positive periodic solutions of singular systems with a parameter abstract the existence and multiplicity of positive periodic solutions for second order non autonomous singular dynamical systems are established with superlinearity or sublinearity assumptions at infinity for an appropriately chosen parameter our results provide a unified treatment for the problem and significantly improve several results in the literature the proof of our results is based on the krasnoselskii fixed point theorem in a cone

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

open quantum random walks ergodicity hitting times gambler s ruin and potential theory in this work we study certain aspects of open quantum random walks oqrws a class of quantum channels described by attal et al j stat phys 147 832 852 2012 as a first objective we consider processes which are nonhomogeneous in time i e at each time step a possibly distinct evolution kernel inspired by a spectral technique described by saloff coste and zuniga stoch proc appl 117 961 979 2007 we define a notion of ergodicity for finite nonhomogeneous quantum markov chains and describe a criterion for ergodicity of such objects in terms of singular values as a second objective and based on a quantum trajectory approach we study a notion of hitting time for oqrws and we see that many constructions are variations of well known classical probability results with the density matrix degree of freedom on each site giving rise to systems which are seen to be nonclassical in this way we are able to examine open quantum versions of the gambler s ruin birth and death chain and a basic theorem on potential theory

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

open quantum random walks on the half line the karlin mcgregor formula path counting and foster s theorem in this work we consider open quantum random walks on the non negative integers by considering orthogonal matrix polynomials we are able to describe transition probability expressions for classes of walks via a matrix version of the karlin mcgregor formula we focus on absorbing boundary conditions and for simpler classes of examples we consider path counting and the corresponding combinatorial tools a non commutative version of the gambler s ruin is studied by obtaining the probability of reaching a certain fortune and the mean time to reach a fortune or ruin in terms of generating functions in the case of the hadamard coin a counting technique for boundary restricted paths in a lattice is also presented we discuss an open quantum version of foster s theorem for the expected return time together with applications

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

kummer congruence for the bernoulli numbers of higher order the authors studied the properties of bernoulli numbers of higher order appl math comput in press bull aust math 65 2002 59 for q 1 we can also find their results proc jangjeon math soc 1 2000 97 arch math 76 2001 190 proc jangjeon math soc 1 2000 161 adv stud contemp math 2 2000 9 proc jangjeon math soc 2 2001 23 j math phys a 34 2001 l643 proc jangjeon math soc 2 2001 19 proc jangjeon math soc 2 2001 9 proc jangjeon math soc 3 2001 63 the authors suggested the question to inquire the proof of kummer congruence for bernoulli numbers of higher order appl math comput in press in this paper we give a proof of kummer type congruence for the bernoulli numbers of higher order which is an answer to a part of the question in appl math comput in press

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

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

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

multiple positive solutions to superlinear periodic boundary value problems with repulsive singular forces abstract the existence of multiple positive solutions to superlinear periodic boundary value problems with repulsive singular forces is discussed in this paper our nonlinearity may be singular in its dependent variable and our analysis relies on a nonlinear alternative of leray schauder type and on a fixed point theorem in cones

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

minimal representations of unitary operators and orthogonal polynomials on the unit circle abstract in this paper we prove that the simplest band representations of unitary operators on a hilbert space are five diagonal orthogonal polynomials on the unit circle play an essential role in the development of this result and also provide a parameterization of such five diagonal representations which shows specially simple and interesting decomposition and factorization properties as an application we get the reduction of the spectral problem of any unitary hessenberg matrix to the spectral problem of a five diagonal one two applications of these results to the study of orthogonal polynomials on the unit circle are presented the first one concerns krein s theorem the second one deals with the movement of mass points of the orthogonality measure under mono parametric perturbations of the schur parameters

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

positive periodic solutions of singular systems of first order ordinary differential equations the existence and multiplicity of positive periodic solutions for first non autonomous singular systems are established with superlinearity or sublinearity assumptions at infinity for an appropriately chosen parameter the proof of our results is based on the krasnoselskii fixed point theorem in a cone

shape fluctuations and random matrices we study a certain random growth model in two dimensions closely related to the one dimensional totally asymmetric exclusion process the results show that the shape fluctuations appropriately scaled converges in distribution to the tracy widom largest eigenvalue distribution for the gaussian unitary ensemble gue

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

on the distribution of the length of the longest increasing subsequence of random permutations let sn be the group of permutations of 1 2 n if 7r e sn we say that 7 i1 7f ik is an increasing subsequence in 7r if il i2 ik and 7r ii 7r i2 7r ik let 1n r be the length of the longest increasing subsequence for example if n 5 and 7r is the permutation 5 1 3 2 4 in one line notation thus 7r 1 5 7r 2 1 then the longest increasing subsequences are 1 2 4 and 1 3 4 and n 3 equip sn with uniform distribution

positive periodic solutions of first order singular systems abstract we establish a complete structure for the existence of positive periodic solutions of first order non autonomous singular system u i t a i t u i t b i t f i u t i 1 n the methods employed are fixed point index theorems leray schauder degree and lower and upper solutions method

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

spectral characteristics of the unitary critical almost mathieu operator we discuss spectral characteristics of a one dimensional quantum walk whose coins are distributed quasi periodically the unitary update rule of this quantum walk shares many spectral characteristics with the critical almost mathieu operator however it possesses a feature not present in the almost mathieu operator namely singularity of the associated cocycles this feature is however present in the so called extended harper s model we show that this operator has empty absolutely continuous spectrum and that the lyapunov exponent vanishes on the spectrum hence this model exhibits cantor spectrum of zero lebesgue measure for all irrational frequencies and arbitrary phase which in physics is known as hofstadter s butterfly in fact we will show something stronger namely that all spectral parameters in the spectrum are of critical type in the language of avila s global theory of analytic quasiperiodic cocycles we further prove that it has empty point spectrum for each irrational frequency and away from a frequency dependent set of phases having lebesgue measure zero the key ingredients in our proofs are an adaptation of avila s global theory to the present setting self duality via the fourier transform and a johnson type theorem for singular dynamically defined cmv matrices which characterizes their spectra as the set of spectral parameters at which the associated cocycles fail to admit a dominated splitting

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

a note on the cochrane sum and its hybrid mean value formula in this paper we use the properties of gauss sums primitive characters and the mean value theorems of dirichlet l functions to study the hybrid mean value of cochrane sums and general kloosterman sums and give two sharp asymptotic formulae

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

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

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

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

rotation number associated with difference equations satisfied by polynomials orthogonal on the unit circle abstract the techniques of topological dynamics and differential dynamical systems are used to study polynomials orthogonal with respect to a measure supported on the unit circle it is assumed that the reflection coefficients associated with these polynomials form a stationary stochastic ergodic process in particular the techniques mentioned above are used to prove a gap labelling result

positive periodic solutions of functional differential equations abstract we consider the existence multiplicity and nonexistence of positive periodic solutions for the periodic equation x t a t g x x t b t f x t t where a b c r 0 are periodic 0 a t dt 0 0 b t dt 0 f g c 0 0 and f u 0 for u 0 g x is bounded t is a continuous periodic function define f 0 lim u 0 f u u f lim u f u u i0 number of zeros in the set f 0 f and i number of infinities in the set f 0 f we show that the equation has i0 or i positive periodic solution s for sufficiently large or small 0 respectively

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

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

algebraic aspects of increasing subsequences we present a number of results relating partial cauchy littlewood sums integrals over the compact classical groups and increasing subsequences of permutations these include integral formulae for the distribution of the longest increasing subsequence of a random involution with constrained number of fixed points new formulae for partial cauchy littlewood sums as well as new proofs of old formulae relations of these expressions to orthogonal polynomials on the unit circle and explicit bases for invariant spaces of the classical groups together with appropriate generalizations of the straightening algorithm

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

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

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

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

bulk edge correspondence of one dimensional quantum walks we outline a theory of symmetry protected topological phases of one dimensional quantum walks we assume spectral gaps around the symmetry distinguished points 1 and 1 in which only discrete eigenvalues are allowed the phase classification by integer or binary indices extends the classification known for translation invariant systems in terms of their band structure however our theory requires no translation invariance whatsoever and the indices we define in this general setting are invariant under arbitrary symmetric local perturbations even those that cannot be continuously contracted to the identity more precisely we define two indices for every walk characterizing the behavior far to the right and far to the left respectively their sum is a lower bound on the number of eigenstates at 1 and 1 for a translation invariant system the indices add up to zero so one of them already characterizes the phase by joining two bulk phases with different indices we get a walk in which the right and left indices no longer cancel so the theory predicts bound states at 1 or 1 this is a rigorous statement of bulk edge correspondence the results also apply to the hamiltonian case with a single gap at zero

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

five diagonal matrices and zeros of orthogonal polynomials on the unit circle it is shown that monic orthogonal polynomials on the unit circle are the characteristic polynomials of certain five diagonal matrices depending on the schur parameters this result is achieved through the study of orthogonal laurent polynomials on the unit circle more precisely it is a consequence of the five term recurrence relation obtained for these orthogonal laurent polynomials and the one to one correspondence established between them and the orthogonal polynomials on the unit circle as an application some results relating the behaviour of the zeros of orthogonal polynomials and the location of schur parameters are obtained

mean value of higher order generalized bernoulli numbers gauss sums and general kloosterman sums the main purpose of this paper is using the properties of higher order gen eralized bernoulli numbers and the mean value theorems of dirichlet l function to study the mean value of gauss sums general kloosterman sums and higher order generalized bernoulli numbers and give two interesting asymptotic formulae for it

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

recurrence and polya number of quantum walks we analyze the recurrence probability polya number for d dimensional unbiased quantum walks a sufficient condition for a quantum walk to be recurrent is derived as a by product we find a simple criterion for localization of quantum walks in contrast with classical walks where the polya number is characteristic for the given dimension the recurrence probability of a quantum walk depends in general on the topology of the walk choice of the coin and the initial state this allows us to change the character of the quantum walk from recurrent to transient by altering the initial state

on a class of quantum channels open random walks and recurrence we study a particular class of trace preserving completely positive maps called pq channels for which classical and quantum evolutions are isolated in a certain sense by combining open quantum random walks with a notion of recurrence we are able to describe criteria for recurrence of the walk related to this class of channels positive recurrence for open walks is also discussed in this context

on the number of positive solutions of nonlinear systems abstract we prove that appropriate combinations of superlinearity and sublinearity of f u with respect to at zero and infinity guarantee the existence multiplicity and nonexistence of positive solutions to boundary value problems for the n dimensional system u h t f u 0 0 is defined by u u 1 u n where u u 1 u n and covers the two important cases u u and u u p 2 u p 1 our methods employ fixed point theorems in a cone

quantal phase factors accompanying adiabatic changes a quantal system in an eigenstate slowly transported round a circuit c by varying parameters r in its hamiltonian r will acquire a geometrical phase factor exp i c in addition to the familiar dynamical phase factor an explicit general formula for c is derived in terms of the spectrum and eigenstates of r over a surface spanning c if c lies near a degeneracy of c takes a simple form which includes as a special case the sign change of eigenfunctions of real symmetric matrices round a degeneracy as an illustration c is calculated for spinning particles in slowly changing magnetic fields although the sign reversal of spinors on rotation is a special case the effect is predicted to occur for bosons as well as fermions and a method for observing it is proposed it is shown that the aharonov bohm effect can be interpreted as a geometrical phase factor

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

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 higher order generalized bernoulli numbers in this paper we define higher order generalized bernoulli numbers in order to give the values of a series attached to dirichlet character at non positive integers and investigate the arithmetic properties of them in particular we obtain euler summation formula involving higher order generalized bernoulli numbers also in the p adic cyclotomic field we construct witt s type formula involving higher order generalized bernoulli numbers

half line schrodinger operators with no bound states acting in l2 0 c with the boundary condition r for convenience we require that the potential v be uniformly locally square integrable we write l l 2 for the banach space of such functions the free operators that is when v 0 can be diagonalized by the fourier transform this shows that they have spectra 2 2 and 0 co respectively and that in both cases the spectrum is purely absolutely continuous for a general discrete operator the mere fact that the spectrum is contained in 2 2 forces it to be purely absolutely continuous this is our first main result

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

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

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

inverse problem for polynomials orthogonal on the unit circle polynomials orthogonal on the unit circle with random recurrence coefficients and finite band spectrum are investigated it is shown that the coefficients are in fact quasi periodic the measures associated with these quasi periodic coefficients are exhibited and necessary and sufficient conditions relating quasi periodicity and spectral measures of this type are given analogs for polynomials orthogonal on subsets of the real line are also presented

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

periodic solutions of second order non autonomous singular dynamical systems abstract in this paper we establish two different existence results of positive periodic solutions for second order non autonomous singular dynamical systems the first one is based on a nonlinear alternative principle of leray schauder and the result is applicable to the case of a strong singularity as well as the case of a weak singularity the second one is based on schauder s fixed point theorem and the result sheds some new light on problems with weak singularities and proves that in some situations weak singularities may help create periodic solutions recent results in the literature are generalized and significantly improved

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