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 unifying convergence analysis of second order methods for secular equations existing numerical methods of second order are considered for a so called secular equation we give a brief description of the most important of these methods and show that all of them can be interpreted as improvements of newton s method for an equivalent problem for which newton s method exhibits convergence from any point in a given interval this interpretation unifies the convergence analysis of these methods provides convergence proofs where they were lacking and furnishes ways to construct improved methods in addition we show that some of these methods are in fact equivalent a second secular equation is also briefly considered

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

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

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

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

a very singular solution of the porous media equation with absorption turns out to play an important role in the description of both the short and the long time behaviour of positive solutions of eq 1 1 for the short time behaviour it provides together with the one parameter family of singular solutions i of problem i a complete description of the possible types of singular behaviour at the point 0 o 121 for the long time behaviour of solutions u x t of the cauchy problem for 1 1 1 2 it provides the asymptotic form when the initial function decays sufficiently fast as 1x1 co t p u x t w x t 0 ast co uniformly in r 7 81 in this paper we shall show that if

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

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

travelling waves and finite propagation in a reaction diffusion equation abstract we study the existence of travelling wave solutions and the property of finite propagation for the reaction diffusion equation ut um xx un x t r 0 with m 1 0 n r and u u x t 0 we show that travelling waves exist globally only if m n 2 and only for velocities c c 2 m in the study of propagation we must take into account that solutions of the cauchy problem can be nonunique for n c 0 for m n 2 maximal solutions propagate instantly to the whole space if n

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

eigenvalues of a symmetric tridiagonal matrix a divide and conquer approach the symmetric tridiagonal eigenproblem has been the topic of some recent work many methods have been advanced for the computation of the eigenvalues of such a matrix in this paper we present a divide and conquer approach to the computation of the eigenvalues of a symmetric tridiagonal matrix via the evaluation of the characteristic polynomial the problem of evaluation of the characteristic polynomial is partitioned into smaller parts which are solved and these solutions are then combined to form the solution to the original problem we give the update equations for the characteristic polynomial and certain auxiliary polynomials used in the computation furthermore this set of recursions can be implemented on a regulartree structure if the concurrency exhibited by this algorithm is exploited it can be shown that thetime for computation of all the eigenvalues becomeso nlogn instead ofo n2 as is the case for the approach where the order is increased by only one at every step we address the numerical problems associated with the use of the characteristic polynomial and present a numerically stable technique for the eigenvalue computation

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

analysis of third order methods for secular equations third order numerical methods are analyzed for secular equations these equations arise in several matrix problems and numerical linear algebra applications a closer look at an existing method shows that it can be considered as a classical method for an equivalent problem this not only leads to other third order methods it also provides the means for a unifying convergence analysis of these methods and for their comparisons finally we consider approximated versions of the aforementioned methods

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 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

maximal functions on classical lorentz spaces and hardy s inequality with weights for nonincreasing functions a characterization is given of a class of classical lorentz spaces on which the hardy littlewood maximal operator is bounded this is done by determining the weights for which hardy s inequality holds for nonincreasing functions an alternate characterization valid for nondecreasing weights is also derived

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

self similar solutions satisfying or not the equation of the interface abstract we study the existence of self similar solutions for the porous medium equation with reaction and convection u t u m 1 u x x u n 1 u x ku p in r 0 where m n 1 0 0 we are in particular interested in compactly supported self similar solutions satisfying some good equation at the interface the same equation appearing in the pure diffusion case we prove that there exist such self similar solutions only if k 1 4n an infinity of solutions with bad behaviour at least at one interface also exist there exist no self similar solutions with support arbitrarily small we complete the study by considering the case 0

hardy and rellich inequalities identities and sharp remainders on homogeneous groups abstract we give sharp remainder terms of l p and weighted hardy and rellich inequalities on one of most general subclasses of nilpotent lie groups namely the class of homogeneous groups as consequences we obtain analogues of the generalised classical hardy and rellich inequalities and the uncertainty principle on homogeneous groups we also prove higher order inequalities of hardy rellich type all with sharp constants a number of identities are derived including weighted and higher order types

global travelling waves in reaction convection diffusion equations we study the existence and properties of solutions in travelling wave form u x t x st defined for every z x st r for the reaction convection diffusion equationut a um xx b un x kupforx r t 0 with a m n 0 b k p r in the reaction case k 0 we prove that there exist travelling waves vanishing for z if and only if b 0 and min 1 n m p2 max 1 n p max 1 n moreover if m p 2n there exists a minimal velocity s a b k m n p 0 for which there are travelling waves only with s s while in the case m p 2n there are travelling waves only when 4amk b2n and for every velocity s 0 some properties of the function s are established all the waves are decreasing in their support and waves having bounded support from the right exist if and only if m min 1 n also the absorption case k 0 is treated where we find that for different values of the parameters there exists a unique travelling wave for every velocity s r but for some case where only negative velocities exist the cases b 0 or k 0 are well known in the literature

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

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

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

the characterization of reaction convection diffusion processes by travelling waves it has long been known that the heat equation displays infinite speed of propagation this is to say that if the initial data are nonnegative and have nonempty compact support the solution of an initial value problem is positive everywhere after any infinitesimal time however since the nineteen fifties it has also been known that certain nonlinear diffusion equations of degenerate parabolic type do not display this phenomenon for these equations the generalized solution of an initial value problem with compactly supported initial data will have bounded support with respect to the spatial variable at all times in this paper the necessary and sufficient criterion for finite speed of propagation for the general nonlinear reaction convection diffusion equation formula is determined the assumptions on the coefficientsa bandcare such that the classification unifies and generalizes previously known results the technique employed is comparision of an arbitrary solution of the equation with suitably constructed travelling wave solutions and subsolutions basically the central conclusion is that the equation exhibits finite speed of propagation if and only if it admits a travelling wave solution with bounded support concurrently the search for a travelling wave solution with bounded support can be reduced to the study of a singular nonlinear integral equation whose solution must satisfy a certain constraint

hardy inequalities on metric measure spaces in this note we give several characterizations of weights for two weight hardy inequalities to hold on general metric measure spaces possessing polar decompositions since there may be no differen

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

similarity solutions of the porous medium equation with sign changes with m 1 here u is a function of x and t for nonnegative u eq 1 1 arises in the theory of gas flow through a one dimensional porous medium and an extensive theory has been developed over the last decades see e g l 3 151 more recently 4 5 12 131 applied and pure mathematicians have also become interested in solutions of 1 1 with sign changes the most striking property of 1 1 is that solutions may have compact support whose free boundaries or interfaces move outwards with a finite possibly zero speed proportional to the one sided x derivative of 1 1 at the free boundary taken from the interior of the support this is usually called the interface condition the expression between brackets in l l i e 1 1 ux l m iu u which we shall think of as theflux because of the divergence form of l l is continuous with respect to x in zeros of u if it is nonzero we shall call this the sign change condition we observe that for u bounded away from zero and infinity eq 1 1 is uniformly parabolic and local smoothness follows from standard regularity theory for the development of the theory of 1 1 similarity solutions play an important role essentially these are solutions whose profiles remain the same as t varies in the nonnegative case they were studied extensively in 7 91 the most familiar of them are of course the burenblatt pattle solutions given by

a new method for computing the stable invariant subspace of a real hamiltonian matrix abstract a new backward stable structure preserving method of complexity o n3 is presented for computing the stable invariant subspace of a real hamiltonian matrix and the stabilizing solution of the continuous time algebraic riccati equation the new method is based on the relationship between the invariant subspaces of the hamiltonian matrix h and the extended matrix 0 h h 0 and makes use of the symplectic urv like decomposition that was recently introduced by the authors

sharp hardy adams inequalities for bi laplacian on hyperbolic space of dimension four abstract we establish sharp hardy adams inequalities on hyperbolic space b 4 of dimension four namely we will show that for any 0 there exists a constant c 0 such that b 4 e 32 2 u 2 1 32 2 u 2 d v 16 b 4 e 32 2 u 2 1 32 2 u 2 1 x 2 4 d x c for any u c 0 b 4 with b 4 h 9 4 h u u d v 1 as applications we obtain a sharpened adams inequality on hyperbolic space b 4 and an inequality which improves the classical adams inequality and the hardy inequality simultaneously the later inequality is in the spirit of the hardy trudinger moser inequality on a disk in dimension two given by wang and ye in 46 and on any convex planar domain by lu and yang in 33 the fourier analysis techniques on hyperbolic and symmetric spaces play an important role in our work

blow up and global existence for solutions to the porous medium equation with reaction and slowly decaying density abstract we study existence of global solutions and finite time blow up of solutions to the cauchy problem for the porous medium equation with a variable density x and a power like reaction term x u p with p 1 this is a mathematical model of a thermal evolution of a heated plasma see 29 the density decays slowly at infinity in the sense that x x q as x with q 0 2 we show that for large enough initial data solutions blow up in finite time for any p 1 on the other hand if the initial datum is small enough and p p for a suitable p depending on m n then global solutions exist in addition if p p for a suitable p p depending on m n then the solution blows up in finite time for any nontrivial initial datum we need the extra hypothesis that q 0 for 0 small enough when m p p observe that p p if x is a multiple of x q for x large enough such results are in agreement with those established in 48 where x 1 and are related to some results in 32 33 the case of fast decaying density at infinity i e q 2 is examined in 36

thermal waves in absorbing media abstract we discuss the existence of travelling wave solutions with interfaces for the nonlinear heat equation with absorption u t a u m xx bu n with a b 0 and m n r several situations occur depending on the relative strength of the diffusion and absorption terms reflected by their exponents m and n we characterize the existence of finite travelling waves in terms of m and n show their uniqueness up to translations in space and time and derive their velocity from the wave profile near the interface or front

regularity of interfaces in diffusion processes under the influence of strong absorption we present a method of analysis which allows us to establish the interface equation and to prove lipschitz continuity of interfaces and solutions which appear in a large class of nonlinear parabolic equations and conservation laws posed in one space dimension its main feature is intersection comparison with travelling waves the method is explained on the following study case we consider the cauchy problem for the diffusion absorption model u t left u m right xx u p quad u ge 0 in the range of parameters m 1 0 1 where the support of the solution expands with time and the motion is governed by darcy s law in the strong absorption range there might appear shrinking interfaces and the interface evolution obeys a different mechanism previous methods have failed to provide an adequate analysis of the interface motion and regularity in such a situation

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