inverse eigenvalue problems for extended hessenberg and extended tridiagonal matrices in inverse eigenvalue problems one tries to reconstruct a matrix satisfying some constraints given some spectral information here two inverse eigenvalue problems are solved first given the eigenvalues and the first components of the associated eigenvectors called the weight vector an extended hessenberg matrix with prescribed poles is computed possessing these eigenvalues and satisfying the eigenvector constraints the extended hessenberg matrix is retrieved by executing particularly designed unitary similarity transformations on the diagonal matrix containing the eigenvalues this inverse problem closely links to orthogonal rational functions the extended hessenberg matrix contains the recurrence coefficients given the nodes eigenvalues poles poles of the extended hessenberg matrix and a weight vector first eigenvector components determining the discrete inner product moreover it is also sort of the inverse of the rational arnoldi algorithm instead of using the rational arnoldi method to compute a krylov basis to approximate the spectrum we will reconstruct the orthogonal krylov basis given the spectral info in the second inverse eigenvalue problem we do the same but refrain from unitarity as a result we execute possibly non unitary similarity transformations on the diagonal matrix of eigenvalues to retrieve a non symmetric extended tridiagonal matrix the algorithm will be less stable but it will be faster as the extended tridiagonal matrix admits a low cost factorization of o n n equals the number of eigenvalues whereas the extended hessenberg matrix does not again there is a close link with orthogonal function theory the extended tridiagonal matrix captures the recurrence coefficients of bi orthogonal rational functions moreover it is again sort of inverse of the nonsymmetric lanczos algorithm given spectral properties we reconstruct the two basis krylov matrices linked to the nonsymmetric lanczos algorithm

accurate recovery of recursion coefficients from gaussian quadrature formulas abstract we present an algorithm for the accurate recovery of recursion coefficients from quadrature formulas with positive weights based on the differential form of the quotient difference algorithm the process is slightly faster than the gragg harrod algorithm and is forward stable in the sense of componentwise relative error when the gaps between the nodes are available as data this result shows constructively that the converse problem is well posed when the data are required to be accurate floating point numbers this is not a contradiction of the examples given by gragg and harrod which are ill posed in a vector norm because in that case very small numbers are not required to have any significant digits

inverse eigenvalue problems associated with spring mass systems it is well known that given ai unless the spectral data satisfy some additional conditions we give a related theorem where instead of prescribing o t and u t 1 we prescribe u t and c t e with e a certain rank one matrix interest in t h e construction of jacobi matrices from spectral data is motivated partly by discrete spring mass systems given the principal frequencies of vibration of such a system and the frequencies of a modification of the same system then constructing a jacobi matrix from those spectral data may lead to determining the masses and spring constants of the system we provide general tools for converting jacobi matrix results into spring mass system results illustrate these techniques with examples among them are new spring mass reconstruction results which follow from our jacobi matrix theorem 0 elsevier science inc 1997

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

solving the inverse eigenvalue problem via the eigenvalue matrix abstract a numerical algorithm for the inverse eigenvalue problem for symmetric matrices is developed based on continually updating the eigenvector matrix using plane rotations a fundamental tool in the algorithm is a matrix formed from the rayleigh quotients of the eigenvectors with respect to each of the basis matrices involved with the basis orthonormalized with respect to the frobenius inner product two criteria of closeness to a solution are defined either of which makes it possible to monitor progress towards a solution the computational questions involved in this approach are examined in detail numerical examples are given for toeplitz and centrosymmetric tridiagonal matrices the new algorithm is much more robust than newton s method

a new parallel chasing algorithm for transforming arrowhead matrices to tridiagonal form rutishauser gragg and harrod and finally h y zha used the same class of chasing algorithms for transforming arrowhead matrices to tridiagonal form using a graphical theoretical approach we propose new chasing algorithm although this algorithm has the same sequential computational complexity and backward error properties as the old algorithms it is better suited for a pipelined approach the parallel algorithm for this new chasing method is described with performance results on the paragon and ncube comparison results between the old and the new algorithms are also presented

the numerically stable reconstruction of jacobi matrices from spectral data we present an expose of the elementary theory of jacobi matrices and in particular their reconstruction from the gaussian weights and abscissas many recent works propose use of the diagonal hermitian lanczos process for this purpose we show that this process is numerically unstable we recall rutishauser s elegant and stable algorithm of 1963 based on plane rotations implement it efficiently and discuss our numerical experience we also apply rutishauser s algorithm to reconstruct a persymmetric jacobi matrix from its spectrum in an efficient and stable manner

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

bi orthonormal polynomial basis function framework with applications in system identification numerical aspects are of central importance in identification and control many computations in these fields involve approximations using polynomial or rational functions that are obtained using orthogonal or oblique projections the aim of this paper is to develop a new and general theoretical framework to solve a large class of relevant problems the proposed method is built on the introduction of bi orthonormal polynomials with respect to a data dependent bi linear form this bi linear form generalises the conventional inner product and allows for asymmetric and indefinite problems the proposed approach is shown to lead to optimal numerical conditioning kappa 1 in a recent frequency domain instrumental variable system identification algorithm in comparison it is shown that these recent algorithms exhibit extremely poor numerical properties when solved using traditional approaches

a modified method for reconstructing periodic jacobi matrices in this note we discuss the reconstruction of periodic jacobi matrices from spectral data the method combines ideas and techniques from the algorithms given by boley and golub 1 2 and ferguson 3 resulting in a numerically stable algorithm applicable to a larger class of problems the number of initial data items needed for this method equals the number of items in the resulting matrix namely 2n

recursive approximation of the dominant eigenspace of an indefinite matrix we consider here the problem of tracking the dominant eigenspace of an indefinite matrix by updating recursively a rank k approximation of the given matrix the tracking uses a window of the given matrix which increases at every step of the algorithm therefore the rank of the approximation increases also and hence a rank reduction of the approximation is needed to retrieve an approximation of rank k in order to perform the window adaptation and the rank reduction in an efficient manner we make use of a new anti triangular decomposition for indefinite matrices all steps of the algorithm only make use of orthogonal transformations which guarantees the stability of the intermediate steps we also show some numerical experiments to illustrate the performance of the tracking algorithm

fast and stable eigendecomposition of symmetric banded plus semi separable matrices abstract a new fast and stable algorithm to reduce a symmetric banded plus semi separable matrix to tridiagonal form via orthogonal similarity transformations is presented

an inverse eigenvalue problem for the finite element model of a vibrating rod an inverse eigenvalue problem for the finite element model of a longitudinally vibrating rod whose one end is fixed and the other end is supported on a spring is considered it is known that the mass and stiffness matrices are both tridiagonal for the finite element model of the rod based on linear shape functions it is shown that the cross section areas can be determined from the spectrum of the rod the inverse vibration problem can be recast into an inverse eigenvalue problem of a special jacobi matrix the necessary and sufficient conditions for the construction of a physically realizable rod with positive cross section areas are established a numerical method is presented and an illustrative example is given

on an inverse eigenvalue problem for unitary hessenberg matrices abstract we show that a unitary upper hessenberg matrix with positive subdiagonal elements is uniquely determined by its eigenvalues and the eigenvalues of a modified principal submatrix this provides an analog of a well known result for jacobi matrices

orthogonal polynomials of equilibrium measures supported on cantor sets the equilibrium measure of a compact set is a fundamental object in logarithmic potential theory we compute numerically this measure and its orthogonal polynomials when the compact set is a cantor set defined by an iterated function system we first construct sequences of discrete measures via the solution of large systems of non linear equations that converge weakly to the equilibrium measure successively we compute their jacobi matrices via standard procedures enhanced for the scope numerical estimates of the convergence rate to the limit jacobi matrix are provided which show stability and efficiency of the whole procedure as a companion result we also compute jacobi matrices in two other cases equilibrium measures on finite sets of intervals and balanced measures of iterated function systems these algorithms can reach large polynomial orders therefore we study the asymptotic behavior of the orthogonal polynomials and by a natural extension of the concept of regular root asymptotics we derive an efficient scheme for the computation of complex green s functions and of related conformal mappings

low rank update of singular values the notion of a low rank update arises in many important applications this paper deals with the inverse problem of updating a rectangular matrix by additive low rank matrices so as to reposition the associated singular values the setting is analogous to the classical pole assignment problem where eigenvalues of a square matrix are relocated precise and easy to check necessary and sufficient conditions under which the problem is solvable are completely characterized generalizing some traditional weyl inequalities for singular values the constructive proof makes it possible to compute such a solution numerically a pseudo algorithm is outlined

computation of gauss type quadrature formulas gaussian formulas for a linear functional l such as a weighted integral are best computed from the recursion coefficients relating the monic polynomials orthogonal with respect to l in gauss type formulas one or more extraneous conditions such as pre assigning certain nodes replace some of the equations expressing exactness when applied to high order polynomials these extraneous conditions may be applied by modifying the same number of recursion coefficients we survey the methods of computing formulas from recursion coefficients methods of obtaining recursion coefficients and modifying them for gauss type formulas and questions of existence and numerical accuracy associated with those computations

hessenberg eigenvalue eigenmatrix relations abstract explicit relations between eigenvalues eigenmatrix entries and matrix elements of unreduced hessenberg matrices are derived the main result is based on the taylor expansion of the adjugate of zi h on the one hand and inherent properties of hessenberg matrix structure on the other hand this result is utilized to construct computable relations between eigenvalues eigenvector components eigenvalues of principal submatrices and products of lower diagonal elements generalizing similar identities for jacobi matrices

a parallel algorithm for discrete least squares rational approximation a new method for discrete least squares linearized rational approximation is presented it generalizes the algorithm of rutishauser gragg harrod reichel for discrete least squares polynomial approximation to the rational case the algorithm is fast in the sense that it requires orderm computation time wherem is the number of data points and is the degree of the approximant we describe how this algorithm can be implemented in parallel

an efficient metamodeling approach for uncertainty quantification of complex systems with arbitrary parameter probability distributions summary this paper proposes an efficient metamodeling approach for uncertainty quantification of complex system based on gaussian process model gpm the proposed gpm based method is able to efficiently and accurately calculate the mean and variance of model outputs with uncertain parameters specified by arbitrary probability distributions because of the use of gpm the closed form expressions of mean and variance can be derived by decomposing high dimensional integrals into one dimensional integrals this paper details on how to efficiently compute the one dimensional integrals when the parameters are either uniformly or normally distributed the one dimensional integrals can be analytically evaluated while when parameters do not follow normal or uniform distributions this paper adopts the effective gaussian quadrature technique for the fast computation of the one dimensional integrals as a result the developed gpm method is able to calculate mean and variance of model outputs in an efficient manner independent of parameter distributions the proposed gpm method is applied to a collection of examples and its accuracy and efficiency is compared with monte carlo simulation which is used as benchmark solution results show that the proposed gpm method is feasible and reliable for efficient uncertainty quantification of complex systems in terms of the computational accuracy and efficiency copyright 2016 john wiley sons ltd

construction of band matrices from spectral data abstract a stable numerical algorithm is presented to generate a symmetric p band matrix from the given eigenvalues of the p greatest leading submatrices the algorithm consists of two parts first a matrix with the given spectral data is constructed then this matrix is transformed into a p band matrix leaving invariant the eigenvalues of the p greatest leading submatrices

a complement about almost controllability subspaces abstract we show that for any almost controllability subspace there exist a maximal input chain and a family of state feedbacks which lead to the well known feedback chain expression for this subspace

optimal panel clustering in the presence of anisotropic mesh refinement in this paper we consider the numerical solution of discrete boundary integral equations on polyhedral surfaces in three dimensions when the solution contains typical edge singularities highly stretched meshes are preferred to uniform meshes since they reduce the number of degrees of freedom needed to obtain a fixed accuracy the classical panel clustering method can still be applied in the presence of such highly stretched meshes however we will show that the savings in computation time and storage become suboptimal because the nearfield matrix arising in the panel clustering algorithm is no longer as sparse as it is in the case of uniform meshes hence a natural question arises as to whether a new enhanced panel clustering algorithm can be designed which performs efficiently even in the presence of highly stretched meshes the main result of this paper is to formulate such an enhanced version of the panel clustering algorithm the key features of the algorithm are i the employment of partial analytic integration in the direction of stretching yielding a new kernel function on a one dimensional manifold where the influence of high aspect ratios in the stretched elements is removed and ii the introduction of a generalized admissibility condition with respect to the partially integrated kernel which ensures that certain stretched clusters which are inadmissible in the classical sense now become admissible in the context of a model problem we prove that our algorithm yields an accurate up to the discretization error matrix vector multiplication which requires cal o n log k n operations where n is the number of degrees of freedom and k is small and independent of the aspect ratio the generalized admissibility condition can be viewed as an addition to the classical method which may be useful in general when stretched meshes are present we also have performed a numerical experiment which shows that the sparsity of the nearfield matrix for the enhanced panel clustering method is not negatively affected by stretched elements and the method will perform optimally

orthogonal polynomials in matlab a suite of matlab programs has been developed as part of the book orthogonal polynomials computation and approximation oxford university press oxford 2004 by gautschi the package contains routines for generating orthogonal polynomials as well as routines dealing with applications in this paper a brief review of the first part of the package is given dealing with procedures for generating the three term recurrence relation for orthogonal polynomials and more general recurrence relations for sobolev orthogonal polynomials moment based methods and discretization methods and their implementation in matlab are among the principal topics discussed

calculation of gauss kronrod quadrature rules the jacobi matrix of the 2n 1 point gauss kronrod quadrature rule for a given measure is calculated efficiently by a five term recurrence relation the algorithm uses only rational operations and is therefore also useful for obtaining the jacobi kronrod matrix analytically the nodes and weights can then be computed directly by standard software for gaussian quadrature formulas

inverse eigenvalue problems a collection of inverse eigenvalue problems are identified and classified according to their characteristics current developments in both the theoretic and the algorithmic aspects are summarized and reviewed in this paper this exposition also reveals many open questions that deserve further study an extensive bibliography of pertinent literature is attached