loewner and bezout matrices abstract the notion of intertwining matrices introduced recently by the authors is used to study loewner matrices as in the case of hankel and bezout matrices compatible with a polynomial f of degree not necessarily equal to the size of the matrices the theory requires a careful handling of polynomials with a root at infinity the authors establish first some technical results essentially of a projective character concerning divisibility of such polynomials these are then used to represent products of the form b1b 12b3b 14 b 12rb2r 1 where the bj are bezoutian matrices corresponding to a polynomial f again in the form of a bezoutian matrix corresponding to the same f this result can be deduced from the barnett formula if the degree of f equals the size of the matrices but requires the intertwining theory if this assumption is dropped among other results a strengthening and a conceptual proof of a recent result of z vavrin on inverses of loewner matrices is given in the last section the authors investigate analogies of barnett type formulae for bezoutian matrices b f g where the degree of f is allowed to be smaller than the size of the matrix the barnett formula does not extend immediately but it can be shown that an analogous formula holds asymptotically in the general case

investigation of nonplanar perfectly matched absorbers for finite element mesh truncation we present a detailed theoretical and numerical investigation of the perfectly matched layer pml concept as applied to the problem of mesh truncation in the finite element method fem we show that it is possible to extend the cartesian pml concepts involving half spaces to cylindrical and spherical geometries appropriate for closed boundaries in two and three dimensions by defining lossy anisotropic layers in the relevant coordinate systems by using the method of separation of variables it is possible to solve the boundary value problems in these geometries the analytical solutions demonstrate that under certain conditions outgoing waves are absorbed with negligible reflection and the transmitted wave is attenuated within the pml to reduce the white space in radiation or scattering problems conformal pmls are constructed via parametric mappings it is also verified that the pml concept which was originally introduced for problems governed by maxwell s equations can be extended to cases governed by the scalar helmholtz equation finally numerical results are presented to demonstrate the use of the pml in fem mesh truncation

lowner based blind signal separation of rational functions with applications a new blind signal separation bss technique is proposed enabling a deterministic separation of signals into rational functions rational functions can take on a wide range of forms such as the well known pole like shape the approach is a possible alternative for the well known independent component analysis when the theoretical sources are not independent such as for frequency spectra or when only a small number of samples is available the technique uses a low rank decomposition on the tensorized version of the observed data matrix the deterministic tensorization with lowner matrices is comprehensively analyzed in this paper uniqueness properties are investigated and a connection with the separation into exponential polynomials is made finally the technique is illustrated for fetal electrocardiogram extraction and with an application in the domain of fluorescence spectroscopy enabling the identification of chemical analytes using only a single excitation emission matrix

inversion of toeplitz matrices with only two standard equations abstract it is shown that the invertibility of a toeplitz matrix can be determined through the solvability of two standard equations the inverse matrix is represented by two of its columns which are the solutions of the two standard equations and the entries of the original toeplitz matrix

bezout hankel and loewner matrices abstract relations between the classes of bezout hankel and loewner matrices and of their inverses are investigated all these classes appear as matrix realizations of one operator in suitable bases different bases in the space of all polynomials of degree not exceeding n 1 are considered these bases correspond to different expansions of the polynomial w z p x p z w x z x we introduce a matrix k corresponding to this polynomial this matrix together with its limit forms may be used to represent all the classes of matrices mentioned in particular we obtain a new interpretation of the variant of generalized loewner matrices introduced recently by the second named author

schur complements of bezoutians and the inversion of block hankel and block toeplitz matrices abstract we derive a closed inversion formula for an np np square block hankel matrix hn 1 wi j with entries wi from the ring of the p p matrices over a field the representation of h 1n 1 relies upon a strong structure preserving property of the schur complements of the nonsingular leading principal submatrices of a certain generalized bezoutian of matrix polynomials these polynomials may be determined by computing iteratively the schur complements of the nonsingular leading principal submatrices of hn 1 in an augmented block hankel matrix h which is a bezoutian matrix up to multiplication by proper reverse matrices since the special structure of the bezoutian matrix is preserved at any step of the computation such method can be efficiently implemented at an overall computational cost of o p3n2 operations in the average case and o p3n3 operations in the worst case in the normal case a suitable modification which uses both divide and conquer strategies and fast polynomial arithmetic reaches the bound of o p3n log2 n arithmetical operations

inversion formulas and fast algorithms for lowner vandermonde matrices abstract matrices consisting of two parts one of vandermonde and the other of lowner type are considered inversion formulas also for one sided inverses are presented moreover in the nonsingular case the parameters of the inversion formulas are described in terms of the evaluation of numerator and denominator polynomials which constitute the solution of associated rational interpolation problems these results can be used to establish an algorithm for the solution of lowner vandermonde systems of equations with o n log 2 n complexity

the general rational interpolation problem in the scalar case and its hankel vector abstract this paper deals with the general rational interpolation problem grip in the scalar case in the recent work of antoulas ball kang and willems the general solution to the grip has been derived in the framework of linear fractions using the so called generating matrix as the main tool within this framework the contribution here consists in a new approach of computing a particular 2 2 polynomial generating matrix based on the deep connection between loewner and hankel matrices it turns out that given a fixed grip we can readily obtain a unique vector called the hankel vector such that both of the allowable mcmillan degrees of interpolants to this grip and the corresponding polynomial generating matrix are completely determined by this hankel vector combined with its characteristic degrees and characteristic polynomials

measured equation of invariance a new concept in field computations numerical computations of frequency domain field problems or elliptical partial differential equations may be based on differential equations or integral equations the new concept of field computation presented in this paper is based on the postulate of the existence of linear equations of the discretized nodal values of the fields different from the conventional equations but leading to the same solutions the postulated equations are local and invariant to excitation it is shown how the equations can be determined by a sequence of measures the measured equations are particularly useful at the mesh boundary where the finite difference methods fail the measured equations do not assume the physical condition of absorption so they are also applicable to concave boundaries using the measured equations one can terminate the finite difference mesh very close to the physical boundary and still obtain robust solutions it will definitely make a great impact on the way one applies finite difference and finite element methods in many problems computational results using the measured equations of invariance in two and three dimensions are presented

on the solution of the minimal rational interpolation problem it is shown that the reachability and the observability indices of appropriately defined pairs of matrices provide the key for solving the minimal rational bitangen tial interpolation problem

the truncated hamburger matrix moment problems in the nondegenerate and degenerate cases and matrix continued fractions abstract the present paper deals simultaneously with the nondegenerate and degenerate truncated hamburger matrix moment problems in a unified way based on the use of the schur algorithm involving matrix continued fractions a full analysis of them together with a relative matrix moment problem on the real axis is given with the help of the correspondence between the moment problem on the real axis and the nevanlinna pick np interpolation the solutions of the nontangential np interpolation in the nevanlinna class are derived as an application

large deviations for local time fractional brownian motion and applications abstract let w h w h t t r be a fractional brownian motion of hurst index h 0 1 with values in r and let l l t t 0 be the local time process at zero of a strictly stable levy process x x t t 0 of index 1 2 independent of w h the stable local time fractional brownian motion z h z h t t 0 is defined by z h t w h l t the process z h is self similar with self similarity index h 1 1 and is related to the scaling limit of a continuous time random walk with heavy tailed waiting times between jumps p becker kern m m meerschaert h p scheffler limit theorems for coupled continuous time random walks ann probab 32 2004 730 756 m m meerschaert h p scheffler limit theorems for continuous time random walks with infinite mean waiting times j appl probab 41 2004 623 638 however z h does not have stationary increments and is non gaussian in this paper we establish large deviation results for the process z h as applications we derive upper bounds for the uniform modulus of continuity and the laws of the iterated logarithm for z h

absorbing boundary conditions on arbitrary boundaries for the scalar and vector wave equations the solution of open region scattering problems involving inhomogeneous arbitrarily shaped objects may be performed through the use of partial differential equation techniques which require enclosing the scatterer by an outer boundary on which an absorbing boundary condition abc is applied in order to minimize the size of the domain to be meshed and consequently the number of unknowns if may be advisable to implement abc s devised for outer boundaries of arbitrary shapes such abc s are obtained for the 3d scalar and vector wave equations they incorporate most of existing boundary conditions when used in conjunction with a finite element technique the numerical results derived by using a simplified form of these abc s compare favourably to those obtained by using a rigorous hybrid finite element integral equation formulation these boundary conditions have been obtained in the frequency domain framework they may however be used in time domain calculations

characterizations of bezout and hankel bezout matrices abstract a characterization of bezout matrices is presented as matrices the entries of which satisfy a certain system of homogeneous quadratic equations also the class of matrices which are both hankel and bezout is characterized and formulae for the inversion of such matrices are given

boundary conditions for the numerical solution of elliptic equations in exterior regions elliptic equations in exterior regions frequently require a boundary condition at infinity to ensure the weill posedness of the problem examples of practical applications include the helmholtz equation and laplace s equation computational procedures based on a direct discretization of the elliptic problem require the replacement of the condition at infinity by a boundary condition on a finite artificial surface direct imposition of the condition at infinity along the finite boundary results in large errors a sequence of boundary conditions is developed which provides increasingly accurate approximations to the problem in the infinite domain estimates of the error due to the finite boundary are obtained for several cases computations are presented which demonstrate the increased accuracy that can be obtained by the use of the higher order boundary conditions the examples are based on a finite element formulation but finite difference methods can also be used

a subclass of symmetric loewner matrices abstract the subclass of real symmetric loewner matrices with fixed interpolation nodes and row sums zero is studied it is shown that it forms a commutative semigroup with respect to multiplication and contains with each element its moore penrose inverse as well

a fast hankel solver based on an inversion formula for loewner matrices abstract we propose a new o n2 algorithm for solving complex n n linear systems that have hankel structure via ffts the hankel system is transformed into a loewner system an inversion formula enables us to calculate the inverse of the loewner matrix explicitely the parameters that occur in this inversion formula are calculated by solving two rational interpolation problems on the unit circle we present an o n2 algorithm to solve these interpolation problems one of the advantages of this algorithm is that it incorporates pivoting we have implemented our hankel solver in fortran 90 numerical examples are included they show the effectiveness of our pivoting strategy

a theoretical study of numerical absorbing boundary conditions electromagnetic field computation involving inhomogeneous arbitrarily shaped objects may be carried out conveniently by using partial differential equation techniques e g the finite element method fem when solving open region problems using these techniques it becomes necessary to enclose the scatterer with an outer boundary on which an absorbing boundary condition abc is applied and analytically derived abcs e g the bayliss gunzburger turkel and engquist majda boundary conditions have been used extensively for this purpose numerical absorbing boundary conditions nabcs have been proposed as alternatives to analytical abcs and they are based upon a numerically derived relationship that links the values of the field at the boundary nodes to those at the neighboring nodes in the paper the authors demonstrate analytically that these nabcs become equivalent to many of the existing analytical abcs in the limit as the cell size tends to zero in addition one can evaluate the numerical efficiency of these nabcs by using as an indicator the reflection coefficient for plane and cylindrical waves incident upon an arbitrary boundary

on invertibility of nonsquare generalized bezoutians abstract conditions for inverting h bezoutians of nonsquare matrix polynomials are studied necessary and sufficient conditions for invertibility of the anderson jury bezoutian are obtained an application to the inversion of block hankel matrices is given

inverses of lowner matrices lowner matrices are special matrices which were first used by lowner in 4 to characterize the so called monotone matrix functions lowner showed also their connection with the problem of rational interpolation for these questions see also donoghue 2 belevitch 1 and fiedler 3 lowner matrices also occur in applications for example in the theory of electric networks in this paper we show that for a nonsingular lowner matrix l ci djyi zjn 1i j 0 the matrix l 1 t can be written in the form l 1 t d1l d2 where d1 and d2 are diagonal matrices and l is again a lowner matrix of the form yi jyi zjn 1i j 0 for a more detailed description of all forms l 1 d1l d2 possible we introduce the notion of d classes of the lowner matrices

confluent cauchy and cauchy vandermonde matrices abstract the s c confluent cauchy and confluent cauchy vandermonde matrices are introduced corresponding to an interpolation problem with multiple interpolation nodes inversion formulas are derived and a factorization of confluent loewner matrices by means of confluent cauchy vandermonde matrices is described

hankel and loewner matrices abstract mutual relations between the hankel toeplitz bezout and loewner matrices as well as further connections to rational interpolation and projective geometry are investigated

a stabilized superfast solver for indefinite hankel systems abstract we present a stabilized superfast solver for indefinite hankel systems whose size is a power of 2 the hankel system is transformed into a loewner system which is solved by using an inversion formula for loewner matrices this explicit formula for the inverse of a loewner matrix contains certain parameters that are computed by solving two linearized rational interpolation problems on the unit circle the heart of our hankel solver is a superfast algorithm to solve these interpolation problems this algorithm is stabilized via pivoting iterative improvement and by giving the so called difficult interpolation points an adequate treatment we have implemented our algorithm in fortran 90 numerical examples illustrate the effectiveness of our approach

product integration methods for solving a system of nonlinear volterra integral equations in this paper the technique of subtracting out singularities is used to derive explicit and implicit product euler schemes with order one convergence and a product trapezoidal scheme with order two convergence for a system of volterra integral equations with a weakly singular kernel the convergence proofs of the numerical schemes are presented these are nonstandard since the nonlinear function involved in the integral equation system does not satisfy a global lipschitz condition

the infinite companion matrix abstract a systematic account is given of the properties and applications of the notion of infinite companion matrix of a polynomial p it turns out to be a particular case of a companion matrix corresponding to a pair of polynomials p w of degree n if w is taken to be zn connections with bezoutians projection operators reproducing kernels and dilation theory are explained

more on loewner matrices abstract the study of loewner matrices is continued in a direction close to the recent work by fiedler ptak and vav in all results are stated for the general case of multiple interpolation nodes i e for generalized loewner matrices both the loewner bezoutian connection of antoulas and anderson and the loewner hankel connection of vav in in the general case which are re proved in a simple way provide a good opportunity for finding various properties of loewner matrices some of them being generalizations of results published by other authors for simple nodes only there are also results which are new especially the barnett like formulas and an intertwining relation characteristic for loewner matrices as well as the loewner matrix hankel vector connection in order to handle loewner matrices further on the authors introduce and deal with a class of so called r matrices it turns out that some known results concerning bezoutians hankel matrices and their products can in a natural way be lifted to the level of r matrices and of the products of loewner and r matrices

rational interpolation and state variable realizations abstract the problem is considered of passing from interpolation data for a real rational transfer function matrix to a minimal state variable realization of the transfer function matrix the tool is a loewner matrix which is a generalization of the standard hankel matrix of linear system realization theory and which possesses a decomposition into a product of generalized observability and controllability matrices

a framework for the solution of the generalized realization problem in this paper we present a novel way of constructing generalized state space representations e a b c d of interpolants matching tangential interpolation data the loewner and shifted loewner matrices are the key tools in this approach

on two variable rational interpolation abstract the goal of this contribution is to investigate interpolation of two variable rational functions the tool is the two variable loewner matrix which is an extension of its single variable counterpart the main property of the loewner matrix is that its rank encodes the information concerning minimal complexity interpolants both polynomial and generalized state space descriptor realizations of interpolants are presented examples illustrate how two variable rational functions can be recovered from appropriate measurements

the multipoint pade approximation problem and its hankel vector abstract the multipoint pade approximation problem and the structure of multipoint pade approximation table have been investigated thoroughly in the literature in this paper we show that each multipoint pade approximation problem can be reduced to what amounts to a certain pade approximation problem at infinity by using the so called hankel vector method that was introduced in chen et al 1996 for the general rational interpolation problem this connection offers a new way to study the properties of multipoint pade approximants and the structure and computation of entries of multipoint pade approximation table based on the characteristic polynomials of hankel matrices

a unified approach to loewner and hankel matrices abstract we introduce so called generalized loewner matrices the definition is almost the same as antoulas and anderson s our class of generalized loewner matrices contains both classical loewner and hankel matrices as special cases the properties of such matrices are studied together with the associated interpolation problem the detailed characterization of its nonsolvable cases is given by means of the s c minimal generalized solution by means of rank properties of the loewner matrix and by means of the s c l polynomial introduced by fiedler

the general rational interpolation problem and its connection with the nevanlinna pick interpolation and power moment problem abstract the hankel vector approach in a recent work of the author with zhao and zhang on the general rational interpolation problem grip allows the problem to be reduced to what amounts to a limiting case i e a high order rational interpolation at only a single node infinity in particular it gives rise to a new method to find the coefficient matrix of the linear fractional parametrization of interpolants to the grip this approach is extended in the present paper to the nevanlinna pick np interpolation problem with multiple nodes in the class of nevanlinna functions which can be in essence considered as a grip with certain symmetries the hankel vector suitably adapted to the present situation is in fact a finite nonnegative sequence relative to an axis if the problem is solvable in particular a finite positive sequence in the indeterminate case this development leads to an intrinsic connection between the np problem and problems of the truncated power moment and reconstructing generators of the hankel vector in question and therefore opens up certain ways for the discussion and solution of the np problem and its relatives on the basis of the theories of the rational interpolation and algebraic moment problems

large deviations for subordinated fractional brownian motion and applications abstract let w h w h t t r be a real valued fractional brownian motion with hurst index h 0 1 and let t t t t 0 be an inverse stable subordinator independent of w h the inverse stable subordinator fractional brownian motion z h z h t t 0 is defined by z h t w h t t which may arise as scaling limit of ctrw or random walk in a random environment in this paper we establish large deviation results for the process z h and its supremum process and we also give asymptotic properties of the tail probability of the supremum process

quasidirect decompositions of hankel and toeplitz matrices decompositions over an algebraically closed field of a hankel matrix into a sum of hankel matrices the sum of the ranks of which is equal to the rank of the original matrix are completely described similar results hold for toeplitz matrices

inversion of a block lo wner matrix in this paper we give a fast algorithm to compute the parameters of an inversion formula for any nonsingular block lowner matrix the connection with matrix rational interpolation is given

a least squares approach to reduce stable discrete linear systems preserving their stability abstract a new stability preserving model reduction algorithm for discrete linear siso systems based on a least squares approach is proposed similar to the pade approximation an equation system for the markov parameters involving a high dimensional hankel matrix is considered it is proved that approximate solutions computed via the moore penrose pseudo inverse give rise to a stability preserving reduction scheme furthermore the proposed algorithm is compared to the balanced truncation method showing comparable performance of the reduced systems

on bezoutian reduction with the vandermonde matrix abstract the vandermonde matrix reduces by congruence the bezoutian matrix when the zeros of the highest degree associated polynomial are simple in this paper a generalization of this result to the case of multiple zeros is presented

the nevanlinna pick interpolation problems and power moment problems for matrix valued functions abstract this paper is concerned with the study of several types of matrix interpolation problems including the matrix nevanlinna pick interpolation problem in the nevanlinna class n p the matrix nevanlinna pick interpolation problem in the stieltjes class l p the truncated matrix power moment problem on the real axis the stieltjes matrix power moment problem we use the so called block hankel vector approach for the treatment of these problems and make explicit the correspondence between solutions to the nevanlinna pick interpolation problem and solutions to an associated power moment problem with the block hankel vector under consideration this enables one not only to find existence criteria for the solutions of the nevanlinna pick interpolation problem but also to describe the solutions based on the results of power moment problem and vice versa

note on products of bezoutians and hankel matrices abstract a quantitative theorem due to fiedler and ptak on products of bezoutians and their inverses is extended to a more general case using as the main tool an alternative to fiedler s assertion on the quasidirect decomposition of hankel matrices

polynomials and hankel matrices abstract compatibility of a hankel n n matrix h and a polynomial f of degree m m n is defined if m n compatibility means that hc t f c f h where c f is t companion matrix of f with a suitable generalization of c f this theorem is generalized to the case that m n

polynomial bezoutian matrix with respect to a general basis we introduce a class of so called polynomial bezoutian matrices the operator representation relative to a pair of dual bases and the generalized barnett factorization formula for this kind of matrix are derived an intertwining relation and generalized bezoutian reduction via polynomial vandermonde matrix are presented

on bezoutians van der monde matrices and the lienard chipart stability criterion we show how the bezoutian can be reduced by congruence to diagonal form with the congruence given by a van der monde matrix this result is applied to obtain new proofs of some classical stability criteria and in particular of the lienardchipart theorem

multiple nevanlinna pick interpolation with both interior and boundary data and its connection with the power moment problem abstract the so called hankel vector approach is used to handle three boundary versions of the multiple nevanlinna pick interpolation problem in the nevanlinna class n involving both interior and boundary data it turns out that each of these boundary interpolation problems can be reduced to what amounts to a certain truncated standard or nonstandard hamburger moment problem associated with the hankel vector of the former with some possible constraint on distribution functions that assign no mass to any of the real nodes in particular this leads to solvability criteria for each of these interpolation problems and the description of solutions by using results from theory of moments

a fast domain decomposition method for the solution of electromagnetic scattering by large objects a nonoverlapping domain decomposition method is proposed for the finite element solution of the scattering problem by electrically large inhomogeneous infinite cylinders of arbitrary cross section to minimize the size of the total computational domain a second order absorbing boundary condition abc is applied upon an outer boundary of arbitrary shape which may be conformal to the surface of the scatterer this domain is then partitioned into concentric subdomains circumscribing the object a second order transmission condition derived from the abc is prescribed upon the interfaces between two adjacent subdomains this particular configuration is responsible for the fast convergence of the domain decomposition iterative algorithm which is parallelizable numerical results obtained with a nonparallelized computer code are presented which emphasize the superiority of this technique in terms of memory storage requirements and computing times over the standard finite element approach as well as over the rigorous hybrid finite element integral equation formulation

on the multiple nevanlinna pick matrix interpolation in the class cp and the caratheodory matrix coefficient problem abstract in the present paper an intrinsic one to one correspondence between the finite multiple nevanlinna pick matrix interpolation np problem in the caratheodory class and the caratheodory matrix coefficient cc problem or equivalently the truncated trigonometric matrix moment tm problem is established thanks to this correspondence the np problem is reduced to what amounts to solving the cc problem or the tm problem associated with the so called toeplitz block vector of the former in both the nondegenerate and degenerate cases simultaneously

polynomials compatible with a symmetric loewner matrix abstract several mutually equivalent conditions are proved for a polynomial of degree n and a symmetric n n matrix which form a basis for so called compatibility of these two objects one of the conditions yields a new parametric expression for a symmetric loewner matrix

numerical absorbing boundary conditions for the scalar and vector wave equations electromagnetic field computation may be carried out conveniently by using the finite element method fem when solving open region problems using this technique it becomes necessary to enclose the scatterer with an outer boundary upon which an absorbing boundary condition abc is applied analytically derived abcs have been used extensively for this purpose numerical absorbing boundary conditions nabcs have been proposed as alternatives to analytical abcs for the two dimensional 2 d helmholtz equation it has been demonstrated analytically that these nabcs become equivalent to many of the existing analytical abs in the limit as the cell size tends to zero in addition the numerical efficiency of these nabcs has been evaluated by using as an indicator the reflection coefficient for plane and cylindrical waves incident upon an arbitrary boundary we have extended this procedure to the study of the nabcs derived for the three dimensional 3 d scalar and vector wave equations from the point of view of their numerical implementation in node and edge based fem formulations respectively

vandermonde factorization and canonical representations of block hankel matrices abstract we study to which extent well known facts concerning vandermonde factorization or canonical representation of scalar hankel matrices transfer to block hankel matrices with p q blocks it is shown that nonsingular block hankel matrices can be factored like in the scalar case into nonconfluent vandermonde matrices and that the theorem on full rank factorization of arbitrary hankel matrices transfers in a weak version to the 2 2 block case but not to larger block sizes in general the minimal rank of a vandermonde factorization both with finite nodes and affine is described in terms of the hankel matrix the main tools are realization partial realization and moebius transformations

bezoutian representation via vandermonde matrices abstract the result on benoutian representation via vandermonde matrices is provided with a proof which is based on polynomial models

intertwining and testing matrices corresponding to a polynomial abstract given a natural number n and a polynomial of degree m n a description is given of a class r n of n by n matrices the intertwining matrices for with the following properties let h n be the family of all n by n hankel matrices compatible with and consider the relation hr t rh then an n by n matrix h belongs to h n if and only if this relation holds for all r r n a characterization is also given of testing matrices a matrix r is said to be a testing matrix if the fulfillment of the above relation already implies that h h n

remarks on the inversion of hankel matrices the paper continues the investigation into the links between algebraic system theory more specifically the partial realization problem and the problem of hankel matrix inversion the representation of the inverse of a hankel matrix as a bezoutiant and the bezout equation play a principal role

rational interpolation and recursive solution of lowner vandermonde systems of equations abstract rational interpolation problems for functions having numerator degree higher than denominator degree are connected with solutions of systems of equations the coefficient matrix of which has a mixed structure the first columns are of vandermonde type whereas the last columns form a lowner matrix here three term recursions for the rational interpolants are developed which can be translated into recurrence formulas for the solutions of homogeneous systems with such a coefficient matrix on this base an o n2 algorithm for the solution of n n nonhomogeneous lowner vandermonde systems is obtained

a generalized bezoutian matrix with respect to a polynomial sequence of interpolatory type in this note a generalized bezoutian matrix with respect to a polynomial sequence of interpolatory type is introduced the operator representation relative to a pair of dual bases and the generalized barnett type factorization formula are derived an intertwining relation and a bezoutian reduction to a block diagonal form by congruence via a generalized vandermonde matrix are presented fujiwara hermite and routh hurwitz criteria in terms of this generalized bezout matrix are obtained

generalized confluent cauchy and cauchy vandermonde matrices abstract the so called generalized confluent cauchy vandermonde gccv matrices of the form c v consisting of a generalized confluent cauchy part c and a generalized confluent vandermonde part v are considered a simple relationship between gccv and classical confluent cauchy vandermonde ccv matrices is given this leads to the reduction of the displacement structure inversion formulas and factorizations of gccv matrices and the interpolation interpretations of linear systems with such gccv coefficient matrices as tangential interpolation problems to the corresponding results of ccv matrices the criteria of invertibility and left right inversion formulas for such matrices are given all results are stated for the general case of multiple interpolation nodes extending the work of heinig and vavrin among others

recursive solution of lowner vandermonde systems of equations ii abstract an o n 2 algorithm for the solution of a linear system the n n coefficient matrix of which consists of a vandermonde and a lowner part is presented in the first part of the algorithm the corresponding vandermonde system is solved in the second part the last rows of the vandermonde matrix are successively replaced by rows of the lowner structure the algorithm is convenient if the vandermonde part is dominant the main tool is the connection of such systems with rational interpolation problems

generalized cauchy vandermonde matrices abstract matrices of the form c v consisting of a generalized cauchy matrix and a generalized vandermonde matrix are considered using the displacement structure of these matrices inversion formulas and criteria are presented the interpretation of linear systems with such a coefficient matrix as tangential interpolation problems leads to the concept of fundamental matrix which is basic in this approach for fundamental matrices recursion formulas are established from them fast inversion algorithms emerge that work for arbitrary nonsingular matrices of this kind

the bezoutian the hankel matrix output feedback and system eigenstructure abstract given a minimal linear time invariant system for which the transfer matrix has a right rp 1 and a left d 1 n coprime matrix fraction description an inverse relation is established between the block hankel matrix formed from the markov parameters of the eigenstructure of p and d and the generalized bezoutian matrix formed from the polynomial matrices r p n and d furthermore it is shown that all eigenstructures that are assignable by output feedback form the same markov parameters and this suggests a new approach to eigenstructure assignment

transformation techniques for toeplitz and toeplitz plus hankel matrices part i transformations transformations of the form a c1ac2 are investigated that transform toeplitz and toeplitz plus hankel matrices into generalized cauchy matrices c1 and c2 are matrices related to the discrete fourier transformation or to various real trigonometric transformations combining these results with pivoting techniques in part ii algorithms for toeplitz and toeplitz plus hankel systems will be presented that are more stable than classical algorithms

scattering of time harmonic electromagnetic waves by anisotropic inhomogeneous scatterers or impenetrable obstacles we investigate an overlapping solution technique to compute the scattering of time harmonic electromagnetic waves in two dimensions the technique can be used to compute waves scattered by penetrable anisotropic inhomogeneous scatterers or impenetrable obstacles the major focus is on implementing the method using finite elements we prove existence of a unique solution to the discretized problem and derive an optimal convergence rate for the scheme which is verified by numerical examples

displacement structure theory and applications in this survey paper we describe how strands of work that are important in two different fields matrix theory and complex function theory have come together in some work on fast computational algorithms for matrices with what we call displacement structure in particular a fast triangularization procedure can be developed for such matrices generalizing in a striking way an algorithm presented by schur 1917 j reine angew math 147 1917 pp 205 232 in a paper on checking when a power series is bounded in the unit disc this factorization algorithm has a surprisingly wide range of significant applications going far beyond numerical linear algebra we mention among others inverse scattering analytic and unconstrained rational interpolation theory digital filter design adaptive filtering and state space least squares estimation

stability of absorbing boundary conditions higher order absorbing boundary conditions abcs exhibit instabilities that can be detrimental to a wide class of finite difference time domain fdtd open region simulations earlier works attributed the cause of instabilities to the intrinsic construction or makeup of the abcs and consequently to the pole zero distribution of the transfer function that characterizes the boundary condition we investigate the cause of instability we focus on axial boundary conditions such as higdon 1986 1990 bayliss turkel 1980 and liao and show through an empirical study that these abcs are not intrinsically unstable in their original unmodified forms furthermore we show that the instability typically observed in fdtd open region simulations is caused by an artifact of the rectangular computational domain contrary to previously conjectured hypotheses or theories these findings will have strong implications that can aid in the construction of stable fdtd schemes

on recursiveness and related topics in linear systems a decomposition of matrix formal power series linear systems is developed the interconnections between formal power series realization pade approximation recursiveness berlekamp massey algorithm feedback interconnections matrix linear fractional representations matrix continued fractions and the matrix euclidean algorithm are thereby revealed

remarks on complexity of polynomial and special matrix computations abstract we prove some inequalities on the asymptotic orders of complexity of the problems of polynomial and rational interpolation polynomial evaluation and inversion of loewner and toeplitz or hankel matrices in this connection an open problem remains whether the whole group of these related problems has the same order of complexity or not

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

inversion of a generalized block loewner matrix the minimal partial realization and matrix rational interpolation problem abstract inversion theorems for generalized block loewner matrices with non square blocks are presented connections are made to the general i e hermite matrix rational interpolation theory of partial realization and its extensions theoretical and computational aspects of many other special types of matrices such as block hankel block toeplitz vandermonde block bezout etc

application and optimization of pml abc for the 3 d wave equation in the time domain a three dimensional algorithm with the perfectly matched layer pml absorbing boundary condition abc for the scalar wave equation in the time domain is presented for general inhomogeneous lossy or loss free problems the proposed pml abc is applicable to practical finite difference schemes treating the time domain wave equation such as the time domain wave potential tdwp technique and the time domain scalar wave equation approaches to the analysis of optical structures the time domain wave equation for lossy media is expressed in terms of stretched coordinate variables the algorithm is tested for homogeneous and inhomogeneous media we demonstrate applications to open radiation problems and to port terminations in high frequency circuit problems new pml conductivity profiles are developed for use with the second order wave equation which offer lower reflections in a wider frequency band in comparison with the commonly used in finite difference time domain fdtd algorithms profiles the effect of the termination walls on the overall pml performance is studied and the best choices are singled out

a note on a toeplitz inversion formula abstract it is shown that a formula of heinig for the inverse of a toeplitz matrix follows from a formula of the author for the inverse of a hankel matrix

multiplication of diagonal transforms of loewner matrices abstract a new formula for multiplication of matrices arising as appropriate diagonal transformations of loewner matrices is derived in the case of symmetric loewner matrices it is shown that special subclasses form multiplicative groups some extensions of known results about polynomials associated with hankel and loewner matrices are given also a previously found formula of the author describing the inverses of loewner matrices is shown to be a consequence of the present results

numerical implementation of second and third order conformal absorbing boundary conditions for the vector wave equation a rigorous implementation in an edge based finite element formulation of second and third order conformal absorbing boundary conditions abcs is presented for the solution of three dimensional 3 d scattering problems when the boundary s terminating the mesh is the surface of a parallelepiped a special treatment is provided for the singularities edges of s a systematic numerical study is carried out that compares the performances of these abcs with those of the standard zero order abc as well as of a more simple though less rigorous implementation of the second order abc when s is separated from the scatterer by only one or two layers of elements the numerical results that are presented demonstrate the good level of numerical accuracy achieved when the second and third order abcs are employed and the singularities of s are appropriately dealt with