the schur algorithm applied to the one dimensional continuous inverse scattering problem the one dimensional continuous inverse scattering problem can be solved by the schur algorithm in the discrete time domain using sampled scattering data the sampling rate of the scattering data should be increased to reduce the discretization error but the complexity of the schur algorithm is proportional to the square of the sampling rate to improve this tradeoff between the complexity and the accuracy we propose a schur algorithm with the richardson extrapolation sare the asymptotic expansion of the schur algorithm necessary for the richardson extrapolation is derived in powers of the discretization step which shows that the accuracy order with respect to the discretization step of the schur algorithm is 1 the accuracy order of the sare with the n step richardson extrapolation is increased to n 1 with comparable complexity to the schur algorithm therefore the discretization error of the schur algorithm can be decreased in a computationally efficient manner by the sare

displacement structure for hankel vandermonde and related derived matrices abstract we introduce some generalized concepts of displacement structure for structured matrices obtained as products and inverses of toeplitz hankel and vandermonde matrices the toeplitz case has already been studied at some length and the corresponding matrices have been called near toeplitz or toeplitz like or toeplitz derived we focus on hankel and vandermonde like matrices and in particular show how the appropriately defined displacement structure yields fast triangular and orthogonal factorization algorithms for such matrices the main contribution of this paper is presenting a unified framework rather than obtaining the fastest algorithm for each special matrix

on orthogonal polynomials transformed by the qr algorithm abstract in this paper we study two equivalent ways of transforming a system of orthogonal polynomials one is the so called lifting of the recurrence relation associated with an orthogonal polynomial system ops and the other consists of applying the qr algorithm to the jacobi matrix of the ops these transformations are useful for among other things identifying the sobolev orthogonal polynomials with respect to certain measures which were previously investigated by iserles et al 1991 as a by product we obtain a new class of modified lommel polynomials

lossless inverse scattering digital filters and estimation theory methods to construct rational solutions of the lossless inverse scattering lis problem for one port passive digital systems are described the first method is recursive and can be viewed as a generalization of the celebrated schur algorithm the second method is global and leads to a parametrization of what we call fundamental solutions from which all lis solutions may be constructed quite a few classical problems in estimation theory and network theory may be viewed as special cases of the lis problem with each fundamental solution there is a solution of a corresponding estimation problem leading to a prediction and a modeling filter for a given stochastic process

a new framework for convergence analysis and algorithm development of adaptive iir filters a parameterization of an adaptive infinite impulse response iir filter s poles is developed based on balanced realization theory from this we develop a local approximation of the actual adapted pole parameters in which convergence speed is related to a certain eigenvalue spread this in turn is shown to relate to the hankel singular values of the system to be identified as well as certain coefficient sensitivity functions of the adapted filter the local approximation is not restricted to stationary points at these points however it is equivalent to a hessian approximation with the benefit of decomposing the hessian matrix into terms related to the aforementioned singular values and sensitivity functions the description of the adaptation process by means of the developed approximation leads to a greater understanding of the effects on convergence speed of factors such as the hankel singular values of the system its order the distribution of its poles and the choice of adapted parameters in particular the use of direct form and lattice parameters are compared in detail based on these properties a new adaptive iir algorithm with faster convergence and relatively low computational complexity has recently been proposed which is briefly mentioned results also indicate a potential for variable gain algorithms

calculation of gauss quadratures with multiple free and fixed knots algorithms are derived for the evaluation of gauss knots in the presence of fixed knots by modification of the jacobi matrix for the weight function of the integral simple gauss knots are obtained as eigenvalues of symmetric tridiagonal matrices and a rapidly converging simple iterative process based on the merging of free and fixed knots of quadratic convergence is presented for multiple gauss knots the procedures also allow for the evaluation of the weights of the quadrature corresponding to the simple gauss knots a new characterization of simple gauss knots as a solution of a partial inverse eigenvalue problem is derived

optimal steady states excessive functions and deterministic dynamic programs this paper studies the relationship between optimal steady states and excessive functions in undiscounted deterministic dynamic programming models optimal steady states arise in the turnpike theorems of optimal economic growth and the steady state policies of 3 4 81 excessive functions which play a critical role in constructing bounds on the performance of steady state policies in 3 81 are generalizations of lagrangian saddlepoints section 2 presents the dynamic programming model section 3 defines optimal steady states and discusses their importance section 4 characterizes the relationship between them and lagrangian saddle points section 5 introduces excessive functions and uses them to derive bounds on the optimal finite stage costs section 6 shows that the set of excessive functions associated with an oss contains a greatest element and a least element and proves that for s to be an optimal steady state it is necessary and sufficient that s be associated with an excessive function that is bounded above note that our model is quite general and does not assume convexity

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

algebraic methods for modified orthogonal polynomials some algebraic methods are given to implement uvarov s extended christoffel theorem the stability of the algorithms is discussed

differential methods in inverse scattering this paper discusses a new set of differential methods for solving the inverse scattering problem associated to the propagation of waves in an inhomogeneous medium by writing the medium equations in the form of a two component system describing the interaction of rightward and leftward propagating waves the causality of the propagation phenomena is exploited in order to identify the medium layer by layer the recursive procedure that we obtain constitutes a continuous version of an algorithm first derived by schur in order to test for the boundedness of functions analytic inside the unit circle it recovers the local reflectivity function of the medium using similar ideas some other differential methods can also be derived to reconstruct alternative parametrizations of the layered medium in terms of the local impedance or of the potential function the differential inverse scattering methods turn out to be very efficient since in some sense they let the medium perform the inversion by itself and thus

accurate singular values and differential qd algorithms in september 1991 j w demmel and w m kahan were awarded the second siam prize in numerical linear algebra for their paper accurate singular values of bidiagonal matrices 1 referred to as dk hereafter among several valuable results was the observation that the standard bidiagonal qr algorithm used in linpack 2 and in many other svd programs can be simplified when the shift is zero and of greater importance no subtractions occur the last feature permits very small singular values to be found with almost all the accuracy permitted by the data and at no extra cost

an algorithm for a dynamic programming model of fractional flows in an earlier paper the author introduced an n dimensional dynamic programming model of fractional flows characterized its solutions under general conditions and developed methods for their computation the latter which resemble algorithms for n state markovian decision processes break down when the costs are undiscounted and the flow matrices are nontransient here an algorithm is presented that computes many of the quantities of interest more efficiently under milder assumptions in addition some of our earlier results for the undiscounted case are extended to models with nontransient flow matrices

some functions that generalize the krall laguerre polynomials let l alpha be the semi infinite tridiagonal matrix associated with the three term recursion relation satisfied by the laguerre polynomials with weight function 1 gamma alpha 1 z alpha e z alpha 1 on the interval 0 infinity we show that when alpha is a positive integer by performing at most alpha successive darboux transformations from l alpha we obtain orthogonal polynomials on 0 infinity with weight distribution 1 gamma alpha k 1 z alpha k e z sigma j 1 k s j delta k j z with 1 less than or equal to k less than or equal to alpha we prove that as a consequence of the rational character of the darboux factorization these polynomials are eigenfunctions of a finite order differential operator our construction calls for a natural bi infinite extension of these results with polynomials replaced by functions of which the semi infinite case is a limiting situation c 1999 elsevier science b v all rights reserved

realizable spatio temporal tomlinson harashima precoders theory and fast computation we derive the optimal realizable tomlinson harashima precoder for frequency selective multiple input multiple output mimo channels with respect to a simplified mean square error mse criterion realizability means that in contrast to other works we do not restrict the internal filters of the precoder to have finite impulse responses firs but nevertheless ensure that the precoder can be operated in real time subject to a finite latency time which can be chosen by the system designer in particular this allows us to consider channels with infinite impulse responses iirs as they occur e g in digital subscriber lines the feedforward filter is located at the transmitter in our system model and an additional scalar gain is employed at the receiver the relocation of the feedforward filter allows us to consider decentralized receivers however it also makes it necessary to impose a transmit power constraint the power constraint often has the effect that the input signals at the receiver have to be rescaled we argue that the inclusion of the additional scalar gain allows us to incorporate the effect of the rescaling operation into the optimization special emphasis is put on the fast computation of the realizable tomlinson harashima precoder via displacement structure theory we propose a fast algorithm which includes a successive construction of a close to optimal ordering of the data streams the complexity of this algorithm is only cubic in the number of channel inputs and quadratic in the latency time the algorithm can also be applied in situations where certain fir constraints have to be fulfilled because we find that the optimal realizable tomlinson harashima precoder involves only fir filters as soon as the channel is fir our results are reproducible

a more accurate algorithm for computing the christoffel transformation a monic jacobi matrix is a tridiagonal matrix which contains the parameters of the three term recurrence relation satisfied by the sequence of monic polynomials orthogonal with respect to a measure the basic christoffel transformation with shift a transforms the monic jacobi matrix associated with a measure email protected into the monic jacobi matrix associated with email protected email protected this transformation is known for its numerous applications to quantum mechanics integrable systems and other areas of mathematics and mathematical physics from a numerical point of view the christoffel transformation is essentially computed by performing one step of the lr algorithm with shift but this algorithm is not stable we propose a more accurate algorithm estimate its forward errors and prove that it is forward stable i e that the obtained forward errors are of similar magnitude to those produced by a backward stable algorithm this means that the magnitude of the errors is the best one can expect because it reflects the sensitivity of the problem to perturbations in the input data

threshold algorithms for the prediction of reflection coefficients in a layered medium an algorithm is presented for the solution of the inverse problem of reflection seismology in the presence of noise the algorithm is based on a new representation of reflection coefficients in terms of the recorded seismogram this representation allows use of matrix perturbation methods for the analysis of error magnification in the recursive reconstruction of a stratified acoustic medium our analysis indicates that one of the main reasons for uncontrollable noise magnification is the assignment of significant values to very small reflection coefficients most of which reflect only noise in the data rather than reflection information our analysis also allows one to decide when a small computed reflection coefficient should be set to zero the strategy of setting small reflection coefficients to zero which is called thresholding has a stabilizing effect on inverse scattering algorithms the threshold algorithm also permits adaptive change of noise barriers which can be used for more detailed exposur

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

the stagewise kuhn tucker condition and differential dynamic programming the intention of this work is to describe and examine a differential dynamic programming ddp algorithm for constrained discrete time optimal control this algorithm has performed successfully on a large scale reservoir control problem 11 the present paper gives conditions under which convergence to the stationary policy is assured the convergence demonstration hinges upon a notion which we refer to as the stagewise kuhn tucker condition strategies generated to satisfy this condition determine policies which satisfy the conventional kuhn tucker condition this observation may be of wider importance in discrete optimal control theory for the stagewise condition might be a convenient criterion for constructing strategies

darboux transformation and perturbation of linear functionals abstract let l be a quasi definite linear functional defined on the linear space of polynomials with real coefficients in the literature three canonical transformations of this functional are studied xl l c x and 1 x l c x where x denotes the linear functional x xk k 0 and k 0 is the kronecker symbol let us consider the sequence of monic polynomials orthogonal with respect to l this sequence satisfies a three term recurrence relation whose coefficients are the entries of the so called monic jacobi matrix in this paper we show how to find the monic jacobi matrix associated with the three canonical perturbations in terms of the monic jacobi matrix associated with l the main tools are darboux transformations in the case that the lu factorization of the monic jacobi matrix associated with xl does not exist and darboux transformation does not work we show how to obtain the monic jacobi matrix associated with x 2 l as a limit case we also study perturbations of the functional l that are obtained by combining the canonical cases finally we present explicit algebraic relations between the polynomials orthogonal with respect to l and orthogonal with respect to the perturbed functionals

spectral transformations self similar reductions and orthogonal polynomials we study spectral transformations in the theory of orthogonal polynomials which are similar to darboux transformations for the schr dinger equation linear transformations of the stieltjes function with coefficients that are rational in the argument are constructed as iterations of the christoffel and geronimus transformations we describe a characteristic property of semi classical orthogonal polynomials scop on the uniform and the exponential lattice namely that all these polynomials can be obtained through simple quasi periodic and q periodic self similar closures of the chain of linear spectral transformations in the self similar setting a characterization of the laguerre hahn polynomials on linear and q linear lattices is obtained by considering rational transformations of the stieltjes function generated by transitions to the associated polynomials

displacement structure theory and applications in this survey paper we describe how strands of work that are important in two different fields matrix theory and complex function theory have come together in some work on fast computational algorithms for matrices with what we call displacement structure in particular a fast triangularization procedure can be developed for such matrices generalizing in a striking way an algorithm presented by schur 1917 j reine angew math 147 1917 pp 205 232 in a paper on checking when a power series is bounded in the unit disc this factorization algorithm has a surprisingly wide range of significant applications going far beyond numerical linear algebra we mention among others inverse scattering analytic and unconstrained rational interpolation theory digital filter design adaptive filtering and state space least squares estimation

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