quadratures and orthogonality associated with the cayley transform abstract in this paper we are dealing with the approximate calculation of weighted integrals over the whole real line the method is based in passing to the unit circle by means of the so called cayley transform z i x i x and then making use of a szeg or interpolatory type quadrature formula on the unit circle in order to obtain a gauss like quadrature rule on the real line some properties concerning orthogonality maximal domains of validity of the quadratures and connections with certain orthogonal rational functions are presented finally some numerical experiments are also carried out

a generalized unitary hessenberg matrix this paper describes the use of a generalized isometric arnoldi algorithm to reduce a unitary matrix via unitary similarity to a product of elementary reflectors and permutations the computation is analogous to the reduction of a unitary matrix to a unitary hessenberg matrix using the isometric arnoldi algorithm in the case in which a is a shift matrix the reduction provides a novel recurrence for the factor r in the qr factorization of a toeplitz like matrix

a general orthogonalization technique with applications to time series analysis and signal processing a new orthogonalization technique is presented for computing the qr factorization of a general n x p matrix of full rank p n 2 p the method is based on the use of projections to solve increasingly larger subproblems recursively and has an o np2 operation count for general matrices the technique is readily adaptable to solving linear least squares problems if the initial matrix has a circulant structure the algorithm simplifies significantly and gives the so called lattice algorithm for solving linear prediction problems from this point of view it is seen that the lattice algorithm is really an efficient way of solving specially structured least squares problems by orthogonalization as opposed to solving the normal equations by fast toeplitz algorithms

poncelet s theorem paraorthogonal polynomials and the numerical range of compressed multiplication operators abstract there has been considerable recent literature connecting poncelet s theorem to ellipses blaschke products and numerical ranges summarized for example in the recent book 16 we show how those results can be understood using ideas from the theory of orthogonal polynomials on the unit circle opuc and in turn can provide new insights to the theory of opuc

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

matrices of sinc methods this paper gives a brief review of sinc methods with emphasis on the matrices of sinc methods a novel procedure is presented based on sinc convolution for solving a poisson problem over a rectangular region although some of the work of gragg may already be applied to the solution of sinc matrix problems this paper also points to new directions of matrix research

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

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

an inexact krylov schur algorithm for the unitary eigenvalue problem abstract we present an efficient inexact implicitly restarted arnoldi algorithm to find a few eigenpairs of large unitary matrices the approximating krylov spaces are built using short term recurrences derived from gragg s isometric arnoldi process the implicit restarts are done by the krylov schur methodology of stewart all of the operations of the restart are done in terms of the schur parameters generated by the isometric arnoldi process numerical results confirm the effectiveness of the algorithm

a cs decomposition for orthogonal matrices with application to eigenvalue computation abstract we show that a schur form of a real orthogonal matrix can be obtained from a full cs decomposition based on this fact a cs decomposition based orthogonal eigenvalue method is developed we also describe an algorithm for orthogonal similarity transformation of an orthogonal matrix to a condensed product form and an algorithm for full cs decomposition the latter uses mixed shifted and zero shift iterations for high accuracy numerical examples are presented

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

convergence of the shifted qr algorithm for unitary hessenberg matrices this paper shows that for unitary hessenberg matrices the qr algorithm with an exceptional initial value modification of the wilkinson shift gives global convergence moreover the asymptotic rate of convergence is at least cubic higher than that which can be shown to be quadratic only for hermitian tridiagonal matrices under no further assumption a general mixed shift strategy with global convergence and cubic rates is also presented

a new integrable two component system with cubic nonlinearity in this paper a new integrable two component system mt m uxvx uv uvx uxv x nt n uxvx uv uvx uxv x where m u uxx and n v vxx is proposed our system is a generalized version of the integrable system mt m ux2 u2 x which was shown having cusped solution cuspon and w m shape soliton solutions by qiao j math phys 47 112701 2006 the new system is proven integrable not only in the sense of lax pair but also in the sense of geometry namely it describes pseudospherical surfaces accordingly infinitely many conservation laws are derived through recursion relations furthermore exact solutions such as cuspons and w m shape solitons are also obtained

polynomial zerofinders based on szego polynomials abstract the computation of zeros of polynomials is a classical computational problem this paper presents two new zerofinders that are based on the observation that after a suitable change of variable any polynomial can be considered a member of a family of szeg polynomials numerical experiments indicate that these methods generally give higher accuracy than computing the eigenvalues of the companion matrix associated with the polynomial

zero curvature formulations of dual hierarchies zero curvature formulations are given for the dual hierarchies of standard soliton equation hierarchies recently introduced by olver and rosenau including the physically interesting fuchssteiner fokas camassa holm hierarchy

quadrature formulas associated with rogers szegő polynomials this paper is concerned with the estimation of integrals of email protected functions with respect to the hermite weight function by passing to the unit circle of the complex plane and considering szego and interpolatory type quadrature formulas with respect now to the rogers szego weight function several numerical experiments are given along with estimations of the error

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

fast qr iterations for unitary plus low rank matrices some fast algorithms for computing the eigenvalues of a block companion matrix have recently appeared in the literature in this paper we generalize the approach to encompass unitary plus low rank matrices of the form a u xy h where u is a general unitary matrix three important cases for applications are u unitary diagonal u unitary block hessenberg and u unitary in block cmv form our extension exploits the properties of a larger matrix hat a obtained by a certain embedding of the hessenberg reduction of a suitable to maintain its structural properties we show that hat a can be factored as product of lower and upper unitary hessenberg matrices possibly perturbed in the first k rows and moreover such a data sparse representation is well suited for the design of fast eigensolvers based on the qr iteration the resulting eigenvalue algorithm is fast and backward stable

anti szego quadrature rules szego quadrature rules are discretization methods for approximating integrals of the form f e t du t this paper presents a new class of discretization methods which we refer to as anti szego quadrature rules anti szego rules can be used to estimate the error in szego quadrature rules under suitable conditions pairs of associated szego and anti szego quadrature rules provide upper and lower bounds for the value of the given integral the construction of anti szego quadrature rules is almost identical to that of szego quadrature rules in that pairs of associated szego and anti szego rules differ only in the choice of a parameter of unit modulus several examples of szego and anti szego quadrature rule pairs are presented

proprietes d intersection des marches aleatoires i convergence vers le temps local d intersection we study intersection properties of multi dimensional random walks letx andy be two independent random walks with values in d d 3 satisfying suitable moment assumptions and letin denote the number of common points to the paths ofx andy up to timen the sequence in suitably normalized is shown to converge in distribution towards the intersection local time of two independent brownian motions results are applied to the proof of a central limit theorem for the range of a two dimensional recurrent random walk thus answering a question raised by n c jain and w e pruitt

error bounds in the isometric arnoldi process abstract error bounds for the eigenvalues computed in the isometric arnoldi method are derived the arnoldi method applied to a unitary matrix u successively computes a sequence of unitary upper hessenberg matrices h k k 1 2 the eigenvalues of the h k s are increasingly better approximations to eigenvalues of u an upper bound for the distance of the spectrum of h k from the spectrum of u and an upper bound for the distance between each individual eigenvalue of h k and one of u are given between two eigenvalues of h k on the unit circle there is guaranteed to lie an eigenvalue of u the results are applied to a problem in signal processing

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

computations with quasiseparable polynomials and matrices in this paper we survey several recent results that highlight an interplay between a relatively new class of quasiseparable matrices and univariate polynomials quasiseparable matrices generalize two classical matrix classes jacobi tridiagonal matrices and unitary hessenberg matrices that are known to correspond to real orthogonal polynomials and szego polynomials respectively the latter two polynomial families arise in a wide variety of applications and their short recurrence relations are the basis for a number of efficient algorithms for historical reasons algorithm development is more advanced for real orthogonal polynomials recent variations of these algorithms tend to be valid only for the szego polynomials they are analogues and not generalizations of the original algorithms herein we survey several recent results for the superclass of quasiseparable matrices which includes both jacobi and unitary hessenberg matrices as special cases the interplay between quasiseparable matrices and their associated polynomial sequences which contain both real orthogonal and szego polynomials allows one to obtain true generalizations of several algorithms specifically we discuss the bjorck pereyra algorithm the traub algorithm certain new digital filter structures as well as qr and divide and conquer eigenvalue algorithms

continuation methods for the computation of zeros of szego polynomials abstract let j j 0 be a family of monic polynomials that are orthogonal with respect to an inner product on the unit circle the polynomials j arise in time series analysis and are often referred to as szego polynomials or levinson polynomials knowledge about the location of their zeros is important for frequency analysis of time series and for filter implementation we present fast algorithms for computing the zeros of the polynomials n based on the observation that the zeros are eigenvalues of a rank one modification of a unitary upper hessenberg matrix h n 0 of order n the algorithms first determine the spectrum of h n 0 by one of several available schemes that require only o n 2 arithmetic operations the eigenvalues of the rank one perturbation are then determined from the eigenvalues of h n 0 by a continuation method the computation of the n zeros of n in this manner typically requires only o n 2 arithmetic operations the algorithms have a structure that lends itself well to parallel computation the latter is of significance in real time applications

a covariance extension approach to identification of time series in this paper we consider a three step procedure for identification of time series based on covariance extension and model reduction and we present a complete analysis of its statistical convergence properties a partial covariance sequence is estimated from statistical data then a high order maximum entropy model is determined which is finally approximated by a lower order model by stochastically balanced model reduction such procedures have been studied before in various combinations but an overall convergence analysis comprising all three steps has been lacking supposing the data is generated from a true finite dimensional system which is minimum phase it is shown that the transfer function of the estimated system tends in h to the true transfer function as the data length tends to infinity if the covariance extension and the model reduction is done properly the proposed identification procedure and some variations of it are evaluated by simulations

moment modification multipeakons and nonisospectral generalizations abstract firstly a formal correspondence is established between the camassa holm ch equation and a two component modified ch or called sqq equation according to the method of moment modification for multipeakon formulae secondly based on the generalized nonisospectral ch equation in chang et al 2014 14 and the interlacing multipeakons of the two component modified ch equation in chang et al 2016 15 we propose a new generalized two component modified ch equation with two parameters which possesses a nonisospectral lax pair the proposed equation still admits multipeakon solutions of explicit and closed form sufficient conditions for global existence of solutions are given and two concrete examples with certain interesting phenomenon are presented last of all as a by product a generalized nonisospectral modified ch equation is deduced together with its lax pair

convergence of the unitary qr algorithm with a unimodular wilkinson shift in applying the qr algorithm to compute the eigenvalues of a unitary hessenberg matrix a projected wilkinson shift of unit modulus is proposed and proved to give global convergence with at least a quadratic asymptotic rate for the qr iteration experimental testing demonstrates that the unimodular shift produces more efficient numerical convergence

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

a bjorck pereyra type algorithm for szego vandermonde matrices based on properties of unitary hessenberg matrices abstract in this paper we carry over the bjorck pereyra algorithm for solving vandermonde linear systems to what we suggest to call szego vandermonde systems v x i e polynomial vandermonde systems where the corresponding polynomial system is the szego polynomials the properties of the corresponding unitary hessenberg matrix allow us to derive a fast o n 2 computational procedure we present numerical experiments that indicate that for ill conditioned matrices the new algorithm yields better forward accuracy than gaussian elimination

the arnoldi process short recursions and displacement ranks abstract we introduce two new methods for generating an orthonormal basis of a krylov subspace spanned by the columns of k n a r r ar a n 1 r with a an arbitrary matrix we show that both of the methods produce short recursions when a h p a where p is a polynomial of low degree this property of a also leads to the low displacement rank structure of the cross product matrix k n a r h k n a r

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

on the construction of szego polynomials abstract the chebyshev and stieltjes procedures are algorithms for computing recursion coefficients for polynomials that are orthogonal with respect to an inner product defined on part of the real axis the chebyshev procedure is an implementation of a map from moments to recursion coefficients of orthogonal polynomials the modified chebyshev procedure is an implementation of a map from modified moments to recursion coefficients the latter map is generally much better conditioned than the former one the conditioning of these maps has been studied by gautschi this paper is concerned with analogues of the chebyshev and stieltjes procedures when the inner product is defined on the unit circle polynomials orthogonal with respect to such an inner product are known as szeg polynomials and the analogue of the chebyshev procedure is known as schur s algorithm this algorithm implements a map from moments to recursion coefficients for szeg polynomials our analysis shows that this map generally is much better conditioned than the map implemented by the chebyshev procedure and suggests that schur s algorithm is as insensitive to errors in the data as the modified chebyshev procedure thus roughly moments associated with inner products defined on the unit circle correspond to modified moments associated with inner products defined on the real axis

sieved para orthogonal polynomials on the unit circle abstract sieved orthogonal polynomials on the unit circle were introduced independently by ismail and li 1992 15 and marcellan and sansigre 1991 19 we look at the para orthogonal polynomials chain sequences and quadrature formulas that follow from the kernel polynomials of sieved orthogonal polynomials on the unit circle

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

signal recovery by discrete approximation and a prony like method abstract we introduce an algorithm which combines ideas of prony s approach to recover signals from given samples with approximation methods we solve two overdetermined systems of linear equations with linear programming methods and calculate the zeros of a suitable prony like polynomial we get the bandwidth m the frequencies as well as the amplitudes and some other characteristics of the signal especially it is reconstructed if sufficient many at least 2 m samples are given if we have too few samples or if they are too much noised we get an approximation of the original noiseless signal even if we have sufficient many but possible erroneous samples the obtained signal interpolates at least m of them usually the low noised or noiseless ones and of course all samples if the signal is recovered the described method behaves well to moderate sampling errors and is resistant to outliers in the samples which can be detected filtered off or corrected during the calculations to improve the quality of the computed signal

szegő lobatto quadrature rules gauss type quadrature rules with one or two prescribed nodes are well known and are commonly referred to as gauss radau and gauss lobatto quadrature rules respectively efficient algorithms are available for their computation szego quadrature rules are analogs of gauss quadrature rules for the integration of periodic functions they integrate exactly trigonometric polynomials of as high degree as possible szego quadrature rules have a free parameter which can be used to prescribe one node this paper discusses an analog of gauss lobatto rules i e szego quadrature rules with two prescribed nodes we refer to these rules as szego lobatto rules their properties as well as numerical methods for their computation are discussed

new 2 1 dimensional nonisospectral toda lattice hierarchy we give a new 2 1 dimensional nonisospectral generalization of the toda lattice hierarchy reductions yield a variety of new integrable hierarchies along with their underlying linear problems including new 1 1 dimensional differential delay hierarchies nonisospectral and isospectral new ordinary differential delay hierarchies and new discrete painleve hierarchies we also show that a reduction in components yields our previously obtained 2 1 dimensional nonisospectral volterra lattice hierarchy

on toda lattices and orthogonal polynomials first we derive a simple connection between toda and langmuir lattices and give a characterization of toda lattices with the help of stieltjes functions then it is shown how to generate by orthogonal polynomials in an elementary way periodic and almost periodic toda lattices the particles of the toda lattice are not even restricted as usual to move on the real line they may also move in the complex plane with the help of this result for special cases explicit solutions are obtained in terms of elliptic functions

symplectic structures their backlund transformations and hereditary symmetries abstract it is shown that compatible symplectic structures lead in a natural way to hereditary symmetries we recall that a hereditary symmetry is an operator valued function which immediately yields a hierarchy of evolution equations each having infinitely many commuting symmetries all generated by this hereditary symmetry furthermore this hereditary symmetry usually describes completely the soliton structure and the conservation laws of these equations this result then provide us with a method for constructing hereditary symmetries and hence exactly solvable evolution equations in addition we show how symplectic structures transform under backlund transformations this leads to a method for generating a whole class of symplectic structures from a given one several examples and applications are given illustrating the above results also the connection of our results with those of gelfand and dikii and of magri is briefly pointed out

further optimized look ahead recurrences for adjacent rows in the pade table and toeplitz matrix factorizations abstract in a recent paper we introduced a new look ahead algorithm for recursively computing pade approximants this algorithm generates a subsequence of the pade approximants on two adjacent rows defined by fixed numerator degree of the pade table its two basic versions reduce to the classical levinson and schur algorithms if no look ahead is required in this paper we show that the computational overhead of the look ahead steps in the o n2 versions of the look ahead levinson and the look ahead schur type algorithm can be further reduced if the algorithms are used to solve toeplitz systems of equations tx b then the corresponding block ldu decompositions of t 1 or t respectively can be found with less computational effort than with any other look ahead algorithm available today

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

product eigenvalue problems many eigenvalue problems are most naturally viewed as product eigenvalue problems the eigenvalues of a matrix a are wanted but a is not given explicitly instead it is presented as a product of several factors a a k a k 1 cdots a 1 usually more accurate results are obtained by working with the factors rather than forming a explicitly for example if we want eigenvalues vectors of b t b it is better to work directly with b and not compute the product the intent of this paper is to demonstrate that the product eigenvalue problem is a powerful unifying concept diverse examples of eigenvalue problems are discussed and formulated as product eigenvalue problems for all but a couple of these examples it is shown that the standard algorithms for solving them are instances of a generic gr algorithm applied to a related cyclic matrix

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

trigonometric orthogonal systems and quadrature formulae quadrature rules with maximal even trigonometric degree of exactness are considered we give a brief historical survey on such quadrature rules special attention is given on an approach given by turetzkii a h turetzkii on quadrature formulae that are exact for trigonometric polynomials east j approx 11 3 2005 337 359 translation in english from uchenye zapiski vypusk 1 149 seria math theory of functions collection of papers izdatel stvo belgosuniversiteta imeni v i lenina minsk 1959 pp 31 54 the main part of the topic is orthogonal trigonometric systems on 0 email protected or on email protected p with respect to some weight functions w x we prove that the so called orthogonal trigonometric polynomials of semi integer degree satisfy a five term recurrence relation in particular we study some cases with symmetric weight functions also we present a numerical method for constructing the corresponding quadratures of gaussian type finally we give some numerical examples also we compare our method with other available methods

generalized averaged szegź quadrature rules szeg quadrature rules are commonly applied to integrate periodic functions on the unit circle in the complex plane however often it is difficult to determine the quadrature error recently spalevic introduced generalized averaged gauss quadrature rules for estimating the quadrature error obtained when applying gauss quadrature over an interval on the real axis we describe analogous quadrature rules for the unit circle that often yield higher accuracy than szeg rules using the same moment information and may be used to estimate the error in szeg quadrature rules

wave breaking and peakons for a modified camassa holm equation in this paper we investigate the formation of singularities and the existence of peaked traveling wave solutions for a modified camassa holm equation with cubic nonlinearity the equation is known to be integrable and is shown to admit a single peaked soliton and multi peakon solutions of a different character than those of the camassa holm equation singularities of the solutions can occur only in the form of wave breaking and a new wave breaking mechanism for solutions with certain initial profiles is described in detail

rational approximants associated with measures supported on the unit circle and the real line abstract as a continuation of the well known connection between the theory of orthogonal polynomials on the unit circle and the interval 1 1 in this paper properties concerning error and convergence of certain rational approximants associated with the measures d t and d d cos supported on 1 1 and the unit circle respectively are deduced numerical illustrations are also given

rational szego quadratures associated with chebyshev weight functions in this paper we characterize rational szeg o quadrature formulas associated with chebyshev weight functions by giving explicit expressions for the corresponding para orthogonal rational functions and weights in the quadratures as an application we give characterizations for szeg o quadrature formulas associated with rational modifications of chebyshev weight functions some numerical experiments are finally presented

a generalized isometric arnoldi algorithm abstract this paper describes a generalization of the isometric arnoldi algorithm and shows that it can be interpreted as a structured form of modified gram schmidt given an isometry a the algorithm efficiently orthogonalizes the columns of a sequence of matrices m j for j 0 with m 1 0 for which the columns of m j am j 1 are in a fixed finite dimensional subspace for each j 0 the dimension of the subspace is analogous to displacement rank in the generalized schur algorithm the algorithm is described in terms of projections and inner products this is in contrast to orthogonalization methods based on the generalized schur algorithm for which cholesky factorization is central to the computation numerical experiments suggest that relative to a generalized schur algorithm the new algorithm improves the numerical orthogonality of the computed orthonormal sequence

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

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

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

eigenvector computation for almost unitary hessenberg matrices and inversion of szegő vandermonde matrices via discrete transmission lines abstract in this paper we use a discrete transmission line model known to geophysicists as a layered earth model to derive several computationally efficient solutions for the following three problems i as is well known a hessenberg matrix capturing recurrence relations for szeg polynomials differs from unitary only by its last column hence the first problem is how to rapidly evaluate the eigenvectors of this almost unitary hessenberg matrix ii the second problem is to design a fast o n 2 algorithm for inversion of szeg vandermonde matrices generalizing the well known traub algorithm for inversion of the usual vandermonde matrices iii finally the third problem is to extend the well known horner rule to evaluate a polynomial represented in the basis of szeg polynomials as we shall see all three problems are closely related and their solutions can be computed by the same family of fast algorithms although all the results can be derived algebraically here we reveal a connection to system theory to deduce these algorithms via elementary operations on signal flow graphs for digital filter structures including the celebrated markel gray filter widely used in speech processing and certain other filter structures this choice not only clarifies the derivation and suggests a variety of possible computational schemes but it also makes an interesting connection to many other results related to szeg polynomials which have already been interpreted via signal flow graphs for generalized lattice filter structures including the formulas of the gohberg semencul type for inversion of toeplitz like matrices schur type and levinson type algorithms etc for example this connection allows us to show that moment matrices corresponding to what we called horner szeg polynomials though not toeplitz are quasi toeplitz i e they have a certain shift invariance property

a new integrable equation with cuspons and w m shape peaks solitons in this paper we propose a new completely integrable wave equation mt mx u2 ux2 2m2ux 0 m u uxx the equation is derived from the two dimensional euler equation and is proven to have lax pair and bi hamiltonian structures this equation possesses new cusp solitons cuspons instead of regular peakons ce x ct with speed c through investigating the equation we develop a new kind of soliton solutions w m shape peaks solitons there exist no smooth solitons for this integrable water wave equation