a new class of gradient adaptive step size lms algorithms the gradient adaptive step size least mean square lms algorithms an important family of variable step size lms vslms algorithms are revisited we propose a simplification to a class of the studied algorithms and show that this leads to a new class of vslms algorithms with reduced complexity but with no observable loss in performance

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

recurrence for discrete time unitary evolutions we consider quantum dynamical systems specified by a unitary operator u and an initial state vector phi in each step the unitary is followed by a projective measurement checking whether the system has returned to the initial state we call the system recurrent if this eventually happens with probability one we show that recurrence is equivalent to the absence of an absolutely continuous part from the spectral measure of u with respect to phi we also show that in the recurrent case the expected first return time is an integer or infinite for which we give a topological interpretation a key role in our theory is played by the first arrival amplitudes which turn out to be the complex conjugated taylor coefficients of the schur function of the spectral measure on the one hand this provides a direct dynamical interpretation of these coefficients on the other hand it links our definition of first return times to a large body of mathematical literature

cubic theta functions abstract the theory of the cubic theta function q m 2 m n n 2 z 1 m z 2 n is developed analogously to the theory of the classical jacobian theta functions many inversion formulas addition formulas and modular equations are derived and these results extend previous work on cubic theta functions

on optimal short recurrences for generating orthogonal krylov subspace bases we analyze necessary and sufficient conditions on a nonsingular matrix a such that for any initial vector r 0 an orthogonal basis of the krylov subspaces cal k n a r 0 is generated by a short recurrence orthogonality here is meant with respect to some unspecified positive definite inner product this question is closely related to the question of existence of optimal krylov subspace solvers for linear algebraic systems where optimal means the smallest possible error in the norm induced by the given inner product the conditions on a we deal with were first derived and characterized more than 20 years ago by faber and manteuffel siam j numer anal 21 1984 pp 352 362 their main theorem is often quoted and appears to be widely known its details and underlying concepts however are quite intricate with some subtleties not covered in the literature we are aware of our paper aims to present and clarify the existing important results in the context of the faber manteuffel theorem furthermore we review attempts to find an easier proof of the theorem and explain what remains to be done in order to complete that task

positive definite toeplitz matrices the arnoldi process for isometric operators and gaussian quadrature on the unit circle abstract we show that the well known levinson algorithm for computing the inverse cholesky factorization of positive definite toeplitz matrices can be viewed as a special case of a more general process the latter process provides a very efficient implementation of the arnoldi process when the underlying operator is isometric this is analogous with the case of hermitian operators where the hessenberg matrix becomes tridiagonal and results in the hermitian lanczos process we investigate the structure of the hessenberg matrices in the isometric case and show that simple modifications of them move all their eigenvalues to the unit circle these eigenvalues are then interpreted as abscissas for analogs of gaussian quadrature now on the unit circle instead of the real line the trapezoidal rule appears as the analog of the gauss legendre formula

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

cmv the unitary analogue of jacobi matrices we discuss a number of properties of cmv matrices by which we mean the class of unitary matrices studied recently by cantero moral and velazquez we argue that they play an equivalent role among unitary matrices to that of jacobi matrices among all hermitian matrices in particular we describe the analogues of well known properties of jacobi matrices foliation by co adjoint orbits a natural symplectic structure algorithmic reduction to this shape lax representation for an integrable lattice system ablowitz ladik and the relation to orthogonal polynomials as offshoots of our analysis we will construct action angle variables for the finite ablowitz ladik hierarchy and describe the long time behavior of this system 2006 wiley periodicals inc

understanding the qr algorithm the qr algorithm is currently the most popular method for finding all eigenvalues of a full matrix while qr is now well understood by specialists in eigenvalue computations this understanding is not being conveyed effectively to the mathematical public many accounts present wilkinson s 1965 convergence proof others establish some of the connections between the qr algorithm the power method and inverse iteration usually much emphasis is rightly placed on the refinements such as shifts of origin which are required to make the algorithm competitive but practically all accounts fail to explain the basic meaning of qr iterations as a consequence the qr algorithm is widely thought to be difficult to understand the purpose of this paper is to try to convince the reader that the opposite is true in fact the qr algorithm is neither more nor less than a clever implementation of simultaneous iteration which is itself a natural easily understood extension of the power method this point

sensitivity analysis for szegő polynomials szeg polynomials are orthogonal with respect to an inner product on the unit circle numerical methods for weighted least squares approximation by trigonometric polynomials conveniently can be derived and expressed with the aid of szeg polynomials this paper discusses the conditioning of several mappings involving szeg polynomials and thereby sheds light on the sensitivity of some approximation problems involving trigonometric polynomials

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

cmv block matrices for symmetric matrix measures on the unit circle abstract in this contribution we study the relationship between the cmv block matrices associated with two symmetric positive definite matrix measures supported on the unit circle we also consider certain transformations of an orthogonality matrix measure supported on the unit circle and deduce connection formulas for the corresponding orthogonal families and their verblunsky matrix coefficients

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

is a complex valued stepsize advantageous in complex valued gradient learning algorithms complex gradient methods have been widely used in learning theory and typically aim to optimize real valued functions of complex variables the stepsize of complex gradient learning methods cglms is a positive number and little is known about how a complex stepsize would affect the learning process to this end we undertake a comprehensive analysis of cglms with a complex stepsize including the search space convergence properties and the dynamics near critical points furthermore several adaptive stepsizes are derived by extending the barzilai borwein method to the complex domain in order to show that the complex stepsize is superior to the corresponding real one in approximating the information in the hessian a numerical example is presented to support the analysis

on the calculation of jacobi matrices abstract given a jacobi matrix the problem in question is to find the jacobi matrix corresponding to the weight function modified by a polynomial r galant and gautschi derived algorithms based on the generalized christoffel theorem of uvarov applicable when the roots of r are known in this paper we present two methods not requiring the explicit knowledge of the roots of r we also obtain various properties of the similarity transformations between jacobi matrices which we prove by simple matrix calculus without using the generalized christoffel theorem

a quantum dynamical approach to matrix khrushchev s formulas khrushchev s formula is the cornerstone of the so called khrushchev theory a body of results which has revolutionized the theory of orthogonal polynomials on the unit circle this formula can be understood as a factorization of the schur function for an orthogonal polynomial modification of a measure on the unit circle no such formula is known in the case of matrix valued measures this constitutes the main obstacle to generalize khrushchev theory to the matrix valued setting which we overcome in this paper it was recently discovered that orthogonal polynomials on the unit circle and their matrix valued versions play a significant role in the study of quantum walks the quantum mechanical analogue of random walks in particular schur functions turn out to be the mathematical tool which best codify the return properties of a discrete time quantum system a topic in which khrushchev s formula has profound and surprising implications we will show that this connection between schur functions and quantum walks is behind a simple proof of khrushchev s formula via quantum diagrammatic techniques for cmv matrices this does not merely give a quantum meaning to a known mathematical result since the diagrammatic proof also works for matrix valued measures actually this path counting approach is so fruitful that it provides different matrix generalizations of khrushchev s formula some of them new even in the case of scalar measures furthermore the path counting approach allows us to identify the properties of cmv matrices which are responsible for khrushchev s formula on the one hand this helps to formalize and unify the diagrammatic proofs using simple operator theory tools on the other hand this is the origin of our main result which extends khrushchev s formula beyond the cmv case as a factorization rule for schur functions related to general unitary operators 2015 wiley periodicals inc

contractions with rank one defect operators and truncated cmv matrices the main issue we address in the present paper are the new models for completely nonunitary contractions with rank one defect operators acting on some hilbert space of dimension n these models complement nicely the well known models of livsic and sz nagy foias we show that each such operator acting on some finite dimensional respectively separable infinite dimensional hilbert space is unitarily equivalent to some finite respectively semi infinite truncated cmv matrix obtained from the full cmv matrix by deleting the first row and the first column and acting in cn respectively l2 n this result can be viewed as a nonunitary version of the famous characterization of unitary operators with a simple spectrum due to cantero moral and velazquez as well as an analog for contraction operators of the result from yu arlinski e tsekanovski non self adjoint jacobi matrices with a rank one imaginary part j funct anal 241 2006 383 438 concerning dissipative non self adjoint operators with a rank one imaginary part it is shown that another functional model for contractions with rank one defect operators takes the form of the compression f pk f on the hilbert space l2 t d with a probability measure onto the subspace k l2 t d c the relationship between characteristic functions of sub matrices of the truncated cmv matrix with rank one defect operators and the corresponding schur iterates is established we develop direct and inverse spectral analysis for finite and semi infinite truncated cmv matrices in particular we study the problem of reconstruction of such matrices from their spectrum or the mixed spectral data involving schur parameters it is pointed out that if the mixed spectral data contains zero eigenvalue then no solution unique solution or infinitely many solutions may occur in the inverse problem for truncated cmv matrices the uniqueness theorem for recovered truncated cmv matrix from the given mixed spectral data is established in this part the paper is closely related to the results of hochstadt and gesztesy simon obtained for finite self adjoint jacobi matrices

the symplectic eigenvalue problem the butterfly form the sr algorithm and the lanczos method abstract we discuss some aspects of the recently proposed symplectic butterfly form which is a condensed form for symplectic matrices any 2n 2n symplectic matrix can be reduced to this condensed form which contains 8n 4 nonzero entries and is determined by 4n 1 parameters the symplectic eigenvalue problem can be solved using the sr algorithm based on this condensed form the sr algorithm preserves this form and can be modified to work only with the 4n 1 parameters instead of the 4n2 matrix elements the reduction of symplectic matrices to symplectic butterfly form has a close analogy to the reduction of arbitrary matrices to hessenberg form a lanczos like algorithm for reducing a symplectic matrix to butterfly form is also presented

adaptively combined fir and functional link artificial neural network equalizer for nonlinear communication channel this paper proposes a novel computational efficient adaptive nonlinear equalizer based on combination of finite impulse response fir filter and functional link artificial neural network cfflann to compensate linear and nonlinear distortions in nonlinear communication channel this convex nonlinear combination results in improving the speed while retaining the lower steady state error in addition since the cfflann needs not the hidden layers which exist in conventional neural network based equalizers it exhibits a simpler structure than the traditional neural networks nns and can require less computational burden during the training mode moreover appropriate adaptation algorithm for the proposed equalizer is derived by the modified least mean square mlms results obtained from the simulations clearly show that the proposed equalizer using the mlms algorithm can availably eliminate various intensity linear and nonlinear distortions and be provided with better anti jamming performance furthermore comparisons of the mean squared error mse the bit error rate ber and the effect of eigenvalue ratio evr of input correlation matrix are presented

short recurrences for computing extended krylov bases for hermitian and unitary matrices it is well known that the projection of a matrix a a onto a krylov subspace span left mathbf h a mathbf h a 2 mathbf h ldots a k 1 mathbf h right h ah a2h ak 1h with a in mathbb c n times n a cn n and mathbf h in mathbb c n h cn results in a hessenberg matrix we show that the projection of the matrix a a onto an extended krylov subspace which is of the form span left a k r mathbf h ldots a 2 mathbf h a 1 mathbf h mathbf h a mathbf h a 2 mathbf h ldots a k ell mathbf h right a krh a 2h a 1h h ah a2h aklh is a matrix of so called extended hessenberg form which can be characterized uniquely by its qr qr factorization this qr qr factorization will be presented by means of a pattern of 2 times 2 2 2 unitary rotations we will show how this rotation pattern leads to new insights and allows to elegantly predict the structure of the matrix in case a a is hermitian or unitary this extended hessenberg matrix is banded and structured allowing the design of short recurrence relations for the unitary case coupled two term recurrence relations are derived of which the coefficients capture all information necessary for a sparse factorization of the corresponding extended hessenberg matrix

matrix orthogonal laurent polynomials on the unit circle and toda type integrable systems matrix orthogonal laurent polynomials in the unit circle and the theory of toda like integrable systems are connected using the gauss borel factorization of two left and a right cantero morales velazquez block moment matrices which are constructed using a quasi definite matrix measure a block gauss borel factorization problem of these moment matrices leads to two sets of biorthogonal matrix orthogonal laurent polynomials and matrix szeg polynomials which can be expressed in terms of schur complements of bordered truncations of the block moment matrix the corresponding block extension of the christoffel darboux theory is derived deformations of the quasidefinite matrix measure leading to integrable systems of toda type are studied the integrable theory is given in this matrix scenario wave and adjoint wave functions lax and zakharov shabat equations bilinear equations and discrete flows connected with darboux transformations we generalize the integrable flows of the cafasso s matrix extension of the toeplitz lattice for the verblunsky coefficients of szeg polynomials an analysis of the miwa shifts allows for the finding of interesting connections between christoffel darboux kernels and miwa shifts of the matrix orthogonal laurent polynomials

schur parameter pencils for the solution of the unitary eigenproblem abstract let u v be an n n pencil with unitary matrices u and v an algorithm is presented which reduces u and v simultaneously to unitary block diagonal matrices g o q h up and g e q h vp with block size at most two it is an o n 3 process using householder eliminations and it is backward stable in the special case v i the block diagonal matrices g o g h e can be normalized so that their entries are just the schur parameters of the hessenberg condensed form of u we call g o g e a schur parameter pencil it can also be derived from u v by a lanczos like process for the solution of the eigenvalue problem for g o g e a qr type algorithm can be developed based on this unitary reduction of a pencil u v to a schur parameter pencil the condensed form is preserved throughout the process each iteration step needs only o n operations this method of solving the unitary eigenvalue problem seems to be the closest possible analogy to the qr method for the hermitian eigenvalue problem

an exact recursion for the composite nearest neighbor degeneracy for a 2 n lattice space a set theoretic argument is utilized to develop a recursion relation that yields exactly the composite nearest neighbor degeneracy for simple indistinguishable particles distributed on a 2 n lattice space the associated generating functions as well as the expectation of the resulting statistics are also treated

the lanczos algorithm with partial reorthogonalization the lanczos algorithm is becoming accepted as a powerful tool for finding the cigenvalues and for solving linear systems of equations any practical implementation of the algorithm suffers however from roundoff errors which usually cause the lanczos vectors to lose their mutual orthogonality in order to maintain some level of orthogonality full reorthogonalization fro and selective orthogonalization so have been used in the past as a remedy here partial reorthogonalization pro is proposed as a new method for maintain ing semiorthogonality among the lanczos vectors pro is based on a simple recurrence which allows us to monitor the loss of orthogonality among the lanczos vectors directly wvithout computing the inner products based on the information from the recurrence reorthogonalizations occur only when necessary thus substantial savings are made as compared to fro in some numerical examples we apply the lanczos algorithm with pro to the solution of large symmetric systems of linear equations and show that it is a robust and efficient algorithm for maintaining semiorthogonality among the lanczos vectors the results obtained compare favorably with the conjugate gradient method

boundedness and complete stability of complex valued neural networks with time delay in this paper the boundedness and complete stability of complex valued neural networks cvnns with time delay are studied some conditions to guarantee the boundedness of the cvnns are derived using local inhibition moreover under the boundedness conditions a compact set that globally attracts all the trajectories of the network is also given additionally several conditions in terms of real valued linear matrix inequalities lmis for complete stability of the cvnns are established via the energy minimization method and the approach that converts the complex valued lmis to real valued ones examples with simulation results are given to show the effectiveness of the theoretical analysis

multiple recurrences and the associated matrix structures stemming from normal matrices there are many classical results in which orthogonal vectors stemming from krylov subspaces are linked to short recurrence relations e g three term recurrences for hermitian and short rational recurrences for unitary matrices these recurrence coefficients can be captured in a hessenberg matrix whose structure reflects the relation between the spectrum of the original matrix and the recurrences the easier the recurrences the faster the orthogonal vectors can be computed possibly resulting in computational savings in the design of e g iterative solvers in this article we focus on multiple recurrence relations i e the j 1 st orthogonal vector satisfies mathbf q j 1 sum i j m j rho j i a mathbf q i sum i j ell j gamma j i mathbf q i with rho j i gamma j i scalars and a the matrix defining the krylov space though many compelling results are around the structure of the corresponding hessenberg matrix is mostly deduced by analyzing the inner product relatio

an interior point method for large constrained discrete ill posed problems ill posed problems are numerically underdetermined it is therefore often beneficial to impose known properties of the desired solution such as nonnegativity during the solution process this paper proposes the use of an interior point method in conjunction with truncated iteration for the solution of large scale linear discrete ill posed problems with box constraints an estimate of the error in the data is assumed to be available numerical examples demonstrate the competitiveness of this approach

power methods for calculating eigenvalues and eigenvectors of spectral operators on hilbert spaces power inverse power and orthogonal iteration algorithms for determining the eigenstructure of operators on infinite dimensional hilbert spaces are developed convergence properties of the algorithms are established for certain classes of operators associated with the control of systems described by partial differential equations a simple finite difference method is applied to a particular operator to illustrate the utility of the algorithms

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

downdating of szego polynomials and data fitting applications abstract many algorithms for polynomial least squares approximation of a real valuedfunction on a real interval determine polynomials that are orthogonal with respect to a suitable inner product defined on this interval analogously it is convenient to compute szego polynomials i e polynomials that are orthogonal with respect to an inner product on the unit circle when approximating a complex valued function on the unit circle in the least squares sense it may also be appropriate to determine szego polynomials in algorithms for least squares approximation of real valued periodic functions by trigonometric polynomials this paper is concerned with szego polynomials that are defined by a discrete inner product on the unit circle we presenta scheme for downdating the szego polynomials and given least squares approximant when a node is deleted from the inner product our scheme uses the q r algorithm for unitary upper hessenberg matrices we describe a data fitting application that illustrates how our scheme can be combined with the fast fourier transform algorithm when the given nodes are not equidistant application to sliding windows is discussed also

darboux transformations for cmv matrices abstract we develop a theory of darboux transformations for cmv matrices canonical representations of the unitary operators in perfect analogy with their self adjoint version the darboux transformations of jacobi matrices they are equivalent to laurent polynomial modifications of the underlying measures we address other questions which emphasize the similarities between darboux transformations for jacobi and cmv matrices like their almost isospectrality or the relation that they establish between the corresponding orthogonal polynomials showing also that both transformations are connected by the szeg mapping nevertheless we uncover some features of the darboux transformations for cmv matrices which are in striking contrast with those of the jacobi case in particular when applied to cmv matrices the matrix realization of the inverse darboux transformations what we call darboux transformations with parameters leads to spurious solutions whose interpretation deserves future research such spurious solutions are neither unitary nor band matrices so darboux transformations for cmv matrices are much more subject to the subtleties of the algebra of infinite matrices than their jacobi counterparts a key role in our theory is played by the cholesky factorizations of infinite matrices actually the darboux transformations introduced in this paper are based on the cholesky factorizations of degree one hermitian laurent polynomials evaluated on cmv matrices these transformations are also generalized to higher degree laurent polynomials as well as to the extension of cmv matrices to quasi definite functionals what we call quasi cmv matrices furthermore we show that this cmv version of darboux transformations plays a role in integrable systems like the schur flows or the ablowitz ladik model which parallels that of darboux for jacobi matrices in the toda lattice

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

a quasiseparable approach to five diagonal cmv and fiedler matrices abstract recent work in the characterization of structured matrices in terms of characteristic polynomials of principal submatrices is furthered in this paper some classical classes of matrices with quasiseparable structure include tridiagonal related to real orthogonal polynomials and banded matrices unitary hessenberg matrices related to szego polynomials and semiseparable matrices as well as others hence working with the class of quasiseparable matrices provides new results which generalize and unify classical results previous work has focused on characterizing h 1 quasiseparable matrices matrices with order one quasiseparable structure that are also upper hessenberg in this paper the authors introduce the concept of a twist transformation and use such transformations to explain the relationship between h 1 quasiseparable matrices and the subclass of 1 1 quasiseparable matrices without the upper hessenberg restriction which are related to the same systems of polynomials these results generalize the discoveries of cantero fiedler kimura moral and velazquez of five diagonal matrices related to horner and szego polynomials in the context of quasiseparable matrices

green s matrices abstract our goal is to identify and understand matrices a that share essential properties of the unitary hessenberg matrices m that are fundamental for szego s orthogonal polynomials those properties include i recurrence relations connect characteristic polynomials rk x of principal minors of a ii a is determined by generators parameters generalizing reflection coefficients of unitary hessenberg theory iii polynomials rk x correspond not only to a but also to a certain cmv like five diagonal matrix iv the five diagonal matrix factors into a product bc of block diagonal matrices with 2 2 blocks v submatrices above and below the main diagonal of a have rank 1 vi a is a multiplication operator in the appropriate basis of laurent polynomials vii eigenvectors of a can be expressed in terms of those polynomials conditions v connects our analysis to the study of quasi separable matrices but the factorization requirement iv narrows it to the subclass of green s matrices that share properties i vii the key tool is twist transformations that provide 2n matrices all sharing characteristic polynomials of principal minors with a one such twist transformation connects unitary hessenberg to cmv another twist transformation explains findings of fiedler who noticed that companion matrices give examples outside the unitary hessenberg framework we mention briefly the further example of a daubechies wavelet matrix infinite matrices are included

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

inverse scattering for discrete transmission line models this paper presents several methods for the identification of the impedance and reflection coefficient profile of a nonuniform discrete transmission line from its response to a given forcing function this problem is the prototype for a wealth of one dimensional inverse scattering problems arising in various fields a unified straightforward approach is presented providing all known inversion procedures and some novel ones the derivations readily follow from causality of signal propagation on the transmission line it is shown that a direct exploitation of the signal propagation model leads to recursive computationally efficient and reliable inversion algorithms in particular the relationships between layer peeling schur type difference equation and layer adjoining levinson type integral equation algorithms are clearly displayed

matrix valued szegő polynomials and quantum random walks we consider quantum random walks qrw on the integers a subject that has been considered in the last few years in the framework of quantum computation we show how the theory of cmv matrices gives a natural tool to study these processes and to give results that are analogous to those that karlin and mcgregor developed to study classical birth and death processes using orthogonal polynomials on the real line in perfect analogy with the classical case the study of qrws on the set of nonnegative integers can be handled using scalar valued laurent polynomials and a scalar valued measure on the circle in the case of classical or quantum random walks on the integers one needs to allow for matrix valued versions of these notions we show how our tools yield results in the well known case of the hadamard walk but we go beyond this translation invariant model to analyze examples that are hard to analyze using other methods more precisely we consider qrws on the set of nonnegative integers the analysis of these cases leads to phenomena that are absent in the case of qrws on the integers even if one restricts oneself to a constant coin this is illustrated here by studying recurrence properties of the walk but the same method can be used for other purposes the presentation here aims at being self contained but we refrain from trying to give an introduction to quantum random walks a subject well surveyed in the literature we quote for two excellent reviews see 1 9 see also the recent notes 20 2009 wiley periodicals inc

a generalization of schur functions applications to nevanlinna functions orthogonal polynomials random walks and unitary and open quantum walks abstract recent work on return properties of quantum walks the quantum analogue of random walks has identified their generating functions of first returns as schur functions this is connected with a representation of schur functions in terms of the operators governing the evolution of quantum walks i e the unitary operators on hilbert spaces in this paper we propose a generalization of schur functions by extending the above operator representation to arbitrary closed operators on banach spaces such generalized schur functions meet the formal structure of first return generating functions thus we call them fr functions we derive some general properties of fr functions among them a simple relation with an operator version of stieltjes functions which generalizes the renewal equation already known for random and quantum walks we also prove that fr functions satisfy splitting properties which extend useful factorizations of schur functions when specialized to self adjoint operators on hilbert spaces we show that fr functions become nevanlinna functions this allows us to obtain properties of nevanlinna functions which as far as we know seem to be new the fr function structure leads to a new operator representation of nevanlinna functions in terms of self adjoint operators whose spectral measures provide also new integral representations of such functions this allows us to characterize each nevanlinna function by a measure on the real line which we refer to as the measure of the nevanlinna function in contrast to standard operator and integral representations of nevanlinna functions these new ones are exact analogues of those already known for schur functions the above results are also the source of a very simple schur algorithm for nevanlinna functions based on interpolations at points on the real line which we refer to as the schur algorithm on the real line the paper is completed with several applications of fr functions to orthogonal polynomials and random and quantum walks which illustrate their wide interest an analogue for orthogonal polynomials on the real line of the khrushchev formula for orthogonal polynomials on the unit circle and the use of fr functions to study recurrence in random walks quantum walks and open quantum walks these applications provide numerous explicit examples of fr functions clarifying the meaning of these functions as first return generating functions and their splittings which become recurrence splitting rules they also show that these new tools despite being extensions of very classical ones play an important role in the study of physical problems of a highly topical nature

quantum recurrence of a subspace and operator valued schur functions a notion of monitored recurrence for discrete time quantum processes was recently introduced in grunbaum et al commun math phys 2 320 543 569 2013 taking the initial state as an absorbing one we extend this notion of monitored recurrence to absorbing subspaces of arbitrary finite dimension the generating function approach leads to a connection with the well known theory of operator valued schur functions this is the cornerstone of a spectral characterization of subspace recurrence that generalizes some of the main results in grunbaum et al commun math phys 2 320 543 569 2013 the spectral decomposition of the unitary step operator driving the evolution yields a spectral measure which we project onto the subspace to obtain a new spectral measure that is purely singular iff the subspace is recurrent and consists of a pure point spectrum with a finite number of masses precisely when all states in the subspace have a finite expected return time this notion of subspace recurrence also links the concept of expected return time to an aharonov anandan phase that in contrast to the case of state recurrence can be non integer even more surprising is the fact that averaging such geometrical phases over the absorbing subspace yields an integer with a topological meaning so that the averaged expected return time is always a rational number moreover state recurrence can occasionally give higher return probabilities than subspace recurrence a fact that reveals once more the counter intuitive behavior of quantum systems all these phenomena are illustrated with explicit examples including as a natural application the analysis of site recurrence for coined walks

matrix methods for quadrature formulas on the unit circle a survey in this paper we give a survey of some results concerning the computation of quadrature formulas on the unit circle like nodes and weights of gauss quadrature formulas for the estimation of integrals with respect to measures on the real line can be computed from the eigenvalue decomposition of the jacobi matrix szeg quadrature formulas for the approximation of integrals with respect to measures on the unit circle can be obtained from certain unitary five diagonal or unitary hessenberg matrices that characterize the recurrence for an orthogonal laurent polynomial basis these quadratures are exact in a maximal space of laurent polynomials orthogonal polynomials are a particular case of orthogonal rational functions with prescribed poles more general szeg quadrature formulas can be obtained that are exact in certain spaces of rational functions in this context the nodes and the weights of these rules are computed from the eigenvalue decomposition of an operator mobius transform of the same five diagonal or hessenberg matrices

unitary orthogonalization process all methods for solving least squares problems involve orthogonalization in one way or another certain fundamental estimation and prediction problems of signal processing and time series analysis can be formulated as least squares problems in these problems the sequence that is to be orthogonalized is generated by an underlying unitary operator a prime example of an efficient orthogonalization procedure for this class of problems is gragg s isometric arnoldi process which is the abstract encapsulation of a number of concrete algorithms in this paper we discuss a two sided orthogonalization process that is equivalent to gragg s process but has certain conceptual strengths that warrant its introduction the connections with classical algorithms of signal processing are discussed

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

an addition formula for the jacobian theta function and its applications in this paper we prove an addition formula for the jacobian theta function using the theory of elliptic functions it turns out to be a fundamental identity in the theory of theta functions and elliptic function and unifies many important results about theta functions and elliptic functions from this identity we can derive the ramanujan cubic theta function identity winquist s identity a theta function identities with five parameters and many other interesting theta function identities and all of which are as striking as winquist s identity this identity allows us to give a new proof of the addition formula for the weierstrass sigma function a new identity about the ramanujan cubic elliptic function is given the proofs are self contained and elementary

convergence analysis of three classes of split complex gradient algorithms for complex valued recurrent neural networks this letter presents a unified convergence analysis of the split complex nonlinear gradient descent scngd learning algorithms for complex valued recurrent neural networks covering three classes of scngd algorithms standard scngd normalized scngd and adaptive normalized scngd we prove that if the activation functions are of split complex type and some conditions are satisfied the error function is monotonically decreasing during the training iteration process and the gradients of the error function with respect to the real and imaginary parts of the weights converge to zero a strong convergence result is also obtained under the assumption that the error function has only a finite number of stationary points the simulation results are given to support the theoretical analysis

stochastic gradient adaptive complex valued nonlinear neural adaptive filters with a gradient adaptive step size a class of variable step size learning algorithms for complex valued nonlinear adaptive finite impulse response fir filters is proposed to achieve this first a general complex valued nonlinear gradient descent cngd algorithm with a fully complex nonlinear activation function is derived to improve the convergence and robustness of cngd we further introduce a gradient adaptive step size to give a class of variable step size cngd vscngd algorithms the analysis and simulations show the proposed class of algorithms exhibiting fast convergence and being able to track nonlinear and nonstationary complex valued signals to support the derivation an analysis of stability and computational complexity of the proposed algorithms is provided simulations on colored nonlinear and real world complex valued signals support the analysis

eigenvalue computation for unitary rank structured matrices in this paper we describe how to compute the eigenvalues of a unitary rank structured matrix in two steps first we perform a reduction of the given matrix into hessenberg form next we compute the eigenvalues of this resulting hessenberg matrix via an implicit qr algorithm along the way we explain how the knowledge of a certain shift correction term to the structure can be used to speed up the qr algorithm for unitary hessenberg matrices and how this observation was implicitly used in a paper due to william b gragg we also treat an analogue of this observation in the hermitian tridiagonal case

cmv matrices in random matrix theory and integrable systems a survey we present a survey of recent results concerning a remarkable class of unitary matrices the cmv matrices we are particularly interested in the role they play in the theory of random matrices and integrable systems throughout the paper we also emphasize the analogies and connections to jacobi matrices

toda flows with infinitely many variables abstract the toda flow and related flows extend naturally to operators in hilbert space and the purpose of this paper is to describe these flows and to analyse some of their special properties

a stochastic gradient adaptive filter with gradient adaptive step size the step size of this adaptive filter is changed according to a gradient descent algorithm designed to reduce the squared estimation error during each iteration an approximate analysis of the performance of the adaptive filter when its inputs are zero mean white and gaussian noise and the set of optimal coefficients are time varying according to a random walk model is presented the algorithm has very good convergence speed and low steady state misadjustment the tracking performance of these algorithms in nonstationary environments is relatively insensitive to the choice of the parameters of the adaptive filter and is very close to the best possible performance of the least mean square lms algorithm for a large range of values of the step size of the adaptation algorithm several simulation examples demonstrating the good properties of the adaptive filters as well as verifying the analytical results are also presented

perturbation and interlace theorems for the unitary eigenvalue problem abstract two aspects of the perturbation problem for the eigenvalues of a unitary matrix u are treated firstly analogues of the hoffman wielandt theorem and a weyl type theorem proved by bhatia and davis are derived which are based on a different measure of the distance of spectra using a suitable parametrization of the unit circle by an angle the new results are called tangent theorems in contrast to the first mentioned well known results which are sine theorems moreover we illuminate the unknown minimizing permutations in the above weyl type theorems with respect to their angles the eigenvalues of u and the perturbed matrix are naturally ordered on the unit circle counterclockwise after a point is cut on the unit circle we prove a well known open conjecture there exists a cutting point such that the weyl type theorems both sine and tangent are true when the ordered eigenvalues of u and are paired with each other secondly the cauchy interlacing theorem for hermitian matrices is generalized it is shown that certain modified principal submatrices of u called the modified kth leading principal submatrices have the property that their eigenvalues interlace those of u finally we discuss block reflectors appearing in the description of the modified principal submatrices and generalize a result of schreiber and parlett

approximation by fully complex multilayer perceptrons we investigate the approximation ability of a multilayer perceptron mlp network when it is extended to the complex domain the main challenge for processing complex data with neural networks has been the lack of bounded and analytic complex nonlinear activation functions in the complex domain as stated by liouville s theorem to avoid the conflict between the boundedness and the analyticity of a nonlinear complex function in the complex domain a number of ad hoc mlps that include using two real valued mlps one processing the real part and the other processing the imaginary part have been traditionally employed however since nonanalytic functions do not meet the cauchy riemann conditions they render themselves into degenerative backpropagation algorithms that compromise the efficiency of nonlinear approximation and learning in the complex vector field a number of elementary transcendental functions etfs derivable from the entire exponential function ez that are analytic are defined as fully complex activation functions and are shown to provide a parsimonious structure for processing data in the complex domain and address most of the shortcomings of the traditional approach the introduction of etfs however raises a new question in the approximation capability of this fully complex mlp in this letter three proofs of the approximation capability of the fully complex mlp are provided based on the characteristics of singularity among etfs first the fully complex mlps with continuous etfs over a compact set in the complex vector field are shown to be the universal approximator of any continuous complex mappings the complex universal approximation theorem extends to bounded measurable etfs possessing a removable singularity finally it is shown that the output of complex mlps using etfs with isolated and essential singularities uniformly converges to any nonlinear mapping in the deleted annulus of singularity nearest to the origin

a numerical comparison of methods for solving secular equations numerical comparisons are carried out by the most prominent methods both second and higher order for solving secular equations these equations arise in several matrix problems and numerical linear algebra applications in addition to the comparisons we propose a new starting point our results show a clear advantage to higher order methods

a fast algorithm for gaussian type quadrature formulae with mixed boundary conditions and some lumped mass spectral approximations after studying gaussian type quadrature formulae with mixed boundary conditions we suggest a fast algorithm for computing their nodes and weights it is shown that the latter are computed in the same manner as in the theory of the classical gauss quadrature formulae in fact all nodes and weights are again computed as eigenvalues and eigenvectors of a real symmetric tridiagonal matrix hence we can adapt existing procedures for generating such quadrature formulae comparative results with various methods now in use are given in the second part of this paper new algorithms for spectral approximations for second order elliptic problems are derived the key to the efficiency of our algorithms is to find an appropriate spectral approximation by using the most accurate quadrature formula which takes the boundary conditions into account in such a way that the resulting discrete system has a diagonal mass matrix hence our algorithms can be used to introduce explicit resolutions for the time dependent problems this is the so called lumped mass method the performance of the approach is illustrated with several numerical examples in one and two space dimensions

analysis and implementation of variable step size adaptive algorithms stochastic gradient adaptive filtering algorithms using variable step sizes are investigated the variable step size algorithm improves the convergence rate while sacrificing little in steady state error expressions describing the convergence of the mean and mean squared values of the coefficients are developed and used to calculate the mean square error evolution the initial convergence rate and the steady state error are also investigated the performance of the algorithm is studied when a power of two quantizer algorithm is used and finite word length effects are considered the analytical results are verified with simulations encompassing variable applications two cmos implementations of the variable step size power of two quantizer algorithm are presented to demonstrate that the performance gains are attainable with only a modest increase in circuit complexity

qr like algorithms for eigenvalue problems in the year 2000 the dominant method for solving matrix eigenvalue problems is still the qr algorithm this paper discusses the family of gr algorithms with emphasis on the qr algorithm included are historical remarks an outline of what gr algorithms are and why they work and descriptions of the latest highly parallelizable versions of the qr algorithm now that we know how to parallelize it the qr algorithm seems likely to retain its dominance for many years to come

reprint of a quasiseparable approach to five diagonal cmv and fiedler matrices abstract recent work in the characterization of structured matrices in terms of characteristic polynomials of principal submatrices is furthered in this paper some classical classes of matrices with quasiseparable structure include tridiagonal related to real orthogonal polynomials and banded matrices unitary hessenberg matrices related to szego polynomials and semiseparable matrices as well as others hence working with the class of quasiseparable matrices provides new results which generalize and unify classical results previous work has focused on characterizing h 1 quasiseparable matrices matrices with order one quasiseparable structure that are also upper hessenberg in this paper the authors introduce the concept of a twist transformation and use such transformations to explain the relationship between h 1 quasiseparable matrices and the subclass of 1 1 quasiseparable matrices without the upper hessenberg restriction which are related to the same systems of polynomials these results generalize the discoveries of cantero fiedler kimura moral and velazquez of five diagonal matrices related to horner and szego polynomials in the context of quasiseparable matrices

on monotonicity of zeros of paraorthogonal polynomials on the unit circle abstract the purpose of this note is to establish necessary and sufficient conditions for the monotonicity with respect to a real parameter of zeros of paraorthogonal polynomials on the unit circle it is also provided tractable sufficient conditions and an application example

a cubic counterpart of jacobi s identity and the agm we produce exact cubic analogues of jacobi s celebrated theta function identity and of the arithmetic geometric mean iteration of gauss and legendre the iteration in question is a n 1 a n 2b n 3 and b n 1 formula cannot be replicated the limit of this iteration is identified in terms of the hypergeometric function f 1 3 2 3 1 which supports a particularly simple cubic transformation

breakdowns in the implementation of the lanczos method for solving linear systems abstract the lanczos method for solving systems of linear equations is based on formal orthogonal polynomials its implementation is realized via some recurrence relationships between polynomials of a family of orthogonal polynomials or between those of two adjacent families of orthogonal polynomials a division by zero can occur in such recurrence relations thus causing a breakdown in the algorithm which has to be stopped in this paper two types of breakdowns are discussed the true breakdowns which are due to the nonexistence of some polynomials and the ghost breakdowns which are due to the recurrence relationship used among all the recurrence relationships which can be used and all the algorithms for implementing the lanczos method which came out from them the only reliable algorithm is lanczos orthodir which can only suffer from true breakdowns it is shown how to avoid true breakdowns in this algorithm other algorithms are also discussed and the case of near breakdown is treated the same treatment applies to other methods related to lanczos

on the theory of elliptic functions based on _ 2 f_ 1 frac 1 3 frac 2 3 frac 1 2 z based on properties of the hypergeometric series 2 f 1 1 3 2 3 1 2 z we develop a theory of elliptic functions that shares many striking similarities with the classical jacobian elliptic functions

a taxonomy for conjugate gradient methods the conjugate gradient method of hestenes and stiefel is an effective method for solving large sparse hermitian positive definite hpd systems of linear equations ax b generalizations to non hpd matrices have long been sought the recent theory of faber and manteuffel gives necessary and sufficient conditions for the existence of a cg method this paper uses these conditions to develop and organize such methods it is shown that any cg method for ax b is characterized by an hpd inner product matrix b and a left preconditioning matrix c at each step the method minimizes the b norm of the error over a krylov subspace this characterization is then used to classify known and new methods finally it is shown how eigenvalue estimates may be obtained from the iteration parameters generalizing the well known connection between cg and lanczos such estimates allow implementation of a stopping criterion based more nearly on the true error

spectral methods for orthogonal rational functions abstract we present an operator theoretic approach to orthogonal rational functions based on the identification of a suitable matrix representation of the multiplication operator associated with the corresponding orthogonality measure two alternatives are discussed leading to representations which are linear fractional transformations with matrix coefficients acting on infinite hessenberg or five diagonal unitary matrices this approach permits us to recover the orthogonality measure throughout the spectral analysis of an infinite matrix depending uniquely on the poles and the parameters of the recurrence relation for the orthogonal rational functions besides the zeros of the orthogonal and para orthogonal rational functions are identified as the eigenvalues of matrix linear fractional transformations of finite hessenberg or five diagonal matrices as an application we use operator perturbation theory results to obtain new relations between the support of the orthogonality measure and the location of the poles and parameters of the recurrence relation for the orthogonal rational functions

ordinary differential equations and the symmetric eigenvalue problem in this paper the authors develop a general framework for calculating the eigenvalues of a symmetric matrix using ordinary differential equations new algorithms are suggested and old algorithms including qr are interpreted

the qr algorithm revisited the qr algorithm is still one of the most important methods for computing eigenvalues and eigenvectors of matrices most discussions of the qr algorithm begin with a very basic version and move by steps toward the versions of the algorithm that are actually used this paper outlines a pedagogical path that leads directly to the implicit multishift qr algorithms that are used in practice bypassing the basic qr algorithm completely

algorithms for the geronimus transformation for orthogonal polynomials on the unit circle abstract let l be a positive definite bilinear functional on the unit circle defined on p n the space of polynomials of degree at most n then its geronimus transformation l is defined by l p q l z p z z q z for all p q p n c given l there are infinitely many such l which can be described by a complex free parameter the hessenberg matrix that appears in the recurrence relations for orthogonal polynomials on the unit circle is unitary and can be factorized using its associated schur parameters recent results show that the unitary hessenberg matrices associated with l and l respectively are related by a qr step where all the matrices involved are of order n 1 for the analogue on the real line of this so called spectral transformation the tridiagonal jacobi matrices associated with the respective functionals are related by an lr step in this paper we derive algorithms that compute the new schur parameters after applying a geronimus transformation we present two forward algorithms and one backward algorithm the qr step between unitary hessenberg matrices plays a central role in the derivation of each of the algorithms where the main idea is to do the inverse of a qr step making use of the special structure of unitary hessenberg matrices all the algorithms are efficient and need only o n flops we present several numerical experiments to analyse the accuracy and to explain the behaviour of the algorithms

quadrature formulas on the unit circle with prescribed nodes and maximal domain of validity in this paper we investigate the szego radau and szego lobatto quadrature formulas on the unit circle these are n m point formulas for which m nodes are fixed in advance with m 1 and m 2 respectively and which have a maximal domain of validity in the space of laurent polynomials this means that the free parameters free nodes and positive weights are chosen such that the quadrature formula is exact for all powers z j email protected email protected p with p p n m as large as possible

inverse unitary eigenproblems and related orthogonal functions this paper explores the relationship between certain inverse unitary eigenvalue problems and orthogonal functions in particular the inverse eigenvalue problems for unitary hessenberg matrices and for schur parameter pencils are considered the szego recursion is known to be identical to the arnoldi process and can be seen as an algorithm for solving an inverse unitary hessenberg eigenvalue problem reformulation of this inverse unitary hessenberg eigenvalue problem yields an inverse eigenvalue problem for schur parameter pencils it is shown that solving this inverse eigenvalue problem is equivalent to computing laurent polynomials orthogonal on the unit circle efficient and reliable algorithms for solving the inverse unitary eigenvalue problems are given which require only o mn arithmetic operations as compared with o mn 2 operations needed for algorithms that ignore the structure of the problem

convergence of algorithms of decomposition type for the eigenvalue problem we develop the theory of convergence of a generic gr algorithm for the matrix eigenvalue problem that includes the qr lr sr and other algorithms as special cases our formulation allows for shifts of origin and multiple gr steps the convergence theory is based on the idea that the gr algorithm performs nested subspace iteration with a change of coordinate system at each step thus the convergence of the gr algorithm depends on the convergence of certain sequences of subspaces it also depends on the quality of the coordinate transformation matrices as measured by their condition numbers we show that with a certain obvious shifting strategy the gr algorithm typically has a quadratic asymptotic convergence rate for matrices possessing certain special types of structure cubic convergence can be achieved