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

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

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

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

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

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

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

multiple scattering and modified green s functions abstract exterior boundary value problems for the helmholtz equation can be reduced to boundary integral equations it is known that the simplest of these fail to be uniquely solvable at certain irregular frequencies for a single smooth scatterer it is also known that irregular frequencies can be eliminated by using a modified fundamental solution one that has additional singularities inside the scatterer this approach is extended to treat the three dimensional exterior neumann problem for any finite number of disjoint smooth scatterers using a fundamental solution that has additional singularities inside every scatterer

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

solution of second order supersymmetrical intertwining relations in minkowski plane supersymmetrical susy intertwining relations are generalized to the case of quantum hamiltonians in minkowski space for intertwining operators supercharges of second order in derivatives the intertwined hamiltonians correspond to completely integrable systems with the symmetry operators of fourth order in momenta in terms of components the intertwining relations correspond to the system of nonlinear differential equations which are solvable with the simplest constant ansatzes for the metric matrix in second order part of the supercharges the corresponding potentials are built explicitly both for diagonalizable and nondiagonalizable form of metric matrices and their properties are discussed

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

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

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

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

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

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

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

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

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