reliable numerical methods for polynomial matrix triangularization numerical procedures are proposed for triangularizing polynomial matrices over the field of polynomial fractions and over the ring of polynomials they are based on two standard polynomial techniques sylvester matrices and interpolation in contrast to other triangularization methods the algorithms described in this paper only rely on well worked numerically reliable tools they can also be used for greatest common divisor extraction polynomial rank evaluation or polynomial null space computation

ratio asymptotic of hermite pade orthogonal polynomials for nikishin systems ii we prove ratio asymptotic for sequences of multiple orthogonal polynomials with respect to a nikishin system of measures such that for each k sk has constant sign on its support consisting on an interval on which almost everywhere and a set without accumulation points in

uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory we consider asymptotics for orthogonal polynomials with respect to varying exponential weights wn x dx e nv x dx on the line as n the potentials v are assumed to be real analytic with sufficient growth at infinity the principle results concern plancherel rotach type asymptotics for the orthogonal polynomials down to the axis using these asymptotics we then prove universality for a variety of statistical quantities arising in the theory of random matrix models some of which have been considered recently in 31 and also in 4 an additional application concerns the asymptotics of the recurrence coefficients and leading coefficients for the orthonormal polynomials see also 4 the orthogonal polynomial problem is formulated as a riemann hilbert problem following 19 20 the riemann hilbert problem is analyzed in turn using the steepest descent method introduced in 12 and further developed in 11 13 a critical role in our method is played by the equilibrium measure d v for v as analyzed in 8 1999 john wiley sons inc

oversampled filter banks as error correcting codes theory and impulse noise correction oversampled filter banks ofbs provide an overcomplete representation of their input signal this paper describes how ofbs can be considered as error correcting codes acting on real or complex sequences very much like classical binary convolutional codes act on binary sequences the structured redundancy introduced by ofbs in subband signals can be used to increase robustness to noise in this paper we define the notions of code subspace syndrome and parity check polynomial matrix for ofbs furthermore we derive generic expressions for projection based decoding suitable for the case when a simple second order model completely characterizes the noise incurred by subband signals we also develop a nonlinear hypotheses test based decoding algorithm for the case when the noise in subbands is constituted by a gaussian background noise and impulsive errors a model that adequately describes the action of both quantization noise and transmission errors simulation results show that the algorithm effectively removes the effect of impulsive errors occurring with a probability of 10 sup 3

polynomial j spectral factorization several algorithms are presented for the j spectral factorization of a para hermitian polynomial matrix the four algorithms that are discussed are based on diagonalization successive factor extraction interpolation and the solution of an algebraic riccati equation respectively the paper includes a special algorithm for the factorization of unimodular para hermitian polynomial matrices and deals with canonical noncanonical and nearly noncanonical factorizations

asymptotics and integrable structures for biorthogonal polynomials associated to a random two matrix model abstract we give a rigorous construction of complete families of biorthonormal polynomials associated to a planar measure of the form e n v x w y 2 xy d x d y for polynomial v and w we are further able to show that the zeroes of these polynomials are all real and distinct a complex analytical construction of the biorthonormal polynomials is given in terms of a non local riemann hilbert problem which given our prior result provides an avenue for developing uniform asymptotics for the statistical distributions of these zeroes as n becomes large the biorthonormal polynomials considered here play a fundamental role in the analysis of certain random multi matrix models we show that the evolutions of the recursion matrices for the polynomials induced by linear deformations of v and w coincide with a semi infinite generalization of the completely integrable full kostant toda lattice this connection could be relevant for understanding aspects of scaling limits for the multi matrix model

parity check matrix calculation for paraunitary oversampled dft filter banks joint source channel coding schemes based on oversampled filter banks were recently proposed oversampled filter banks are considered as error correcting codes extending the notion of code subspace parity check matrix and syndrome to the domain of real and complex numbers we here present an efficient calculation and implementation of the parity check polynomial matrices for oversampled discrete fourier transform dft filter banks if desired the calculation of the parity check polynomials can be performed as part of the prototype filter design procedure we compare our method to those previously presented in the literature

the isomonodromy approach to matric models in 2d quantum gravity we consider the double scaling limit in the hermitian matrix model for 2d quantum gravity associated with the measure exp sum limits j 1 n t j z 2j n geqq 3 we show that after the appropriate modification of the contour of integration the cross migdal douglas shenker limit to the painleve i equation in the generic case of the pure gravity is valid and calculate the nonperturbative parameters of the corresponding painleve function our approach is based on the wkb analysis of the l a pair corresponding to the discrete string equation in the framework of the inverse monodromy method here we extend our results which were obtained before for the particular casesn 2 3 our analysis complements the isomonodromy approach proposed by g moore to the general string equations that come from the matrix model in the continuous limit and differ in that we apply the isomonodromy technique to investigate the double scaling limit itself

spectral factorization of non symmetric polynomial matrices the topic of the paper is spectral factorization of rectangular and possibly non full rank polynomial matrices to each polynomial matrix we associate a matrix pencil by direct assignment of the coefficients the associated matrix pencil has its finite generalized eigenvalues equal to the zeros of the polynomial matrix the matrix dimensions of the pencil we obtain by solving an integer linear programming ilp minimization problem then by extracting a deflating subspace of the pencil we come to the required spectral factorization we apply the algorithm to most general case of inner outer factorization regardless continuous or discrete time case and to finding the greatest common divisor of polynomial matrices

an improved toeplitz algorithm for polynomial matrix null space computation in this paper we present an improved algorithm to compute the minimal null space basis of polynomial matrices a problem which has many applications in control and systems theory this algorithm takes advantage of the block toeplitz structure of the sylvester matrix associated with the polynomial matrix the analysis of algorithmic complexity and numerical stability shows that the algorithm is reliable and can be considered as an efficient alternative to the well known pencil state space algorithms found in the literature

computing dynamic output feedback laws the pole placement problem asks to find laws to feed the output of a plant governed by a linear system of differential equations back to the input of the plant so that the resulting closed loop system has a desired set of eigenvalues converting this problem into a question of enumerative geometry efficient numerical homotopy algorithms to solve this problem for general multiple input multiple output systems have been proposed recently despite the wider application range of dynamic feedback laws the realization of the output of the numerical homotopies as a machine to control the plant in the time domain has not been addressed before in this note we present symbolic numeric algorithms to turn the solution to the question of enumerative geometry into a useful control feedback machine we report on numerical experiments with our publicly available software phcpack and illustrate its application on various control problems from the literature

riemann hilbert problem for bi orthogonal polynomials the 3 3 matrix riemann hilbert problem for bi orthogonal polynomials with the third degree polynomial potential functions is explicitly constructed the developed approach can be extended to bi orthogonal polynomials with arbitrary polynomial potentials

generic behavior of the density of states in random matrix theory and equilibrium problems in the presence of real analytic external fields the equilibrium measure in the presence of an external field plays a role in a number of areas in analysis for example in random matrix theory the limiting mean density of eigenvalues is precisely the density of the equilibrium measure typical behavior for the equilibrium measure is 1 it is positive on the interior of a finite number of intervals 2 it vanishes like a square root at endpoints and 3 outside the support there is strict inequality in the euler lagrange variational conditions if these conditions hold then the limiting local eigenvalue statistics is loosely described by a bulk in which there is universal behavior involving the sine kernel and edge effects in which there is a universal behavior involving the airy kernel through techniques from potential theory and integrable systems we show that this regular behavior is generic for equilibrium measures associated with real analytic external fields in particular we show that for any one parameter family of external fields v c the equilibrium measure exhibits this regular behavior except for an at most countable number of values of c we discuss applications of our results to random matrices orthogonal polynomials and integrable systems 2000 john wiley sons inc

a modified belief propagation algorithm for code word quantization modern coding advances including dirty paper coding and information hiding require quantizing a given binary word to a code word a good solution would approach the rate distortion bound in lossy source compression here we propose a simple variant on belief propagation which is observed to converge to a solution giving respectable rate distortion performance comparisons with other recently proposed source quantization methods reveal that the proposed algorithm holds particular interest in short block length applications as encountered in packet based communication systems

duality biorthogonal polynomials and multi matrix models the statistical distribution of eigenvalues of pairs of coupled random matrices can be expressed in terms of integral kernels having a generalized christoffel darboux form constructed from sequences of biorthogonal polynomials for measures involving exponentials of a pair of polynomials v1 v2 in two different variables these kernels may be expressed in terms of finite dimensional windows spanned by finite subsequences having length equal to the degree of one or the other of the polynomials v1 v2 the vectors formed by such subsequences satisfy dual pairs of first order systems of linear differential equations with polynomial coefficients having rank equal to one of the degrees of v1 or v2 and degree equal to the other they also satisfy recursion relations connecting the consecutive windows and deformation equations determining how they change under variations in the coefficients of the polynomials v1 and v2 viewed as overdetermined systems of linear difference differential deformation equations these are shown to be compatible and hence to admit simultaneous fundamental systems of solutions the main result is the demonstration of a spectral duality property namely that the spectral curves defined by the characteristic equations of the pair of matrices defining the dual differential systems are equal upon interchange of eigenvalue and polynomial parameters

multiple orthogonal polynomials results on multiple orthogonal polynomials will be surveyed multiple orthogonal polynomials are intimately related to hermite pade approximants and often they are also called hermite pade polynomials special attention will be paid to an application of multiple orthogonal polynomials and to analytic theory of two model families of general multiple orthogonal polynomials referred to as angelesco and nikishin systems among the applications the number theory special functions and spectral analysis of nonsymmetric band operators will be highlighted in the analytic theory results and methods for the study of multiple orthogonal polynomials asymptotics will be reviewed new results on strong asymptotics of multiple orthogonal polynomials for nikishin system will be presented

first order asymptotics of matrix integrals a rigorous approach towards the understanding of matrix models we investigate the large n limit of spectral measures of matrices which relate to the gibbs measures of a number of statistical mechanical systems on random graphs these include the ising and potts models on random graphs for most of these models we prove that the spectral measures converge almost surely and describe their limit via solutions to an euler equation for isentropic flow with negative pressure p 3 1 2 3

algorithms for computing a hermite reduction of a matrix with polynomial coefficients resume nous etudions dans cet article des algorithmes de calcul de la reduction de hermite d une matrice a coefficients polynomiaux qui evitent l explosion de la taille des objets intermediaires et ont une bonne complexite sequentielle le deuxieme et le troisieme algorithmes generalisent la methode des sous resultants pour le calcul du pgcd de deux polynomes le dernier de ces algorithmes semble optimal en ce sens qu il calcule une reduction de hermite avec une matrice de passage de degre minimal en utilisant une methode progressive et economique nous ramenons le calcul de la reduite de hermite a un probleme de triangulation progressive des matrices dont les tailles sont bien controlees et dont les entrees sont des coefficients de polynomes donnes en entree l algorithme est donc faisable des qu on dispose d une methode de triangulation en temps polynomial pour les matrices dans le corps des coefficients il revient au meme de dire que le calcul des determinants est en temps polynomial dans le corps en question pour le codage choisi les resultats theorqiues sont meilleurs que pour les algorithmes connus precedemment et les resultats partiques sont encourageants

convergence analysis of krylov subspace iterations with methods from potential theory krylov subspace iterations are among the best known and most widely used numerical methods for solving linear systems of equations and for computing eigenvalues of large matrices these methods are polynomial methods whose convergence behavior is related to the behavior of polynomials on the spectrum of the matrix this leads to an extremal problem in polynomial approximation theory how small can a monic polynomial of a given degree be on the spectrum this survey gives an introduction to a recently developed technique to analyze this extremal problem in the case of symmetric matrices it is based on global information on the spectrum in the sense that the eigenvalues are assumed to be distributed according to a certain measure then depending on the number of iterations the lanczos method for the calculation of eigenvalues finds those eigenvalues that lie in a certain region which is characterized by means of a constrained equilibrium problem from potential theory the same constrained equilibrium problem also describes the superlinear convergence of conjugate gradients and other iterative methods for solving linear systems

double scaling limit in the random matrix model the riemann hilbert approach we prove the existence of the double scaling limit in the unitary matrix model with quartic interaction and we show that the correlation functions in the double scaling limit are expressed in terms of the integrable kernel determined by the function for the hastings mcleod solution to the painleve ii equation the proof is based on the riemann hilbert approach and the central point of the proof is an analysis of analytic semiclassical asymptotics for the function at the critical point in the presence of four coalescing turning points 2003 wiley periodicals inc

matrices coupled in a chain i eigenvalue correlations the general correlation function for the eigenvalues of p complex hermitian matrices coupled in a chain is given as a single determinant for this we use a slight generalization of a theorem of dyson

a coprime factorisation and its application in ℋ2 control given a plant transfer matrix as a fraction of two polynomial matrices we present a doubly coprime factorisation identity over the proper stable rational matrices of the plant transfer matrix and

polynomial and rational matrix interpolation theory and control applications a generalization of polynomial interpolation to the matrix case is introduced and applied to problems in systems and control it is shown that this generalization is most general and it includes all other such interpolation schemes that have appeared in the literature the polynomial matrix interpolation theory is developed and then applied to solving matrix equations solutions to the diophantine equation are also derived the relation between a polynomial matrix and its characteristic values and vectors is established and it is used in pole assignment and other control problems rational matrix interpolation is discussed it can be seen as a special case of polynomial matrix interpolation it is then used to solve rational matrix equations including the model matching problem

universality in the two matrix model a riemann hilbert steepest descent analysis the eigenvalue statistics of a pair m 1 m 2 of n x n hermitian matrices taken randomly with respect to the measure 1 z n exp n tr v m 1 w m 2 tau m 1 m 2 dm 1 dm 2 can be descri

a toeplitz algorithm for polynomial j spectral factorization a block toeplitz algorithm is proposed to perform the j spectral factorization of a para hermitian polynomial matrix the input matrix can be singular or indefinite and it can have zeros along the imaginary axis the key assumption is that the finite zeros of the input polynomial matrix are given as input data the algorithm is based on numerically reliable operations only namely computation of the null spaces of related block toeplitz matrices polynomial matrix factor extraction and linear polynomial matrix equations solving

diophantine equations in control a survey abstract this survey is also a tutorial whose aim is to explain the role of diophantine equations in the synthesis of feedback control systems these are linear equations in a ring and result from a fractional representation of the systems involved the cornerstone of the exposition is a simple parametrization of all stabilizing controllers for a given plant one can then choose in principle the best controllers for various applications these ideas evolved from early attempts to use polynomial equations in the design of discrete time linear systems by now they have been extended to continuous time infinite dimensional time varying and non linear systems

universality of a double scaling limit near singular edge points in random matrix models we consider unitary random matrix ensembles z n s t 1 e n tr v s t m dm on the space of hermitian n n matrices m where the confining potential vs t is such that the limiting mean density of eigenvalues as n and s t 0 vanishes like a power 5 2 at a singular endpoint of its support the main purpose of this paper is to prove universality of the eigenvalue correlation kernel in a double scaling limit the limiting kernel is built out of functions associated with a special solution of the pi2 equation which is a fourth order analogue of the painleve i equation in order to prove our result we use the well known connection between the eigenvalue correlation kernel and the riemann hilbert rh problem for orthogonal polynomials together with the deift zhou steepest descent method to analyze the rh problem asymptotically the key step in the asymptotic analysis will be the construction of a parametrix near the singular endpoint for which we use the model rh problem for the special solution of the pi2 equation

a method of integration over matrix variables the integral over twon n hermitan matricesz g c dadbexp tr a 2 b 2 2cab g n a 4 b 4 is evaluated in the limit of largen for this purpose use is made of the theory of diffusion equation and that of orthogonal polynomials with a non local weight the above integral arises in the study of the planar approximation to quantum field theory

symmetric matrix polynomial equation interpolation results new numerical procedures are proposed to solve the symmetric matrix polynomial equation a t s x s x t s a s 2b s that is frequently encountered in control and signal processing an interpolation approach is presented that takes full advantage of symmetry properties and leads to an equivalent reduced size linear system of equations it results in a simple and general characterization of all solutions of expected column degrees several new theoretical results concerning stability theory and reduced sylvester resultant matrices are also developed and used to conclude a priori on the existence of a solution by means of numerical experiments it is shown that our algorithms are more efficient than older methods and namely appear to be numerically reliable

semiclassical asymptotics of orthogonal polynomials riemann hilbert problem and universality in the matrix model we derive semiclassical asymptotics for the orthogonal polynomials pn z on the line with respect to the exponential weight exp inv z where v z is a double well quartic polynomial in the limit when n n 1 we assume that n n cri for some 0 where cr is the critical value which separates orthogonal polynomials with two cuts from the ones with one cut simultaneously we derive semiclassical asymptotics for the recursive coe cients of the orthogonal polynomials and we show that these coe cients form a cycle of period two which drifts slowly with the change of the ratio n n the proof of the semiclassical asymptotics is based on the methods of the theory of integrable systems and on the analysis of the appropriate matrix riemannhilbert problem as an application of the semiclassical asymptotics of the orthogonal polynomials we prove the universality of the local distribution of eigenvalues in the matrix model with the double well quartic interaction in the presence of two cuts contents

numerical computation of a coprime factorization of a transfer function matrix abstract a new algorithm is presented for computing a coprime factorization of a transfer function matrix the method is based on reformulating the problem as a problem of constructing a minimal basis of the right nullspace of a matrix pencil b a due to the special structure and rank properties of the pencil obtained after transforming b a to generalized schur form the original problem can be easily solved in a numerically reliable way

two coupled matrices eigenvalue correlations and spacing functions for two n n hermitian matrices a and b with joint probability density proportional to exp tr u a v b 2cab where u and v are polynomials a method is given to calculate all correlation cluster and spacing functions of the eigenvalues of either one or both matrices the method relies on the introduction of two sets of bi orthogonal polynomials with non local weights in a linear chain of coupled matrices if one looks for the statistical properties of the eigenvalues of only one matrix two matrices situated anywhere in the chain then we can proceed as a one matrix two matrices problem