmannian of rank 2 quotient spaces of v and of course g c g received by the editors july 5 1999 2000 mathematics subject classification primary 14j70 secondary 14m15 14n99 2000 american mathematical society

paraexp a parallel integrator for linear initial value problems a novel parallel algorithm for the integration of linear initial value problems is proposed this algorithm is based on the simple observation that homogeneous problems can typically be integrated much faster than inhomogeneous problems an overlapping time domain decomposition is utilized to obtain decoupled inhomogeneous and homogeneous subproblems and a near optimal krylov method is used for the fast exponential integration of the homogeneous subproblems we present an error analysis and discuss the parallel scaling of our algorithm the efficiency of this approach is demonstrated with numerical examples

a unification of unitary similarity transforms to compressed representations in this paper a new framework for transforming arbitrary matrices to compressed representations is presented the framework provides a generic way of transforming a matrix via unitary similarity transformations to e g hessenberg hessenberg like form and combinations of both the new algorithms are deduced based on the qr factorization of the original matrix relying on manipulations with rotations all the algorithms consist of eliminating the correct set of rotations resulting in a matrix obeying the desired structural constraints based on this new reduction procedure we investigate further correspondences such as irreducibility uniqueness of the reduction procedure and the link with rational krylov methods the unitary similarity transform to hessenberg like form as presented here differs significantly from the one presented in earlier work not only does it use less rotations to obtain the desired structure also the convergence to rational ritz values is not observed in the conventional approach

on generic identifiability of symmetric tensors of subgeneric rank we prove that the general symmetric tensor in sdcn 1 of rank r is identifiable provided that r is smaller than the generic rank that is its waring decomposition as a sum of r powers of linear forms is unique only three exceptional cases arise all of which were known in the literature our original contribution regards the case of cubics d 3 while for d 4 we rely on known results on weak defectivity by ballico ciliberto chiantini and mella

high performance implementation of chebyshev filter diagonalization for interior eigenvalue computations we study chebyshev filter diagonalization as a tool for the computation of many interior eigenvalues of very large sparse symmetric matrices in this technique the subspace projection onto the target space of wanted eigenvectors is approximated with filter polynomials obtained from chebyshev expansions of window functions after the discussion of the conceptual foundations of chebyshev filter diagonalization we analyze the impact of the choice of the damping kernel search space size and filter polynomial degree on the computational accuracy and effort before we describe the necessary steps towards a parallel high performance implementation because chebyshev filter diagonalization avoids the need for matrix inversion it can deal with matrices and problem sizes that are presently not accessible with rational function methods based on direct or iterative linear solvers to demonstrate the potential of chebyshev filter diagonalization for large scale problems of this kind we include as an example the computation of the 102 innermost eigenpairs of a topological insulator matrix with dimension 109 derived from quantum physics applications

the rkfit algorithm for nonlinear rational approximation the rkfit algorithm outlined in m berljafa and s ga ttel generalized rational krylov decompositions with an application to rational approximation siam j matrix anal appl 2015 is a krylov based approach for solving nonlinear rational least squares problems this paper puts rkfit into a general framework allowing for its extension to nondiagonal rational approximants and a family of approximants sharing a common denominator furthermore we derive a strategy for the degree reduction of the approximants as well as methods for their conversion to partial fraction form for the efficient evaluation and root finding we also discuss commons and differences of rkfit and the popular vector fitting algorithm a matlab implementation of rkfit is provided and numerical experiments including the fitting of a mimo dynamical system and an optimization problem related to exponential integration demonstrate its applicability

structured perturbation analysis of sparse matrix pencils with s specified eigenpairs abstract we study the structured backward error analysis of s specified 1 s n eigenpairs of n by n matrix pencils the structures we consider include t symmetric t skew symmetric hermitian skew hermitian t even t odd h even h odd t palindromic t anti palindromic h palindromic and h anti palindromic matrix pencils minimal structured perturbations are constructed with respect to frobenius norm such that s specified eigenpairs become exact eigenpairs of an appropriately perturbed matrix pencil which preserves sparsity

error estimates and evaluation of matrix functions via the faber transform the need to evaluate expressions of the form f a or f a b where f is a nonlinear function a is a large sparse n times n matrix and b is an n vector arises in many applications this paper describes how the faber transform applied to the field of values of a can be used to determine improved error bounds for popular polynomial approximation methods based on the arnoldi process applications of the faber transform to rational approximation methods and in particular to the rational arnoldi process also are discussed

a brief proof of a maximal rank theorem for generic double points in projective space we give a simple proof of the following theorem of j alexander and a hirschowitz given a general set of points in projective space the homogeneous ideal of polynomials that are singular at these points has the expected dimension in each degree of 4 and higher except in 3 cases

rational qr iteration without inversion in this manuscript a new method will be presented for performing a qr iteration with a i a i 1 qr without explicit inversion of the factor a i 1 a qr method driven by a rational function is attractive since convergence can occur at both sides of the matrix each step of this new iteration consists of two substeps in the explicit version first an rq factorization of the initial matrix a i rq will be computed followed by a qr factorization of the matrix a i q h the factorization of a i q h can be computed in an intelligent manner exploiting properties of the already known rq factorization of a i the similarity transformation yielding the qr step is defined by the unitary factor q in the qr factorization of the transformed matrix a i q h examples will be given illustrating how to efficiently compute the factorization for some specific classes of matrices the novelties of this approach with respect to these matrix classes will be discussed

inverse system of a symbolic power ii the waring problem for forms abstract the classical waring problem for forms is to determine the smallest length s of an additive decomposition of a general degree d homogeneous polynomial or form f in r variables as sum of s d th powers of linear forms we show that its solution is implied by a result of j alexander and a hirschowitz concerning the hilbert functions of the ideal of functions vanishing to order two at a generic set of s points in p r 1 using macaulay s inverse systems we show that the alexander hirschowitz result is equivalent to determining the number of linear syzygies of s homogeneous forms in r variables that are d th powers of a given set of general linear forms we also determine the dimension of the family of degree d forms that have additive decompositions of length s we then study several notions of length for forms f having to do with the kind of length s zero dimensional schemes z in p r 1 whose defining ideal i z annihilates the inverse system of f when z is to consist of distinct points we obtain the above length of additive decomposition of f when z is smoothable we obtain the smoothable length of f when z is arbitrary we obtain a scheme length of f all these lengths are at least as large as the dimension of the vector space of all order i partial derivates of f for each i the above mentioned length functions are distinct using results about the existence of nonsmoothable gorenstein point singularities in codimension 4 we show that when r 5 there are forms f of scheme length s which are not in the closure of the family of forms having additive decompositions of length s finally we propose a new set of waring problems for forms using these lengths

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

invariant subspaces for operator semigroups with commutators of rank at most one let x be a complex banach space of dimension at least 2 and let s be a multiplicative semigroup of operators on x such that the rank of st ts is at most 1 for all s t s we prove that s has a non trivial invariant subspace provided it is not commutative as a consequence we show that s is triangularizable if it consists of polynomially compact operators this generalizes results from h radjavi p rosenthal from local to global triangularization j funct anal 147 1997 443 456 and g cigler r drnovsek d kokol bukovsek t laffey m omladic h radjavi p rosenthal invariant subspaces for semigroups of algebraic operators j funct anal 160 1998 452 465

on the non hermitian feast algorithms with oblique projection for eigenvalue problems abstract the feast algorithm due to polizzi is a typical contour integral based eigensolver for computing the eigenvalues along with their eigenvectors inside a given region in the complex plane it was formulated under the circumstance that the considered eigenproblem is hermitian the feast algorithm is stable and accurate and has attracted much attention in recent years however yin et al 2017 observed that the feast algorithm may fail to find the target eigenpairs when applied to non hermitian problems this observation motivated them to formulate a non hermitian version for the feast algorithm the key to the success of their method is that they use the oblique projection rather than the orthogonal projection used in the feast algorithm to extract desired eigenpairs the objective of the present work is two folded firstly we would like to present a new non hermitian scheme for the feast algorithm which also uses the oblique projection technique to extract the desired eigenpairs secondly we would like to study the convergence properties of the non hermitian feast algorithms based on oblique projection to illustrate the effectiveness numerical experiments are reported to demonstrate the numerical performance of the non hermitian feast algorithms based on oblique projection and to validate the convergence properties studied

recursive integral method for transmission eigenvalues the transmission eigenvalue problem arose in the inverse scattering theory for inhomogeneous media and has important applications in a variety of inverse problems for target identification and nondestructive testing the problem is numerically challenging because of the nonself adjointness and complicated spectrum in this talk we introduce a novel recursive contour integral method using the spectrum projection the method can compute both real and complex eigenvalues and does not require any a priori information numerical examples show that the method is effective and robust

invariant subspaces for semigroups of algebraic operators abstract t laffey showed linear and multilinear algebra 6 1978 269 305 that a semigroup of matrices is triangularizable if the ranks of all the commutators of elements of the semigroup are at most 1 our main theorem is an extension of hthis result to semigroups of algebraic operators on a banach space we also obtain a related theorem for a pair a b of arbitrary bounded operators satisfying rank ab ba 1 and several related conditions in addition it is shown that a semigroup of algebraically unipotent operators of bounded degree is triangularizable

solution of the k th eigenvalue problem in large scale electronic structure calculations abstract we consider computing the k th eigenvalue and its corresponding eigenvector of a generalized hermitian eigenvalue problem of n n large sparse matrices in electronic structure calculations several properties of materials such as those of optoelectronic device materials are governed by the eigenpair with a material specific index k we present a three stage algorithm for computing the k th eigenpair with validation of its index in the first stage of the algorithm we propose an efficient way of finding an interval containing the k th eigenvalue 1 k n with a non standard application of the lanczos method in the second stage spectral bisection for large scale problems is realized using a sparse direct linear solver to narrow down the interval of the k th eigenvalue in the third stage we switch to a modified shift and invert lanczos method to reduce bisection iterations and compute the k th eigenpair with validation numerical results with problem sizes up to 1 5 million are reported and the results demonstrate the accuracy and efficiency of the three stage algorithm

a high order accurate scheme for maxwell s equations with a generalized dispersive material model abstract a high order accurate scheme for solving the time domain maxwell s equations with a generalized dispersive material model is described the equations for the electric field are solved in second order form and a general dispersion model is treated with the addition of one or more polarization vectors which obey a set of auxiliary differential equations ade numerical methods are developed for both second order and fourth order accuracy in space and time the equations are discretized using finite differences and advanced in time with a single stage three level space time scheme which remains stable up to the usual explicit cfl restriction as proven using mode analysis because the equations are treated in their second order form there is no need for grid staggering and instead a collocated grid is used composite overlapping grids are used to treat complex geometries with boundary conforming grids and a high order upwind dissipation is added to ensure robust and stable approximations on overlapping grids numerical results in two and three space dimensions confirm the accuracy and stability of the new schemes

conversions between barycentric rkfun and newton representations of rational interpolants we derive explicit formulas for converting between rational interpolants in barycentric rational krylov rkfun and newton form we show applications of these conversions when working with rational approximants produced by the aaa algorithm y nakatsukasa o sete l n trefethen arxiv preprint 1612 00337 2016 within the rational krylov toolbox and for the solution of nonlinear eigenvalue problems

restrictions on implicit filtering techniques for orthogonal projection methods abstract we consider the class of the orthogonal projection methods opm to solve iteratively large eigenvalue problems an opm is a method that projects a large eigenvalue problem on a smaller subspace in this subspace an approximation of the eigenvalue spectrum can be computed from a small eigenvalue problem using a direct method examples of opms are the arnoldi and the davidson method we show how an opm can be restarted implicitly and explicitly this restart can be used to remove a specific subset of vectors from the approximation subspace this is called explicit filtering an implicit restart can also be combined with an implicit filtering step i e the application of a polynomial or rational function on the subspace even if inaccurate arithmetic is assumed however the condition for the implicit application of a filter is that the rank of the residual matrix must be small

a filter diagonalization for generalized eigenvalue problems based on the sakurai sugiura projection method the sakurai sugiura projection method which solves generalized eigenvalue problems to find certain eigenvalues in a given domain was reformulated by using the resolvent theory a new interpretation based on filter diagonalization was given and the corresponding filter function was derived explicitly a block version of the method was also proposed which enabled not only resolution of degenerated eigenvalues but also an improvement in numerical accuracy three numerical examples were provided to illustrate the method

canonical curves and varieties of sums of powers of cubic polynomials in this note we show that the apolar cubic forms associated to codimension 2 linear sections of canonical curves of genus g 11 are special with respect to their presentation as sums of cubes

stability analysis of time delay systems using a contour integral method in this paper we propose a contour integral method for computing the rightmost characteristic roots of systems of linear time delay differential equations ddes these roots are very important in the context of stability analysis of the time delay systems the effectiveness of the proposed method is illustrated by some numerical experiments

birational aspects of the geometry of varieties of sums of powers abstract varieties of sums of powers describe the additive decompositions of a homogeneous polynomial into powers of linear forms the study of these varieties dates back to sylvester and hilbert but only few of them for special degrees and number of variables are concretely identified in this paper we aim to understand a general birational behavior of vsp to do this we birationally embed these varieties into grassmannians and prove the rational connectedness of many vsp in arbitrary degrees and number of variables

zolotarev quadrature rules and load balancing for the feast eigensolver the feast method for solving large sparse eigenproblems is equivalent to subspace iteration with an approximate spectral projector and implicit orthogonalization this relation allows to characterize the convergence of this method in terms of the error of a certain rational approximant to an indicator function we propose improved rational approximants leading to feast variants with faster convergence in particular when using rational approximants based on the work of zolotarev numerical experiments demonstrate the possible computational savings especially for pencils whose eigenvalues are not well separated and when the dimension of the search space is only slightly larger than the number of wanted eigenvalues the new approach improves both convergence robustness and load balancing when feast runs on multiple search intervals in parallel

rational approximation preconditioners for sparse linear systems this paper presents a class of preconditioning techniques which exploit rational function approximations to the inverse of the original matrix the matrix is first shifted and then an incomplete lu factorization of the resulting matrix is computed the resulting factors are then used to compute a better preconditioner for the original matrix since the incomplete factorization is made on a shifted matrix a good lu factorization is obtained without allowing much fill in the result needs to be extrapolated to the nonshifted matrix thus the main motivation for this process is to save memory the method is useful for matrices whose incomplete lu factorizations are poor e g unstable

designing rational filter functions for solving eigenvalue problems by contour integration abstract solving nonlinear eigenvalue problems by contour integration requires an effective discretization for the corresponding contour integrals in this paper it is shown that good rational filter functions can be computed using nonlinear least squares optimization techniques as opposed to designing those functions based on a thorough understanding of complex analysis the conditions that such an effective filter function should satisfy are derived and translated in a nonlinear least squares optimization problem solved by optimization algorithms from tensorlab numerical experiments illustrate the validity of this approach

weakly defective varieties a projective variety x is k weakly defective when its intersection with a general k 1 tangent hyperplane has no isolated singularities at the k 1 points of tangency if x is k defective i e if the k secant variety of x has dimension smaller than expected then x is also k weakly defective the converse does not hold in general a classification of weakly defective varieties seems to be a basic step in the study of defective varieties of higher dimension we start this classification here describing all weakly defective irreducible surfaces our method also provides a new proof of the classical terracini s classification of k defective surfaces

waring loci and the strassen conjecture the waring locus of a form f is the collection of the degree one forms appearing in some minimal sum of powers decomposition of f in this paper we give a complete description of waring loci for several family of forms such as quadrics monomials binary forms and plane cubics we also introduce a waring loci version of strassen s conjecture which implies the original conjecture and we prove it in many cases

an adaptive order rational arnoldi method for model order reductions of linear time invariant systems abstract this work proposes a model reduction method the adaptive order rational arnoldi aora method to be applied to large scale linear systems it is based on an extension of the classical multi point pade approximation or the so called multi point moment matching using the rational arnoldi iteration approach given a set of predetermined expansion points an exact expression for the error between the output moment of the original system and that of the reduced order system related to each expansion point is derived first in each iteration of the proposed adaptive order rational arnoldi algorithm the expansion frequency corresponding to the maximum output moment error will be chosen hence the corresponding reduced order model yields the greatest improvement in output moments among all reduced order models of the same order a detailed theoretical study is described the proposed method is very appropriate for large scale electronic systems including vlsi interconnect models and digital filter designs several examples are considered to demonstrate the effectiveness and efficiency of the proposed method