the riemann hilbert approach to strong asymptotics for orthogonal polynomials on 1 1 abstract we consider polynomials that are orthogonal on 1 1 with respect to a modified jacobi weight 1 x 1 x h x with 1 and h real analytic and strictly positive on 1 1 we obtain full asymptotic expansions for the monic and orthonormal polynomials outside the interval 1 1 for the recurrence coefficients and for the leading coefficients of the orthonormal polynomials we also deduce asymptotic behavior for the hankel determinants and for the monic orthogonal polynomials on the interval 1 1 for the asymptotic analysis we use the steepest descent technique for riemann hilbert problems developed by deift and zhou and applied to orthogonal polynomials on the real line by deift kriecherbauer mclaughlin venakides and zhou in the steepest descent method we will use the szeg function associated with the weight and for the local analysis around the endpoints 1 we use bessel functions of appropriate order whereas deift et al use airy functions

an lp analog to aak theory for p 2 we develop an lp analog to aak theory on the unit circle that interpolates continuously between the case p which classically solves for best uniform meromorphic approximation and the case p 2 which is equivalent to h2 best rational approximation we apply the results to the uniqueness problem in rational approximation and to the asymptotic behaviour of poles of best meromorphic approximants to functions with two branch points as pointed out by a referee part of the theory extends to every p 1 when the definition of the hankel operator is suitably generalized this we discuss in connection with the recent manuscript by v a prokhorov submitted for publication

on proof and progress in mathematics author s thurston william p abstract in response to jaffe and quinn math ho 9307227 the author discusses forms of progress in mathematics that are not captured by formal proofs of theorems especially in his own work in the theory of foliations and geometrization of 3 manifolds and dynamical systems

an analysis of the hr algorithm for computing the eigenvalues of a matrix abstract the hr algorithm a method of computing the eigenvalues of a matrix is presented it is based on the fact that almost every complex square matrix a can be decomposed into a product a hr of a so called pseudo hermitian matrix h and an upper triangular matrix r this algorithm is easily seen to be a generalization of the well known qr algorithm it is shown how it is related to the power method and inverse iteration and for special matrices the connection between the lr and hr algorithms is indicated

strong asymptotics of orthogonal polynomials with respect to exponential weights we consider asymptotics of orthogonal polynomials with respect to weights w x dx e q x dx on the real line where q x 2m k 0 qkx k q2m 0 denotes a polynomial of even order with positive leading coefficient the orthogonal polynomial problem is formulated as a riemann hilbert problem following 22 23 we employ the steepest descent type method introduced in 18 and further developed in 17 19 in order to obtain uniform plancherel rotach type asymptotics in the entire complex plane as well as asymptotic formulae for the zeros the leading coefficients and the recurrence coefficients of the orthogonal polynomials c 1999 john wiley sons inc

a restricted signature normal form for hermitian matrices quasi spectral decompositions and applications abstract in recent years a number of results on the relationships between the inertias of hermitian matrices and the inertias of their principal submatrices have appeared in the literature in this paper we study restricted congruence transformations of hermitian matrices m that at the same time induce a congruence transformation of a given principal submatrix a of m such transformations lead to the concept of the restricted signature normal form of m in particular by means of this normal form we obtain new and shorter proofs for several known inertia theorems and also derive some new results of this type for some applications a special class of almost unitary restricted congruence transformations turns out to be useful we show that with such transformations m can be reduced to a quasi diagonal form which in particular displays the eigenvalues of a moreover this quasi spectral decomposition is used to derive a generalized signature formula and to study hermitian matrix pencils

weighted extremal domains and best rational approximation abstract let f be holomorphically continuable over the complex plane except for finitely many poles and branch points contained in the unit disk we prove that best rational approximants to f of degree n in the l 2 sense on the unit circle have poles that asymptotically distribute according to the equilibrium measure on the compact set outside of which f is single valued and which has minimal green capacity in the disk among all such sets this provides us with n th root asymptotics of the approximation error by conformal mapping we deduce further estimates in approximation by rational or meromorphic functions to f in the l 2 sense on more general jordan curves encompassing the poles and branch points the key to these approximation theoretic results is a characterization of extremal domains of holomorphy for f in the sense of a weighted logarithmic potential which is the technical core of the paper

all optimal hankel norm approximations of linear multivariable systems and their l error bounds the problem of approximating a multivariable transfer function g s of mcmillan degree n by s of mcmillan degree k is considered a complete characterization of all approximations that minimize the hankel norm is derived the solution involves a characterization of all rational functions s f s that minimize where s has mcmillan degree k and f s is anticavisal the solution to the latter problem is via results on balanced realizations all pass functions and the inertia of matrices all in terms of the solutions to lyapunov equations it is then shown that where k 1 g s is the k l st hankel singular value of g s and for one class of optimal hankel norm approximations the method is not computationally demanding and is applied to a 12 state model

on reducing infinite eigenvalues of regular pencils by a nonequivalence transformation abstract we develop an algorithm for the deflation of the invariant subspace corresponding to infinite eigenvalues of a regular pencil by a nonequivalence transformation this method is a generalization of wielandt s algorithm for the deflation of the invariant subspace of a constant matrix

the computation of the kronecker canonical form of an arbitrary symmetric pencil abstract we present an algorithm for the computation of the kronecker structure of a symmetric pencil a m this method which preserves the property of symmetry is somewhat different from van dooren s algorithm for the determination of the kronecker structure of an arbitrary pencil we show how to use this method to determine the structure of the infinite elementary divisors of a m and its kronecker blocks which may occur for the case of a singular pencil the cost of computations of our algorithm is cheaper than van dooren s algorithm and the symmetry of the reduced pencils can be preserved the algorithm is fairly stable though we use some nonunitary transformation matrices but the norms of these matrices are bounded under a tolerance the present procedure can also be used to separate from a symmetric pencil a smaller symmetric regular pencil which contains only the finite eigenvalues of the original one so this method can be used as a preprocessing for a qz or hr algorithm in order to get rid of singularities which can be troublesome for those algorithms

an extension problem for h unitary matrices with applications to hermitian toeplitz matrices abstract given a hermitian matrix h a matrix u is said to be h unitary if u h hu h we consider the following extension problem if u 0 is a rectangular matrix such that u h 0 hu 0 a where a is a leading principal submatrix of h can u 0 be extended to an h unitary matrix after presenting necessary conditions for a more general situation we state a necessary and sufficient criterion for this problem and give a description of all its solutions finally these results are used to derive some properties of factorizations of hermitian toeplitz matrices

pade approximants for functions with branch points strong asymptotics of nuttall stahl polynomials let f be a germ of an analytic function at infinity that can be analytically continued along any path in the complex plane deprived of a finite set of points f in mathcal a bar mathbb c setminus a a infty j nuttall has put forward the important relation between the maximal domain of f where the function has a single valued branch and the domain of convergence of the diagonal pade approximants for f the pade approximants which are rational functions and thus single valued approximate a holomorphic branch of f in the domain of their convergence at the same time most of their poles tend to the boundary of the domain of convergence and the support of their limiting distribution models the system of cuts that makes the function f single valued nuttall has conjectured and proved for many important special cases that this system of cuts has minimal logarithmic capacity among all other systems converting the function f to a single valued branch thus the domain of convergence corresponds to the maximal in the sense of minimal boundary domain of single valued holomorphy for the analytic function f in mathcal a bar mathbb c setminus a the complete proof of nuttall s conjecture even in a more general setting where the set a has logarithmic capacity 0 was obtained by h stahl in this work we derive strong asymptotics for the denominators of the diagonal pade approximants for this problem in a rather general setting we assume that a is a finite set of branch points of f which have the algebro logarithmic character and which are placed in a generic position the last restriction means that we exclude from our consideration some degenerated constellations of the branch points

the pseudosymmetric tridiagonalization of an arbitrary real matrix this paper presents an algorithm for similarly reducing an arbitrary real matrix to a pseudosymmetric tridiagonal one discusses the possibility of overcoming breakdowns in the tridiagonalization process and gives some methods it also discusses the perturbation bounds of eigenvalues of a pseudosymmetric tridiagonal matrix some numerical examples are given

simple and fast algorithms for linear and integer programs with two variables per inequality the authors present an o mn 2 log m algorithm for solving feasibility in linear programs with up to two variables per inequality which is derived directly from the fourier motzkin elimination method the number of variables and inequalities are denoted by n and m respectively the running time of the algorithm dominates that of the best known algorithm for the problem and is far simpler integer programming on monotone inequalities i e inequalities where the coefficients are of opposite sign is then considered this problem includes as a special case the simultaneous approximation of a rational vector with specified accuracy which is known to be np complete however it is shown that both a feasible solution and an optimal solution with respect to an arbitrary objective function can be computed in pseudo polynomial time

incremental closure for systems of two variables per inequality subclasses of linear inequalities where each inequality has at most two vari ables are popular in abstract interpretation and model checking because they strike a balance between what can be described and what can be efficiently computed this paper focuses on the tvpi class of inequalities for which each coefficient of each two variable inequality is unrestricted an implied tvpi in equality can be generated from a pair of tvpi inequalities by eliminating a given common variable echoing resolution on clauses this operation called result can be applied to derive tvpi inequalities which are entailed implied by a given tvpi system the key operation on tvpi is calculating closure satisfiability can be observed from a closed system and a closed system also simplifies the calculation of other operations a closed system can be derived by repeatedly applying the result operator the process of adding a single tvpi inequality to an already closed input tvpi system and then finding the closure of this augmented system is called incremental closure this too can be calcu lated by the repeated application of the result operator this paper studies the calculus defined by result the structure of result derivations and how deriva tions can be combined and controlled a series of lemmata on derivations are presented that collectively provide a pathway for synthesising an algorithm for incremental closure the complexity of the incremental closure algorithm is analysed and found to be o n 2 m 2 lg m where n is the number of variables and m the number of inequalities of the input tvpi system

rogers ramanujan and the baker gammel wills pade conjecture in 1961 baker gammel and wills conjectured that for functions f meromorphic in the unit ball a subsequence of its diagonal pade approximants converges uniformly in compact subsets of the ball omitting poles of f there is also apparently a cruder version of the conjecture due to pade himself going back to the early twentieth century we show here that for carefully chosen q on the unit circle the rogers ramanujan continued fraction

simultaneous band reduction of two symmetric matrices abstract in this paper we consider simultaneous band reduction of two dense symmetric matrices by congruent transformations the ideas of simultaneous tridiagonalization are generalized to propose an efficient algorithm for the simultaneous band reduction in contrast to the algorithms of simultaneous tridiagonalization which are mainly based on matrix vector operations the proposed algorithm of simultaneous band reduction has the advantage that matrix matrix operations can be fully used to achieve better performance on modern computer architecture numerical results are presented to illustrate the effectiveness of our proposed algorithm

the computational complexity of simultaneous diophantine approximation problems simultaneous diophantine approximation in d dimensions deals with the approximation of a vector bf alpha alpha 1 cdots alpha d of d real numbers by vectors of rational numbers all having the same denominator this paper considers the computational complexity of algorithms to find good simultaneous approximations to a given vector bf alpha of d rational numbers we measure the goodness of an approximation using the sup norm we show that a result of h w lenstra jr produces polynomial time algorithms to find sup norm best approximations to a given vector bf alpha when the dimension d is fixed we show that a recent algorithm of a k lenstra h w lenstra jr and l lovasz to find short vectors in an integral lattice can be used to find a good approximation to a given vector bf alpha in d dimensions with a denominator q satisfying 1 leqq q leqq 2 d 2 n which is within a factor sqrt 5d 2 d 1 2 of the best approximation with denominato

eigenvalues of ax λbx for real symmetric matrices a and b computed by reduction to a pseudosymmetric form and the hr process abstract the paper presents a method for solving the eigenvalue problem ax bx where a and b are real symmetric but not necessarily positive definite matrices and b is nonsingular the method reduces the general case into a form cz z where c is a pseudosymmetric matrix a further reduction of c produces a tridiagonal pseudosymmetric form to which the iterative hr process is applied the tridiagonal pseudosymmetric form is invariant under the hr transformations the amount of computation is significantly less than in treating the problem by a general method

simultaneous tridiagonalization of two symmetric matrices we show how to simultaneously reduce a pair of symmetric matrices to tridiagonal form by congruence transformations no assumptions are made on the non singularity or definiteness of the two matrices the reduction follows a strategy similar to the one used for the tridiagonalization of a single symmetric matrix via householder reflectors two algorithms are proposed one using non orthogonal rank one modifications of the identity matrix and the other more costly but more stable using a combination of householder reflectors and non orthogonal rank one modifications of the identity matrix with minimal condition numbers each of these tridiagonalization processes requires o n 3 arithmetic operations and respects the symmetry of the problem we illustrate and compare the two algorithms with some numerical experiments

two connections between the sr and hr eigenvalue algorithms abstract the sr and hr algorithms are members of the family of gr algorithms for calculating eigenvalues and invariant subspaces of matrices this paper makes two connections between the sr and hr algorithms 1 an iteration of the sr algorithm on a 2 n 2 n symplectic butterfly matrix using shifts i 1 i i 1 k is equivalent to an iteration of the hr algorithm on an n n tridiagonal sign symmetric matrix using shifts i 1 i i 1 k 2 an iteration of the sr algorithm on a 2 n 2 n j tridagonal hamiltonian matrix using shifts i i i 1 k is equivalent to an iteration of the hr algorithm on an n n tridiagonal sign symmetric matrix using shifts 2 i i 1 k

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

improved algorithms for linear inequalities with two variables per inequality the authors show that a system of m linear inequalities with n variables where each inequality involves at most two variables can be solved in tilde o mn 2 time we denote tilde o f o f polylog n polylog m deterministically and in tilde o n 3 mn expected time using randomization parallel implementations of these algorithms run in tilde o n time where the deterministic algorithm uses tilde o mn processors and the randomized algorithm uses tilde o n 2 m processors the bounds significantly improve over previous algorithms the randomized algorithm is based on novel insights into the structure of the problem

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

self similar flows abstract large classes of self similar isospectral flows can be viewed as continuous analogues of certain matrix eigenvalue algorithms in particular there exist families of flows associated with the qr lr and cholesky eigenvalue algorithms this paper uses lie theory to develop a general theory of self similar flows which includes the qr lr and cholesky flows as special cases also included are new families of flows associated with the sr and hr eigenvalue algorithms the basic theory produces analogues of unshifted single step eigenvalue algorithms but it is also shown how the theory can be extended to include flows which are continuous analogues of shifted and multiple step eigenvalue algorithms

identification and rational l 2 approximation a gradient algorithm this paper deals with the identification of linear constant dynamical systems when formalized as a rational approximation problem the criterion is the l2 norm of the transfer function which is of interest in a stochastic context the problem can be expressed as nonlinear optimization in a hilbert space but standard algorithms are usually not well adapted we present a generic recursive procedure to find a local optimum of the criterion in the case of scalar systems our methods are borrowed from differential theory mixed with a bit of classical complex analysis to our knowledge the algorithm described in this paper is the first that ensures convergence to a local minimum

a symmetric structure preserving γqr algorithm for linear response eigenvalue problems abstract in this paper we present an efficient qr algorithm for solving the linear response eigenvalue problem h x x where h is symmetric with respect to 0 diag i n i n based on newly introduced orthogonal transformations the qr algorithm preserves the symmetric structure of h throughout the whole process and thus guarantees the computed eigenvalues to appear pairwise as they should with the help of a newly established implicit orthogonality theorem we incorporate the implicit multi shift technique to accelerate the convergence of the qr algorithm numerical experiments are given to show the effectiveness of the algorithm