a unifying convergence analysis of second order methods for secular equations existing numerical methods of second order are considered for a so called secular equation we give a brief description of the most important of these methods and show that all of them can be interpreted as improvements of newton s method for an equivalent problem for which newton s method exhibits convergence from any point in a given interval this interpretation unifies the convergence analysis of these methods provides convergence proofs where they were lacking and furnishes ways to construct improved methods in addition we show that some of these methods are in fact equivalent a second secular equation is also briefly considered

efficient arnoldi type algorithms for rational eigenvalue problems arising in fluid solid systems we develop and analyze efficient methods for computing damped vibration modes of an acoustic fluid confined in a cavity with absorbing walls capable of dissipating acoustic energy the discretization in terms of pressure nodal finite elements gives rise to a rational eigenvalue problem numerical evidence shows that there are no spurious eigenmodes for such discretization and also confirms that the discretization based on nodal pressures is much more efficient than that based on raviart thomas finite elements for the displacement field the trimmed linearization method is used to linearize the associated rational eigenvalue problem into a generalized eigenvalue problem gep of the form ax lbx for solving the gep we apply arnoldi algorithm to two different types of single matrices b 1a and ab 1 numerical accuracy shows that the application of arnoldi on ab 1 is better than that on b 1a

absolutely continuous invariant measures that are maximal let a be a certain irreducible 0 1 matrix and let t denote the family of piecewise linear markov maps on 0 1 which are consistent with a the main result of this paper characterizes those maps in t whose unique absolutely continuous invariant measure is maximal and proves that for most of the maps that are consistent with a the absolutely continuous invariant measure is not maximal

quadratically constrained least squares and quadratic problems we consider the following problem compute a vectorx such that ax b 2 min subject to the constraint x 2 a new approach to this problem based on gauss quadrature is given the method is especially well suited when the dimensions ofa are large and the matrix is sparse it is also possible to extend this technique to a constrained quadratic form for a symmetric matrixa we consider the minimization ofx t a x 2b t x subject to the constraint x 2 some numerical examples are given

fast and stable qr eigenvalue algorithms for generalized companion matrices and secular equations we introduce a class of n n structured matrices which includes three well known classes of generalized companion matrices tridiagonal plus rank one matrices comrade matrices diagonal plus rank one matrices and arrowhead matrices relying on the structure properties of we show that if a then a rq where a qr is the qr decomposition of a this allows one to implement the qr iteration for computing the eigenvalues and the eigenvectors of any a with o n arithmetic operations per iteration and with o n memory storage this iteration applied to generalized companion matrices provides new o n2 flops algorithms for computing polynomial zeros and for solving the associated rational secular equations numerical experiments confirm the effectiveness and the robustness of our approach

a matrix method for estimating the liapunov exponent of one dimensional systems let 0 1 0 1 be a piecewise monotonie expanding map then admits an absolutely continuous invariant measure a result of kosyakin and sandler shows that can be approximated by a sequence of absolutely continuous measures n invariant under piecewise linear markov maps itn each itn is constructed on the inverse images of the turning points of the easily computable measures n are used to estimate the liapunov exponent of the idea of using markov maps for estimating the liapunov exponent is applied to both expanding and nonexpanding maps

spectral and computational properties of band symmetric toeplitz matrices abstract we are investigating spectral properties of band symmetric toeplitz matrices bst matrices by giving a suitable representation of a bst matrix we achieve separation results and multiplicity conditions for the eigenvalues of a 7 r 5 diagonal bst matrix and also structural properties of the eigenvectors we give eigenvalue bounds for a 2k 1 diagonal bst matrix and also necessary and sufficient conditions for positive definiteness which are easy to check the same conditions apply in the case of block bst matrices either with full blocks or with bst blocks we exhibit fast computational methods for the evaluation of the determinant and the characteristic polynomial of a bst matrix either for sequential or for parallel computations two algorithms based on the bisection technique and newton s method are shown to be very fast for computing the eigenvalues of a 7 or 5 diagonal bst matrix

simple deterministic dynamical systems with fractal diffusion coefficients we analyze a simple model of deterministic diffusion the model consists of a one dimensional array of scatterers with moving point particles the particles move from one scatterer to the next according to a piecewise linear expanding deterministic map on unit intervals the microscopic chaotic scattering process of the map can be changed by a control parameter the macroscopic diffusion coefficient for the moving particles is well defined and depends upon the control parameter we calculate the diffusion coefficent and the largest eigenmodes of the system by using markov partitions and by solving the eigenvalue problems of respective topological transition matrices for different boundary conditions we find that the largest eigenmodes of the map match the ones of the simple phenomenological diffusion equation our main result is that the diffusion coefficient exhibits a fractal structure as a function of the control parameter we provide qualitative and quantitative arguments to explain features of this fractal structure

a divide and conquer method for unitary and orthogonal eigenproblems leth n xn be a unitary upper hessenberg matrix whose eigenvalues and possibly also eigenvectors are to be determined we describe how this eigenproblem can be solved by a divide and conquer method in which the matrixh is split into two smaller unitary upper hessenberg matricesh 1 andh 2 by a rank one modification ofh the eigenproblems forh 1 andh 2 can be solved independently and the solutions of these smaller eigenproblems define a rational function whose zeros on the unit circle are the eigenvalues ofh the eigenvector ofh can be determined from the eigenvalues ofh and the eigenvectors ofh 1 andh 2 the outlined splitting of unitary upper hessenberg matrices into smaller such matrices is carried out recursively this gives rise to a divide and conquer method that is suitable for implementation on a parallel computer whenh n xn is orthogonal the divide and conquer scheme simplifies and is described separately our interest in the orthogonal eigenproblem stems from applications in signal processing numerical examples for the orthogonal eigenproblem conclude the paper

construction of ergodic transformations classes of piecewise constant functions which can serve as the ergodic densities for constructible piecewise linear transformations are characterized let m denote the class of non negative continuous piecewise monotonic functions f from an interval j into j satisfying i f x m where the derivative exists and ii f x has the value 0 at all the relative minima points then one of the main results is that a class of piecewise constant ergodic densities can be found and the associated piecewise linear transformations under which they are ergodic densities can be constructed which is dense in m with respect to the uniform norm on j

modelling and numerical solution of elastoacoustic vibrations with interface damping a finite element method is applied to compute the vibrations of an elastoacoustic system subject to an external harmonic excitation the effect of a thin layer of noise damping material between the fluid and the solid is taken into account by relaxing the interface kinematic condition of perfect contact we consider the non linear eigenvalue problem arising from the free vibration problem for the damped system for the case of a fluid in a rectangular rigid cavity with one absorbing wall we deduce the dispersion equation which allows computing analytically its eigenvalues and eigenmodes and compare the numerical solution with them finally we apply the finite element method to assess the effect of introducing a real viscoelastic material to damp the elastoacoustic vibrations of a coupled system copyright 1999 john wiley sons ltd

finite element analysis of a quadratic eigenvalue problem arising in dissipative acoustics a quadratic eigenvalue problem arising in the determination of the vibration modes of an acoustic fluid contained in a cavity with absorbing walls is considered the problem is shown to be equivalent to the spectral problem for a noncompact operator and a thorough spectral characterization is given a numerical discretization based on raviart thomas finite elements is analyzed the method is proved to be free of spurious modes and to converge with optimal order implementation issues and numerical experiments confirming the theoretical results are reported

iteration schemes for the divide and conquer eigenvalue solver we propose globally convergent iteration schemes for updating the eigenvalues of a symmetric matrix after a rank 1 modification such calculations are the core of the divide and conquer technique for the symmetric tridiagonal eigenvalue problem we prove the superlinear convergence right from the start of our schemes which allows us to improve the complexity bounds of 3 the effectiveness of our algorithms is confirmed by numerical results which are reported and discussed

finite element vibration analysis of fluid solid systems without spurious modes this paper deals with the finite element approximation of the vibration modes of a problem with fluid structure interaction displacement variables are used for both the fluid and the solid to avoid the typical spurious modes of this formulation we introduce a nonconforming discretization error estimates for the approximation of eigenvalues and eigenvectors are given

restarting iterative projection methods for hermitian nonlinear eigenvalue problems with minmax property in this work we present a new restart technique for iterative projection methods for nonlinear eigenvalue problems admitting minmax characterization of their eigenvalues our technique makes use of the minmax induced local enumeration of the eigenvalues in the inner iteration in contrast to global numbering which requires including all the previously computed eigenvectors in the search subspace the proposed local numbering only requires a presence of one eigenvector in the search subspace this effectively eliminates the search subspace growth and therewith the super linear increase of the computational costs if a large number of eigenvalues or eigenvalues in the interior of the spectrum are to be computed the new restart technique is integrated into nonlinear iterative projection methods like the nonlinear arnoldi and jacobi davidson methods the efficiency of our new restart framework is demonstrated on a range of nonlinear eigenvalue problems quadratic rational and exponential including an industrial real life conservative gyroscopic eigenvalue problem modeling free vibrations of a rolling tire we also present an extension of the method to problems without minmax property but with eigenvalues which have a dominant either real or imaginary part and test it on two quadratic eigenvalue problems

matrices and eigenfunctions induced by markov maps abstract the matrix m induced by a markov map has 1 as the eigenvalue of maximum modulus if m is also irreducible then the algebraic and geometric multiplicities of the eigenvalue 1 are also 1 let g g 1 g 2 g n be an n row vector such that g 1 c and g i c i 2 then every such g is the unique upto constant multiples left eigenvector associated with the eigenvalue 1 of a matrix m induced by a markov map

a modification of the dewilde van der veen method for inversion of finite structured matrices abstract we study a class of block structured matrices r r ij i j 1 n with a property that the solution of the corresponding system rx y of linear algebraic equations may be performed for o n arithmetic operations in this paper for finite invertible matrices we analyze in detail factorization and inversion algorithms these algorithms are related to those suggested by p m dewilde and a j van der veen time varying systems and computations kluwer academic publishers new york 1998 for a class of finite and infinite matrices with a small hankel rank the algorithms presented here are more transparent and are a modification of the algorithms from the above reference the approach and the proofs are essentially different from those in the above mentioned reference the paper contains also analysis of complexity and results of numerical experiments

fast inversion algorithms for a class of structured operator matrices we study linear complexity inversion algorithms for diagonal plus semiseparable operator matrices applications to integral and differential equations and block matrices are obtained comparison of different algorithms is performed complexity choice of parameters and results of numerical experiments are analyzed

restricted rank modification of the symmetric eigenvalue problem theoretical considerations the problem is considered how to obtain the eigenvalues and vectors of a matrix a vvt where a is a symmetric matrix with known spectral decomposition and vvt is a positive semidefinite matrix of low rank it is shown that the eigenvalues of a vvt can easily be located to any desired accuracy by means of the inertia of the matrix i vt a 1v the problem of determining the eigenvalues of a restricted to r v can be treated likewise

inversion formulas and linear complexity algorithm for diagonal plus semiseparable matrices abstract matrices represented as a sum of diagonal and semiseparable ones are considered here these matrices belong to the class of structured matrices which arises in numerous applications fast o n algorithms for their inversion were developed before under additional restrictions which were a source of instability our aim is to eliminate these restrictions and to develop reliable and stable numerical algorithms in this paper the case of semiseparable matrices of order one is considered

low memory and low complexity iterative schemes for a nonsymmetric algebraic riccati equation arising from transport theory abstract we reconsider newton s method and two fixed point methods for finding the minimal positive solution of a nonsymmetric algebraic riccati equation arising from transport theory we rewrite the subproblem of the newton and fixed point iterative schemes into an equivalent form with some special structure by the use of the particular structure of the subproblem we then present low memory and low complexity versions of these iterative methods with a factored alternating direction implicit iteration some properties of eigenvalues for iterative coefficient matrices in solving the subproblem are derived and the convergence of the proposed methods is established numerical experiments show that the new iterative schemes are highly efficient to obtain the minimal positive solution the proposed low memory and low complexity newton s method is particularly efficient for solving large scale riccati equation arising from transport theory

numerical solution of a secular equation a method is proposed for the solution of a secular equation arising in modified symmetric eigenvalue problems and in several other areas this equation has singularities which make the application of standard root finding methods difficult in order to solve the equation a class of transformations of variables is considered which transform the equation into one for which newton s method converges from any point in a certain given interval in addition the form of the transformed equation suggests a convergence accelerating modification of newton s method the same ideas are applied to the secant method and numerical results are presented

on computing the eigenvectors of a class of structured matrices a real symmetric matrix of order n has a full set of orthogonal eigenvectors the most used approach to compute the spectrum of such matrices reduces first the dense symmetric matrix into a symmetric structured one i e tridiagonal matrices or semiseparable matrices this step is accomplished in o n 3 operations once the latter symmetric structured matrix is available its spectrum is computed in an iterative fashion by means of the qr method in o n 2 operations in principle the whole set of eigenvectors of the latter structured matrix can be computed by means of inverse iteration in o n 2 operations the blemish in this approach is that the computed eigenvectors may not be numerically orthogonal if clusters are present in the spectrum to enforce orthogonality the gram schmidt procedure is used requiring o n 3 operations in the worst case in this paper a fast and stable method to compute the eigenvectors of tridiagonal and semiseparable matrices which does not suffer from the loss of orthogonality due to the presence of clusters in the spectrum is presented the algorithm requires o n 2 floating point operations if clusters of small size are present in the spectrum

structural and computational properties of possibly singular semiseparable matrices a classical result of structured numerical linear algebra states that the inverse of a nonsingular semiseparable matrix is a tridiagonal matrix such a property of a semiseparable matrix has been proved to be useful for devising linear complexity solvers for establishing recurrence relations among its columns or rows and moreover for efficiently evaluating its characteristic polynomial in this paper we provide sparse structured representations of a semiseparable matrix a which hold independently of the fact that a is singular or not these relations are found by pointing out the band structure of the inverse of the sum of a plus a certain sparse perturbation of minimal rank further they can be used to determine in a computationally efficient way both a reflexive generalized inverse of a and its characteristic polynomial

solving secular and polynomial equations a multiprecision algorithm we present an algorithm for the solution of polynomial equations and secular equations of the form s x 0 for s x i 1 na ix b i 1 0 which provides guaranteed approximation of the roots with any desired number of digits it relies on the combination of two different strategies for dealing with the precision of the floating point computation the strategy used in the package mpsolve of d bini and g fiorentino d a bini g fiorentino design analysis and implementation of a multi precision polynomial rootfinder numer algorithms 23 2000 127 173 and the strategy used in the package eigensolve of s fortune s fortune an iterated eigenvalue algorithm for approximating the roots of univariate polynomials j symbolic comput 33 5 2002 627 646 the algorithm is based on the ehrlich aberth ea iteration and on several results introduced in the paper in particular we extend the concept and the properties of root neighborhoods from polynomials to secular functions provide perturbation results of the roots obtain an effective stop condition for the ea iteration and guaranteed a posteriori error bounds we provide an implementation released in the package mpsolve 3 0 based on the gmp library from the many numerical experiments it turns out that our code is generally much faster than mpsolve 2 0 and of the package eigensolve for certain polynomials like the mandelbrot or the partition polynomials the acceleration is dramatic the algorithm exploits the parallel architecture of the computing platform

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

iterative projection methods for computing relevant energy states of a quantum dot a computational technique for computing relevant energy levels and corresponding wave functions of an electron confined by a 3d quantum dot embedded in a semiconductor matrix are studied assuming an energy and position dependent electron effective mass approximation this problem is governed by a rational eigenvalue problem we discuss the application of iterative projection method of arnoldi and jacobi davidson type projected problems of small dimension are solved efficiently by safeguarded iteration

rank structures preserved by the qr algorithm the singular case in an earlier paper we introduced the classes of polynomial and rank structures both of them preserved by applying a shifted qr step on a matrix a in the present paper we further investigate the case of rank structures we show that even if a is a singular matrix a new qr iterate can be constructed having the same rank structure as the matrix a itself to this end we introduce the concepts of effectively eliminating qr decompositions and sparse givens patterns both of them being of independent interest

additive preconditioning and aggregation in matrix computations we combine our novel svd free additive preconditioning with aggregation and other relevant techniques to facilitate the solution of a linear system of equations and other fundamental matrix computations our analysis and experiments show the power of our algorithms guide us in selecting most effective policies of preconditioning and aggregation and provide some new insights into these and related subjects compared to the popular svd based multiplicative preconditioners our additive preconditioners are generated more readily and for a much larger class of matrices furthermore they better preserve matrix structure and sparseness and have a wider range of applications e g they facilitate the solution of a consistent singular linear system of equations and of the eigenproblem

eigenvalues of a symmetric tridiagonal matrix a divide and conquer approach the symmetric tridiagonal eigenproblem has been the topic of some recent work many methods have been advanced for the computation of the eigenvalues of such a matrix in this paper we present a divide and conquer approach to the computation of the eigenvalues of a symmetric tridiagonal matrix via the evaluation of the characteristic polynomial the problem of evaluation of the characteristic polynomial is partitioned into smaller parts which are solved and these solutions are then combined to form the solution to the original problem we give the update equations for the characteristic polynomial and certain auxiliary polynomials used in the computation furthermore this set of recursions can be implemented on a regulartree structure if the concurrency exhibited by this algorithm is exploited it can be shown that thetime for computation of all the eigenvalues becomeso nlogn instead ofo n2 as is the case for the approach where the order is increased by only one at every step we address the numerical problems associated with the use of the characteristic polynomial and present a numerically stable technique for the eigenvalue computation

the cal d q r algorithm basic theory convergence and conditional stability we describe a fast matrix eigenvalue algorithm that uses a matrix factorization and reverse order multiply technique involving three factors and that is based on the symmetric matrix factorization as well as on d orthogonal reduction techniques where d mbox diag pm1 is computed from the given matrix a it operates on a similarity reduction of a real matrix a nn to general tridiagonal form t and computes all of a s eigenvalues in frac 4 3 n 3 o n 2 operations where the o n 2 part of the operations is possibly performed over bbb c instead of the 7 8 n 3 real flops required by the qr eigenvalue algorithm potential breakdo wn of the dqr algorithm can occur in the reduction to tridiagonal form and in the d orthogonal dqr reductions both however can be monitored during the computations the former occurs rather rarely for dimensions n leq 300 and can essentially be bypassed while the latter is extremely rare and can be bypassed as well in our conditionally stable implementation of the dqr steps we prove an implicit dq theorem which allows implicit shifts give a convergence proof for the dqr algorithm and show that dqr is conditionally stable for general balanced tridiagonal matrices t rm b

mixed finite element methods in fluid structure systems the eigenvalue problem describing the frequencies of a fluid vibrating in a rigid cavity or within moving boundaries is considered based on the method of lagrange multipliers a three field mixed formulation is introduced in order to avoid the spurious circulating modes stability and optimal error bounds are proved for two choices of finite element spaces

statistical analysis and spectral estimation techniques for one dimensional chaotic signals signals arising out of nonlinear dynamics are compelling models for a wide range of both natural and man made phenomena in contrast to signals arising out of linear dynamics extremely rich behavior is obtained even when we restrict our attention to one dimensional 1 d chaotic systems with certain smoothness constraints an important class of such systems are the so called markov maps we develop several properties of signals obtained from markov maps and present analytical techniques for computing a broad class of their statistics in closed form these statistics include for example correlations of arbitrary order and all moments of such signals among several results we demonstrate that all markov maps produce signals with rational spectra and we can therefore view the associated signals as chaotic arma processes with chaotic white noise as a special case finally we also demonstrate how markov maps can be used to approximate to arbitrary accuracy the statistics any of a broad class of non markov chaotic maps

compactness of eigenvectors for certain families of non negative matrices conditions are given which guarantee that the normalized left eigenvectors of certain classes of non negative matrices of unbounded dimension interpreted as functions of 0 1 form precompact sets in l1 these eigenvectors are uniformly bounded above and in some cases uniformly bounded below

computing matrix eigenvalues and polynomial zeros where the output is real surprisingly simple corollaries from the courant fischer minimax characterization theorem enable us to devise a very effective algorithm for the evaluation of a set s interleaving the set e of the eigenvalues of an n x n real symmetric tridiagonal rst matrix tn as well as a point that splits e into two subsets of comparable cardinalities furthermore we extend this algorithm so as to approximate all the n eigenvalues of tn at nearly optimal sequential and parallel cost that is at the cost of staying within polylogarithmic factors from the straightforward lower bounds the resulting improvement of the known processor bound in nc algorithms for the rst eigenproblem is roughly by factor n our approach extends the previous works m ben or and p tiwari j complexity 6 1990 pp 417 442 and m ben or et al siam j comput 17 1988 pp 1081 1092 for the approximation of the zeros of a polynomial having only real zeros and our algorithm leads to an alternative and simplified derivation of the known record parallel and sequential complexity estimates for the latter problem

linear construction of companion matrices this note is concerned with the following problem for a given matrix a e c ix and a vector a e c does there exist a mapping 2 assigning to each manic polynomial f of degree n a vector x f e c such that the matrix b a a x f is a companion matrix off i e the characteristic polynomial of b is t l ftf the classes of suitable matrices a and vectors a are characterized and some properties of b are described the corresponding unique mapping is determined by a system of linear equations the cases of a triangular bidiagonal or diagonal matrix a are discussed explicitly and many known companion matrices are obtained as particular cases then gershgorin s theorem is applied yielding error estimates for polynomial roots finally the extension to block companion matrices and an example of nonlinear construction are discussed

a qr method for computing the singular values via semiseparable matrices the standard procedure to compute the singular value decomposition of a dense matrix first reduces it into a bidiagonal one by means of orthogonal transformations once the bidiagonal matrix has been computed the qr method is applied to reduce the latter matrix into a diagonal one in this paper we propose a new method for computing the singular value decomposition of a real matrix in a first phase an algorithm for reducing the matrix a into an upper triangular semiseparable matrix by means of orthogonal transformations is described a remarkable feature of this phase is that depending on the distribution of the singular values after few steps of the reduction the largest singular values are already computed with a precision depending on the gaps between the singular values an implicit qr method for upper triangular semiseparable matrices is derived and applied to the latter matrix for computing its singular values the numerical tests show that the proposed method can compete with the standard method using an intermediate bidiagonal matrix for computing the singular values of a matrix

displacement pressure based mixed finite element formulations for acoustic fluid structure interaction problems summary we present reliable finite element discretizations based on displacement pressure interpolations for the analysis of acoustic fluid structure interaction problems the finite element interpolations are selected using the inf sup condition and emphasis is given to the fact that the boundary conditions must satisfy the mass and momentum conservation we show that with our analysis procedure no spurious non zero frequencies are encountered as heretofore calculated with other displacement based discretizations 1997 by john wiley sons ltd

generalized updating problems and computation of the eigenvalues of rational toeplitz matrices abstract algebraic and computational properties of the rank one updating of a generalized eigenvalue problem are investigated the results are applied to the computation of the eigenvalues of full toeplitz matrices related to the laurent expansion of a rational function extending a method of handy and barlow already known for the banded toeplitz case

a special class of structured matrices constructed with the kronecker product and its use for difference equations abstract starting from a sequence of specific orthogonal matrices and using the matricial kronecker product we build classes of matrices sk k 1 2 which are shown to possess a certain affiliation with toeplitz and hankel matrices being completely determined by the first row elements their eigenvalues as well as an inversion formula are easily derived therefrom s matrices as they are termed in general may arise through the application of finite difference methods to boundary value problems

sine transform based preconditioners for symmetric toeplitz systems abstract the optimal circulant preconditioner for a given matrix a is defined to be the minimizer of c a f over the set of all circulant matrices c here f is the frobenius norm optimal circulant preconditioners have been proved to be good preconditioners in solving toeplitz systems with the preconditioned conjugate gradient method in this paper we construct an optimal sine transform based preconditioner which is defined to be the minimizer of b a f over the set of matrices b that can be diagonalized by sine transforms we will prove that for general n by n matrices a these optimal preconditioners can be constructed in o n 2 real operations and in o n real operations if a is toeplitz we will also show that the convergence properties of these optimal sine transform preconditioners are the same as that of the optimal circulant ones when they are employed to solve toeplitz systems numerical examples are given to support our convergence analysis

vibration analysis of fluid solid systems using a finite element displacement formulation this report presents a finite element solution for the vibration interaction between an inviscid fluid and a solid the equation of motion governing the inviscid fluid is expressed in terms of the displacements reduced integration is used when computing the element stiffness matrix contributed by the fluid in addition a projection is used on the element mass matrix in order to remove the spurious modes which result from the use of reduced integration

conservative p 1 conforming and nonconforming galerkin fems effective flux evaluation via a nonmixed method approach given a p1 conforming or nonconforming galerkin finite element method gfem solution ph which approximates the exact solution p of the diffusion reaction equation nabla cdot mathcal k nabla p alpha p f with full tensor variable coefficient mathcal k we evaluate the approximate flux uh to the exact flux bf u mathcal k nabla p by a simple but physically intuitive formula over each finite element the flux is sought in the continuous in normal component or the discontinuous raviart thomas space a systematic way of deriving such a formula is introduced this direct method retains local conservation property at the element level typical of mixed methods finite element or finite volume type but avoids solving an indefinite linear system in short the present method retains the best of the gfem and the mixed method but without their shortcomings thus we view our method as a conservative gfem and demonstrate its equivalence to a certain mixed finite volume box method the equivalence theorems explain how the pressure can decouple basically cost free from the mixed formulation the accuracy in the flux is of first order in the h dive omega norm numerical results are provided to support the theory

efficient computation of tridiagonal matrices largest eigenvalue abstract this paper proposes a method for a fast estimation of the largest eigenvalue of an asymmetric tridiagonal matrix the proposed method is based on the power method and the computation of the square of the original matrix the matrix square is computed through a proposed fast algorithm designed specifically for tridiagonal matrices implementations for compressed column ccs and compressed row storage crs formats are provided discussed and compared to a standard scientific library we investigate the roundoff numerical errors showing that the proposed method provides errors no greater than the usual power method we provide numerical results with simulations in c c implementation in order to demonstrate the effectiveness of the proposed method

the qr algorithm for unitary hessenberg matrices abstract let h be an n n unitary right hessenberg matrix with positive subdiagonal elements using what we call the schur parameterization of h we show how one step of the shifted qr algorithm for h can be carried out in o n arithmetic operations coupled with the shift strategy of eberlein and huang 3 this will permit computation of the spectrum of h to machine precision in o n 2 operations one potential application is the computation of gauss szego quadrature formulas 12 given the associated schur parameters 7 the weights can also be computed by direct analogy with 6

the order of convergence of eigenfrequencies in finite element approximations of fluid structure interaction problems in this paper we prove a double order for the convergence of eigenfrequencies in fluid structure vibration problems we improve estimates given recently for compressible and incompressible fluids to do this we extend classical results on finite element spectral approximation to nonconforming methods for noncompact operators

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

a constrained eigenvalue problem in this paper we consider the following mathematical and computational problem given the quantities a n m by n m matrix symmetric n 0 n n m by m matrix with full rank t vector of dimension m with nt t 1 determine an x such that operatorname x t ax min subject to the constraints n t x t i x t x 1 ii variants of this problem occur in many applications 1 5 7 8 11 the problem has been studied previously when t 0 the null vector cf 4 6 when t 0 then the problem becomes more complicated

new progress in real and complex polynomial root finding matrix methods are increasingly popular for polynomial root finding the idea is to approximate the roots as the eigenvalues of the companion or generalized companion matrix associated with an input polynomial the algorithms also solve secular equation qr algorithm is the most customary method for eigen solving but we explore the inverse rayleigh quotient iteration instead which turns out to be competitive with the most popular root finders because of its excellence in exploiting matrix structure to advance the iteration we preprocess the matrix and incorporate newton s linearization repeated squaring homotopy continuation techniques and some heuristics the resulting algorithms accelerate the known numerical root finders for univariate polynomial and secular equations and are particularly well suited for the acceleration by using parallel processing furthermore even on serial computers the acceleration is dramatic for numerical approximation of the real roots in the typical case where they are much less numerous than all complex roots

a composite polynomial zerofinding matrix algorithm abstract a globally convergent matrix algorithm is presented for finding the real and complex zeros of a complex polynomial p x a combination of schmeisser s and fiedler s matrices is used in the algorithm first the greatest common divisor g x of p x and its derivative p x is obtained and used to reduce p x to a polynomial q x p x q x with simple zeros second the zeros of q x are computed either by finding the eigenvalues of the first block of schmeisser s matrix or by applying fiedler s algorithm recursively in either case the eigenvalues are obtained by the qr algorithm third the multiplicity of each zero of p x is calculated by means of lagouanelle s modified limiting formula sample numerical results are given to demonstrate the effectiveness of the algorithm

fast qr factorization of cauchy like matrices abstract in this paper we present two fast numerical methods for computing the qr factorization of an n n cauchy like matrix c c qr with data points lying on the real axis or on the unit circle in the complex plane it is shown that the rows of the q factor of c are the eigenvectors of a rank structured matrix partially determined by some prescribed spectral data this property establishes a basic connection between the computation of q and the solution of an inverse eigenvalue problem for a rank structured matrix exploiting the structure of this problem enables us to develop quadratic time i e o n 2 qr factorization algorithms

a parallel algorithm for computing the eigenvalues of a symmetric tridiagonal matrix a parallel algorithm called polysection is presented for computing the eigenvalues of a symmetric tridiagonal matrix the method is based on a quadratic recurrence in which the characteristic polynomial is constructed on a binary tree from polynomials whose degree doubles at each level intervals that contain exactly one zero are determined by the zeros of polynomials at the previous level which ensures that different processors compute different zeros the signs of the polynomials at the interval endpoints are determined a priori and used to guarantee that all zeros are found the use of finite precision arithmetic may result in multiple zeros however in this case the intervals coalesce and their number determines exactly the multiplicity of the zero for an n x n matrix the eigenvalues can be determined in o log squared n time with n squared processors and o n time with n processors the method is compared with a parallel variant of bisection that requires o n squared time on a single processor o n time with n processors and o log n time with n squared processors

on the fullness of surjective maps of an interval let i 0 1 13 lebesgue measurable subsets of 0 1 and let a denote the lebesgue measure on i i13 let t i i be measurable and surjective we say t is full if for all a e 93 a a 0 t a t 2 a measurable the condition 1 lim ax t a 1 holds we say t is interval full if 1 holds for any interval a c i in this note we give an example of t i i which is continuous and interval full but not full we also show that for a class of transformations t satisfying renyi s condition interval fullness implies fullness finally we show that fullness is not preserved under limits on the surjections

finite element computation of the vibration modes of a fluid solid system abstract in this paper we solve the interior elastoacoustic problem by a finite element method which does not present spurious or circulation modes for nonzero frequencies it consists of classical triangular lagrangian elements for the solid and lowest order triangular raviart thomas elements for the fluid transmission conditions at the fluid solid interface are taken into account in a weak sense numerical results for some test examples are presented which show the good performance of the proposed methodology

numerical evaluation of reproducing kernel hilbert space inner products a new approach to numerically evaluate an inner product or a norm in an arbitrary reproducing kernel hilbert space rkhs is considered the proposed methodology enables us to approximate the rkhs inner product with the desired accuracy avoiding analytical expressions furthermore its implementation is illustrated by means of some classic examples and compared with the standard iterative method provided by weiner for this purpose finally applications in both the problem of representing approximately second order stochastic processes by means of series expansions and in the problem of signal detection are studied

inverse power and durand kerner iterations for univariate polynomial root finding abstract univariate polynomial root finding is the oldest classical problem of mathematics and computational mathematics and is still an important research topic due to its impact on computational algebra and geometry the weierstrass durand kerner approach and its variations as well as matrix methods based on the qr algorithm are among the most popular practical choices for simultaneous approximation of all roots of a polynomial we propose an alternative application of the inverse power iteration to generalized companion matrices for polynomial root finding demonstrate its effectiveness and relate its study to unifying the derivation of the weierstrass durand kerner algorithm having quadratic convergence and its extensions having convergence rates 4 6 8 our experiments show substantial improvement versus the latter algorithm even though the inverse power iteration is most effective for the more limited tasks of approximating a single root or a few selected roots

expressing a polynomial as the characteristic polynomial of a symmetric matrix abstract we show that given a polynomial one can without knowing the roots construct a symmetric matrix whose characteristics polynomial is the given polynomial appropriately normed for a real polynomial the matrix can have purely imaginary off diagonal entries for a polynomial with only real distinct roots the matrix can be chosen real

spectral decomposition for combinations of markov operators abstract let p p1 1 p2 be the convex combination of two constrictive markov operators if p is constrictive then the spectral representation of p is related to the spectral representations of p1 and p2

structured eigenvalue condition numbers for parameterized quasiseparable matrices the development of fast algorithms for performing computations with n times n n n low rank structured matrices has been a very active area of research during the last two decades as a consequence of the numerous applications where these matrices arise the key ideas behind these fast algorithms are that low rank structured matrices can be described in terms of o n parameters and that these algorithms operate on the parameters instead on the matrix entries therefore the sensitivity of any computed quantity should be measured with respect to the possible variations that the parameters defining these matrices may suffer since this determines the maximum accuracy of a given fast computation in other words it is necessary to develop condition numbers with respect to parameters for different magnitudes and classes of low rank structured matrices but as far as we know this has not yet been accomplished in any case in this paper we derive structured relative eigenvalue condition numbers for the important class of low rank structured matrices known as 1 1 1 1 quasiseparable matrices with respect to relative perturbations of the parameters in the quasiseparable and in the givens vector representations of these matrices and we provide fast algorithms for computing them comparisons among the new structured condition numbers and the unstructured one are also presented as well as numerical experiments showing that the structured condition numbers can be small in situations where the unstructured one is huge in addition the approach presented in this paper is general and may be extended to other problems and classes of low rank structured matrices

the restarted qr algorithm for eigenvalue computation of structured matrices the qr algorithm is a popular numerical method for the computation of eigenvalues of matrices all eigenvalues of a general n n upper hessenberg matrix typically can be computed in o n3 arithmetic floating point operations using o n2 storage locations when the upper hessenberg matrix is hermitian or unitary then it can be represented by o n parameters and there are variants of the qr algorithm that reduce the operation count for computing all eigenvalues to o n2 arithmetic floating point operations and the storage requirement to o n locations however for many structured matrices that can be represented with o n storage locations available implementations of the qr algorithm require o n3 arithmetic floating point operations and o n2 storage locations to determine all eigenvalues this note shows that for some of the latter matrices the operation count can be reduced to o n2 arithmetic floating point operations and the memory requirement to o n storage locations by periodically restarting the qr algorithm