equality conditions for matrix kantorovich type inequalities abstract following a recent paper of 7 we present sufficient and necessary conditions snecs under which equalities occur in those corresponding matrix kantorovich type inequalities we also present several relevant inequalities

a connection between state space and doubly coprime fractional representations explicit formulas for a doubly coprime fractional representation of the transfer function of a lumped lti system are given in terms of a stabilizable and detectable state space realization of the transfer function these formulas allow existing computational algorithms to be utilized for the purpose of computing doubly coprime factorizations several additional implications of this result are briefly discussed as are some immediate extensions and future research directions

the numerical evaluation of very oscillatory infinite integrals by extrapolation recently the author has given two modifications of a nonlinear extrapolation method due to levin and sidi which enable one to accurately and economically compute certain infinite integrals whose integrands have a simple oscillatory behavior at infinity in this work these modifications are extended to cover the case of very oscillatory infinite integrals whose integrands have a complicated and increasingly rapid oscillatory behavior at infinity the new method is applied to a number of complicated integrals among them the solution to a problem in viscoelasticity some convergence results for this method are presented

matrix fraction construction of linear compensators a construction is given using polynomial matrix fractions of the compensator resulting from a combination of a linear feedback control law and estimator design

numerical solution of systems of ordinary differential equations with the tau method an error analysis the recursive formulation of the tau method is extended to the case of systems of ordinary differential equations and an error analysis is given upper and lower error bounds are given in one of the examples considered the asymptotic behavior of the error compares in this case with that of the best approximant by algebraic polynomials for each of the components of the vector solution 1 introduction interest in the tau method see 2 3 5 for a long time regarded only as a tool for the construction of accurate approximations of a very restricted class of functions has been enhanced by the availability of software for its computer implementation and by the possibility of using it in the numerical solution of complex nonlinear differential equations over extended intervals the approxima tion of the solution of such type of equations is achieved as a result of finding tau approximants of a sequence of problems defined by linear differential equations details of this technique are given in 6 the subject of this paper is the extension of ortiz recursive formulation of lanczos tau method 5 to the case of systems of differential equations and more particularly to its error analysis for such systems our error estimation technique is applied to three model examples for which the exact solution is readily available it is discussed in general and with more detail when applied to the first of these examples for the second example we show how to get upper and lower error bounds we then compare these bounds with those given by meinardus 4 for the best uniform approximation of each of the components of the vector solution by algebraic polynomials to find that they are asymptotically equivalent the third example is a differential equation with variable coefficients and a nonempty subspace of residuals see 5 results of numerical experiments on the use of the tau method for the approxi mate solution of systems of ordinary differential equations with particular reference to stiff systems are reported in 8 the problems discussed in this paper can also be considered in the framework of simultaneous approximation of a function and its derivatives with the tau method see 1

computation of the modified bessel function of the third kind of imaginary orders uniform airy type asymptotic expansion the use of a uniform airy type asymptotic expansion for the computation of the modified bessel functions of the third kind of imaginary orders kia x near the transition point x a is discussed in a gil et al evaluation of the modified bessel functions of the third kind of imaginary orders j comput phys 17 2002 398 411 an algorithm for the evaluation of kia x was presented which made use of series a continued fraction method and nonoscillating integral representations the range of validity of the algorithm was limited by the singularity of the steepest descent paths near the transition point we show how uniform airy type asymptotic expansions fill the gap left by the steepest descent method

fast and accurate computation of bessel functions with large complex order and argument abstract a fast and highly accurate computational scheme to compute a product of bessel functions with large complex order and argument is of utmost importance in some problems in physics involving cylindrical configurations since these products can occur in the kernel of multiple integrals and are subject to mutual cancellations speed and accuracy of their numerical computation are key moreover the order and argument can vary over a vast range of several orders of magnitude to deliver a high relative accuracy 10 11 over the entire range we use the integral representations of the bessel functions and introduce a deformation into steepest descent paths the numerical computation is accelerated by three orders of magnitude if these paths are replaced by well chosen piecewise linear segments while maintaining the set relative accuracy

algorithms and codes for the macdonald function recent progress and comparisons the modified bessel function kiv x also known as the macdonald function finds application in the kontorovich lebedev integral transform when x and v are real and positive in this paper a comparison of three codes for computing this function is made these codes differ in algorithmic approach timing and regions of validity one of them can be tested independent of the other two through wronskian checks and therefore is used as a standard against which the others are compared

several inequalities involving khatri rao products of positive semidefinite matrices abstract the khatri rao and tracy singh products for partitioned matrices are viewed as generalized hadamard and generalized kronecker products respectively based on the connection between the khatri rao and tracy singh products we establish new inequalities involving khatri rao products of positive semidefinite matrices the results lead to inequalities involving hadamard products as a special case

zeros of the macdonald function of complex order the z zeros of the modified bessel function of the third kind k n z also known as modified hankel function or macdonald function are considered for arbitrary complex values of the order n approximate expressions for the zeros applicable in the cases of very small or very large n are given the behaviour of the zeros for varying n or arg n obtained numerically is illustrated by means of some graphics

several matrix kantorovich type inequalities 1994 217 221 7 h neudecker the hessian matrix for image factor analysis in psychometric methodol ogy r steyer k f wender and k f widaman eds pp 361 365 gustav fischer stuttgart 1993 8 h neudecker w polasek and s liu the heteroskedastic linear regression model andthe hadamard product a note

laplace s method on a computer algebra system with an application to the real valued modified bessel functions we examine a maple implementation of two distinct approaches to laplace s method used to obtain asymptotic expansions of laplace type integrals one algorithm uses power series reversion whereas the other expands all quantities in taylor or puiseux series these algorithms are used to derive asymptotic expansions for the real valued modified bessel functions of pure imaginary order and real argument that mimic the well known corresponding expansions for the unmodified bessel functions

the numerical inversion of two classes of kontorovich lebedev transform by direct quadrature abstract the inversion of the conventional kontorovich lebedev transform with the macdonald function as kernel and also with a kelvin function as kernel is considered strategies are presented first for computing values of these functions for purely imaginary order the interior strategy and these are tested in the subsequent inversion routines the exterior strategy which are developed for use where high speed mainframe or workstation computing is available both the interior and exterior strategies can exploit the w transformation and the w algorithm pioneered by sidi it is shown that conventional nag quadrature software is often incapable of carrying out the inversion without an implementation of this type examples considered show that relative error can generally be kept to o 1 d 8 although for work requiring hundreds of full inversion computations this would have to be relaxed somewhat to obviate the need for overnight computer runs application to a divergent inversion integral summable in the sense of abel is made with satisfactory results a case with weaker summability is also considered with somewhat inferior results

the heteroskedastic linear regression model and the hadamard product a note a unified matrix approach to the heteroskedastic linear regression model and its estimation is presented the hadamard product plays an essential role our approach creates the possibility of treating not only the standard linear but also nonlinear specifications special attention is being paid to maximum likelihood estimation

a quantitative version of the observation that the hadamard product is a principal submatrix of the kronecker product abstract the hadamard and kronecker products of two n m matrices a b are related by a b p t 1 a b p 2 where p 1 p 2 are partial permutation matrices after establishing several properties of the p matrices this relationship is employed to demonstrate how a simplified theory of the hadamard product can be developed during this process the well known result a b a b aa bb is extended to a b a b 1 2 aa bb ab ba aa bb showing an inherent link between the hadamard product and conventional product of two matrices this leads to a sharper bound on the spectral norm of a b a b 1 2 a 2 b 2 ab 2 1 2 a b and an improvement on the weak majorization of a b 2 a b w 1 2 2 a 2 b 2 ab w 2 a 2 b for a real non singular matrix x and invertible diagonal matrices d e the spectral condition number is shown to be if scaled bounded below as follows dxe 2 x x t 2 x x t 2 1 2 x x t 1 for a 0 we have i a 2 1 2 i a 2 a a i a 2 and a 1 2 a 1 2 2 1 2 i a a 1 a a 1 when a 0 the latter inequality is compared to styan s inequality a a 1 1 2 i a a 1 when a is a correlation matrix and is shown to possess stronger properties of ordering finally the relationship a b p t 1 a b p 2 is applied to determine conditions of singularity of certain orderings of the hadamard products of matrices

an algorithm for a special case of a generalization of the richardson extrapolation process a special case of a generalization of the richardson extrapolation process is considered and its complete solution is given in closed form using this an algorithm for implementing the extrapolation is devised it is shown that this algorithm needs a very small amount of arithmetic operations and very little storage convergence and stability properties for some cases are also considered

parameterization of reduced order compensator in the frequency domain after a brief survey of the relations between time and frequency domain representations of the observer based linear state feedback loop the parameterization of the compensator is discussed it is shown that the free parameters specified by the state feedback matrix k and the output error feedback matrix l in the time domain appear as coefficients in the polynomial matrices dtilde s and s characterizing the closed loop dynamics in the frequency domain these polynomial matrices can either be chosen arbitrarily pole placement problem or they may be derived as solutions from the corresponding optimal control or the optimal estimation problems respectively starting from the polynomial matrices dtilde s and s the doubly coprime fractional representations of the system and of the compensator transfer matrices can be computed directly without recurrence to the time domain results from these representations the right and left coprime compensator matrix fraction descriptions can easily be evalu

evaluation of the modified bessel function of the third kind of imaginary orders the evaluation of the modified bessel function of the third kind of purely imaginary order kia x is discussed we also present analogous results for the derivative the methods are based on the use of maclaurin series nonoscillatory integral representations asymptotic expansions and a continued fraction method depending on the ranges of x and a we discuss the range of applicability of the different approaches considered and conclude that power series the continued fraction method and the nonoscillatory integral representation can be used to accurately compute the function kia x in the range 0 a 200 0 x 100 using a similar scheme the derivative k ia x can also be computed within these ranges

summability experiments with a class of divergent inverse kontorovich lebedev transforms abstract the computation of inverse kontorovich lebedev transforms is examined with the help of the extrapolative m w transformation first presented by sidi in user friendly form in sidi 1988 recently published improved values for the macdonald function of purely imaginary order have enabled this strategy to give 10 digit accuracy in the inversion including an example with competing oscillations having different phase functions moreover the method is extended with similar accuracy to cases where the integral diverges but is summable in the sense of abel further probing with divergent integrals that are not abel summable but instead summable in the sense defined by jones 1980 is also yielding encouraging results and some analysis culminating in a formal theorem of convergence in that sense is presented to support the interpretation and computation of these

design of observer based compensators in the frequency domain the discrete time case for nth order systems with m outputs the design of compensators using state observers of order n x with 0 x m is presented in the z domain the design is carried out in equivalence with the usual time domain approach and the relations between the time and the frequency domain results are presented the state feedback and the observer are characterized by polynomial matrices in the frequency domain and the corresponding compensator matrix fraction descriptions are obtained from the solution of a linear diophantine equation in view of the optimal state estimation problem the use of an updated state estimate is of interest it is shown how this problem can also be solved in the frequency domain

canonical vector polynomials for the computation of complex order bessel functions with the tau method abstract a modification of the lanczos tau method for the approximate solution of second order differential equations with polynomial coefficients was proposed by the author earlier this modification chooses a minimum perturbation among all possible on the right hand side of the corresponding volterra integral equation a tau method computational scheme is applied to the approximate solution of a system of differential equations related to the differential equation of hypergeometric type various vector perturbations are discussed our choice of the perturbation term is a shifted chebyshev polynomial with a special form of selected transition and normalization the minimality conditions for the perturbation term are found for one equation they are sufficiently simple for verification in a number of important cases ortiz s recursive approach to the tau method is extended to canonical vector polynomial coefficients and to the coefficients of the linear combination that generate an approximate solution the advantages of the proposed approach are shown for modified bessel functions computation seven digit tables of modified bessel function k 1 2 i x have been computed and published the computer programs for the calculations may be found in the state fund of algorithms and programs and in the united library of numerical analysis of the moscow state university

extrapolation methods for divergent oscillatory infinite integrals that are defined in the sense of summability in a recent work by the author an extrapolation method the w transformation was developed by which a large class of oscillatory infinite integrals can be computed very efficiently the results of this work are extended to a class of divergent oscillatory infinite integrals in the present paper it is shown in particular that these divergent integrals exist in the sense of abel summability and that the w transformation can be applied to them without any modifications convergence results are stated and numerical examples given

optimal reduced order estimators in the frequency domain the discrete time case the discrete time stationary kalman filter problem is solved in the z domain for nth order systems with m outputs where 0 k m measurements are noise free the design equations are very similar to the continuous time cage and they can be solved by spectral factorization giving the polynomial matrix d z which parametrizes the reduced order optimal estimator in the frequency domain by solving a single linear equation the equivalent time domain representation for the optimal filter can also be obtained a simple example demonstrates the filter design in the frequency domain

on computing uniformly valid approximations for viscous waves on a plane beach abstract existing expansions for the near and far field flow of linear waves in a viscous liquid on a plane beach are supplemented by an approximation in the intermediate region which enables the flow to be matched from the shore line to infinity methods of computation are discussed and incorporated are recent improvements on the performance of numerical computation of oscillatory integrals sidi s user friendly w transform 1988 and kohler s parameter optimisation this journal 1993 for generalised newton cotes quadrature are found to be significant elements particularly with regard to the inversion of the kontorovich lebedev transform comprehensive computations are undertaken and diagrams shown to display the behaviour of the stream function for all depths an attempt is also made to construct a composite expansion of the vorticity and this is computed on a ray bisecting the wedge

solution in the z domain to the discrete time stationary kalman filtering problem for systems with perfect measurements the discrete time stationary kalman filtering problem is solved in the z domain for general right invertible linear systems whose measurements are all noise free a simple expression in closed form is obtained for the transfer function of the optimal filter both for the uniform and the general nonuniform rank cases it provides an optimal solution also for systems with uniform and nonuniform time delays the minimum variance error of the estimate is analyzed and the independent contributions to this error of the system zeros outside the unit circle finite and infinite and the excess of the number of inputs over the number of measurements are investigated

kantorovich and cauchy schwarz inequalities involving positive semidefinite matrices and efficiency comparisons for a singular linear model abstract matrix kantorovich inequalities involving two positive semidefinite matrices are presented corresponding cauchy schwarz inequalities are discussed some of these are used to compare several efficient and inefficient estimators for a singular linear model

matrix inequalities for convex functions abstract many converses of jensen s inequality for convex functions can be found in the literature here we give matrix versions with matrix weights of these inequalities some applications to the hadamard product of matrices are also given

a user friendly extrapolation method for oscillatory infinite integrals developpement d une methode d extrapolation flexible pour les integrales infinies oscillatoires

inequalities involving hadamard products of positive semidefinite matrices abstract an inequality established by g p h styan 1973 linear algebra appl 6 217 240 is on the hadamard product and a correlation matrix an inequality obtained by b y wang and f zhang 1997 linear and multilinear algebra 43 315 326 involves the hadamard product and schur complements these two inequalities hold in the positive definite matrix case based on albert s theorem we present their extensions to cover the positive semidefinite matrix case we also give relevant inequalities

numerical solution of stiff and singularly perturbed boundary value problems with a segmented adaptive formulation of the tau method this paper concerns the application of ortiz recursive formulation of the tau method to the construction of piecewise polynomial approximations to the solution of linear and nonlinear boundary value problems for ordinary differential equations a practical error estimation technique related to the concept of correction in zadunaisky s sense is considered and used in the design of an adaptive approach to the tau method it proves efficient in the numerical treatment of problems with rapid functional variations stiff and singularly per turbed problems a technique of increased accuracy at matching points of segmented tau approximants is also discussed and successfully applied to several problems numerical examples show that for a given degree of approximation our segmented tau approximant gives an accuracy comparable to that of the best segmented approximation of the exact solution by means of algebraic polynomials 1 introduction we discuss the use of ortiz recursive formulation of the tau method 23 25 in the numerical solution of boundary value problems for linear and nonlinear differential equations defined over an interval a x b we con sider global approximations over a b with a single polynomial expression and segmented forms based on a step by step formulation of the tau method considered by ortiz in 26 the tau approximate solution of a differential problem defined by a differential operator d is represented in terms of the elements of a sequence q of canonical polynomials such a sequence is uniquely determined by d it is independent of the specific boundary conditions of the problem and of the particular interval a b in which the solution is required these properties make possible the use of segmenta tion within the framework and with the software 32 designed for the recursive formulation of the tau method the concept of correction in zadunaisky s sense 37 see also stetter 36 is discussed in the context of the tau method and related to a practical error estimation technique this technique based on tau estimators introduced here is systematically applied to all examples linear or nonlinear it is