some numerical algorithms for solving the highly oscillatory second order initial value problems abstract in this paper some numerical algorithms spectral collocation method block spectral collocation method boundary value method block boundary value method implicit runge kutta method diagonally implicit runge kutta method and total variation diminishing runge kutta method are used to solve the highly oscillatory second order initial value problems we first derive these methods for the first order initial value problems and then extend these methods to the highly oscillatory nonlinear systems by matrix analysis methods these new methods preserve the accuracy of the original methods and the main advantages of these new methods are low storage requirements and high efficiency extensive numerical results are presented to demonstrate the convergence properties of these methods

two step hybrid methods for periodic initial value problems we consider a family of two step methods for the numerical integration of periodic initial value problems where the required solution has an unknown period we derive efficient p stable formulas of orders 4 and 6 which are particularly appropriate for nonlinear systems as their newton iteration matrices can be expressed as a perfect square and a perfect cube respectively

stability of explicit arkn methods for perturbed oscillators the stability of explicit arkn methods for the numerical integration of perturbed oscillators is analyzed stability regions for the second order homogeneous linear test model are obtained these stability regions represent a generalization of the stability intervals for the classical rkn methods

a qualitative analysis of two and three step methods for stable second order systems a variety of two and three step direct integration methods have been proposed for solving stable systems of linear second order ordinary differential equations we will assess their suitability using as our main criterion their ability to reproduce stable numerical solutions for stable positive definite and symmetric differential systems in addition we determine which of these methods have the ability to reproduce the weak and strong damping properties associated with these differential systems next we investigate their suitability for solving unsymmetric linear gyroscopic systems and finally we show that for a wider class of unsymmetric but stable differential systems only a very restricted class of methods may be stable through this investigation we isolate all the potentially useful two step methods as a subset of the newmark methods similarly we identify a class of useful second order three step methods and this latter investigation provides new insight into the popular houbolt and wilson

general synthesis procedures for fir lossless transfer matrices for perfect reconstruction multirate filter bank applications a recently reported procedure see ibid vol assp 35 p 476 492 apr 1987 and proc ieee int conf acoust speech sig processing apr 1987 p 2169 2172 for the design of m channel perfect reconstruction quadrature mirror filter qmf bands uses the concept of lossless alias component matrices the synthesis of such qmf banks centers around the generation of a lossless m m finite impulse response fir transfer matrix e z recent results for such generation have been somewhat ad hoc i e not sufficiently general a general procedure is outlined for the generation of such transfer matrices the procedure is based on a cascaded lattice structure derived from a state space viewpoint the structure is such that it generates only m m lossless fir transfer matrices regardless of parameter values and conversely any m m lossless fir transfer matrix can be obtained by a suitable choice of parameters these parameters turn out to be angles theta sub k and the structure is such that the number of angles is minimal a design example is presented to demonstrate the main results

h sub spl infin nonlinear filtering of discrete time processes this correspondence investigates the problem of h sub spl infin estimation of a discrete time nonlinear process an estimator which may be nonlinear is introduced so that an h sub spl infin norm like of what we call a generalized estimation error is guaranteed to be bounded by a prescribed level conditions for the existence of such an estimator and formulae for its derivation are obtained utilizing a discrete time analog of the hamilton jacobi inequality an approximate filter based on linearization is developed this filter relates to the extended kalman filter in the same way that the linear h sub spl infin filter relates to the kalman filter

stability of runge kutta nystrom methods in this paper a general and detailed study of linear stability of runge kutta nystrom rkn methods is given in the case that arbitrarily stiff problems are integrated we establish a condition that rkn methods must satisfy so that a uniform bound for stability can be achieved this condition is not satisfied by any method in the literature therefore a stable method is constructed and some numerical comparisons are made

a 6 4 optimized embedded runge kutta nystrom pair for the numerical solution of periodic problems in this paper an optimization of the non fsal embedded rkn 6 4 pair with six stages of moawwad el mikkawy el desouky rahmo is presented the new method is derived after applying phase fitting and amplification fitting and has variable coefficients the preservation of the algebraic order is verified and the principal term of the local truncation error is evaluated furthermore periodicity analysis is performed which reveals that the new method is almost p stable the efficiency of the new method is measured via the integration of several initial value problems

trigonometrically fitted three derivative runge kutta methods for solving oscillatory initial value problems abstract trigonometrically fitted three derivative runge kutta tfthdrk methods for solving numerically oscillatory initial value problems are proposed and developed tfthdrk methods improve three derivative runge kutta thdrk methods numer algor 74 247 265 2017 and integrate exactly the problem whose solutions can be expressed as the linear combinations of functions from the set of exp i w t exp i w t or equivalently the set cos wt sin wt where w approximate the main frequency of the problem the order conditions are deduced by the theory of rooted trees and b series and two new explicit special tfthdrk methods with order five and seven respectively are constructed linear stability of tfthdrk methods is examined numerical results show the superiority of the new methods over other methods from the scientific literature

high orderp stable formulae for the numerical integration of periodic initial value problems recently there has been considerable interest in the approximate numerical integration of the special initial value problemy f x y for cases where it is known in advance that the required solution is periodic the well known class of stormer cowell methods with stepnumber greater than 2 exhibit orbital instability and so are often unsuitable for the integration of such problems an appropriate stability requirement for the numerical integration of periodic problems is that ofp stability however lambert and watson have shown that ap stable linear multistep method cannot have an order of accuracy greater than 2 in the present paper a class of 2 step methods of runge kutta type is discussed for the numerical solution of periodic initial value problems p stable formulae with orders up to 6 are derived and these are shown to compare favourably with existing methods

two channel iir qmf banks with approximately linear phase analysis and synthesis filters perfect linear phase two channel qmf banks require the use of finite impulse response fir analysis and synthesis filters although they are less expensive and yield superior stopband characteristics perfect linear phase cannot be achieved with stable infinite impulse response iir filters thus iir designs usually incorporate a postprocessing equalizer that is optimized to reduce the phase distortion of the entire filter bank however the analysis and synthesis filters of such an iir filter bank are not linear phase in this paper a computationally simple method to obtain iir analysis and synthesis filters that possess negligible phase distortion is presented the method is based on first applying the balanced reduction procedure to obtain nearly allpass iir polyphase components and then approximating these with perfect allpass iir polyphase components the resulting iir designs already have only negligible phase distortion however if required further improvement may be achieved through optimization of the filter parameters for this purpose a suitable objective function is presented bounds for the magnitude and phase errors of the designs are also derived design examples indicate that the derived iir filter banks are more efficient in terms of computational complexity than the fir prototypes and perfect reconstruction fir filter banks although the pr fir filter banks when implemented with the one multiplier lattice structure and iir filter banks are comparable in terms of computational complexity the former is very sensitive to coefficient quantization effects

subband filters optimized for lost coefficient reconstruction transmission of digital subband coded images over lossy packet networks presents a reconstruction problem at the decoder this correspondence addresses the design of a family of quadrature mirror filters qmfs that minimize the mean squared error of mean reconstructed low frequency subband coefficients the resulting filters perform well as qmfs as well as provide the desired reconstruction properties in the event of loss

classical sampling theorems in the context of multirate and polyphase digital filter bank structures the recovery of a signal from so called generalized samples is a problem of designing appropriate linear filters called reconstruction or synthesis filters this relationship is reviewed and explored novel theorems for the subsampling of sequences are derived by direct use of the digital filter bank framework these results are related to the theory of perfect reconstruction in maximally decimated digital filter bank systems one of the theorems pertains to the subsampling of a sequence and its first few differences and its subsequent stable reconstruction at finite cost with no error the reconstruction filters turn out to be multiplierless and of the fir finite impulse response type these ideas are extended to the case of two dimensional signals by use of a kronecker formalism the subsampling of bandlimited sequences is also considered a sequence x n with a fourier transform vanishes for mod omega mod or l pi m where l and m are integers with l

a new collocation method for approximate solution of the pantograph functional differential equations with proportional delay the paper presents a new numerical method for solving functional differential equations with proportional delays of the first and higher orders the method consists of replacing the initial equation by an approximate equation which has an exact analytic solution with a set of free parameters these free parameters are determined by the use of the collocation procedure some examples are given to demonstrate the validity and applicability of the new method and a comparison is made with the existing results the numerical results show that the proposed method is of a high accuracy and is efficient for solving a wide class of functional differential equations with proportional delays including equations of neutral type the method is applicable to both initial and boundary value problems

higher derivative explicit one step methods for non linear dynamic problems part i design and theory a new higher order explicit one step scheme is presented for use in the solution of a certain class of non linear problems in structural dynamics the generalized scheme covers some recently developed multistep algorithms as well as some new one step methods with improved numerical properties a weighted residual approach is used which contains new aspects for design of single step algorithms accuracy and stability are investigated for linear and non linear systems algorithms with higher accurary than the central difference method are developed and optimized to minimize the reduction in the stability limit

on the optimality of ideal filters for pyramid and wavelet signal approximation the reconstructed lowpass component in a quadrature mirror filter qmf bank provides a coarser resolution approximation of the input signal since the outputs of the two qmf branches are orthogonal the transformation that provides the maximum energy compaction in the lowpass channel is also the one that results in the minimum approximation error this property is used as a common strategy for the optimization of qmf banks orthogonal wavelet transforms and least squares pyramids a general solution is derived for the qmf bank that provides the optimal decomposition of an arbitrary wide sense stationary process this approach is extended to the continuous case to obtain the minimum error approximation of a signal at a given sampling rate in particular it is shown that the sine wavelet transform is optimal for the representation at all scales of signals with non increasing spectral density

modified runge kutta methods for the numerical solution of odes with oscillating solutions three new modified runge kutta methods with minimal phase lag are developed for the numerical solution of ordinary differential equations with oscillating solutions these methods are based on the well known runge kutta method of dormand and prince rk5 4 t see 1 of order five numerical and theoretical results show that this new approach is more efficient compared with the well known classical fifth order runge kutta dormand and prince method

on the t method for delay differential equations with infinite lag abstract this paper discusses the method for the numerical solution of delay differential equations with infinite lag our analysis is based on the test equation y t ay t by t where a b c and 0 1 in order to solve the storage problem involved in studying the long time behaviour of the solution we use a grid with increased stepsizes monotonicity uniform boundedness asymptotic stability algebraic decay and global discretisation error of the numerical solution are investigated

an approximate solution for a neutral functional differential equation with proportional delays in this paper a novel algorithm based on reproducing kernel theory for neutral functional differential equation with proportional delays is proposed the advantages of the presented method are the establishment of complete approximate solution theory and high precision of absolute error some examples are given to demonstrate the validity and applicability of the new method and some comparisons are made with the existing results

high order hierarchical a and l stable integration methods numerical integration of stiff first order systems of differential equations is considered it is shown how a c continuous polynomial time discretization in conjunction with a weighted residual method can be used to derive methods corresponding to the diagonal and first subdiagonal pade approximants in this manner a stable schemes of order 2k and l stable schemes of order 2k l are obtained at least for k 4 the methods are hierarchical in the sense that a scheme with a given accuracy embraces the equations of all lower order methods that have the same stability type

exponentially fitted two step hybrid methods for y f x y it is the purpose of this paper to derive two step hybrid methods for y f x y with oscillatory or periodic solutions specially tuned to the behaviour of the solution through the usage of the exponential fitting technique the construction of two step exponentially fitted hybrid methods is shown and their properties are discussed some numerical experiments confirming the theoretical expectations are provided

high order p stable multistep methods abstract recently numerical integration of the special ivp y f t y whose solutions are of periodic type have been considered of great interest p stability is the appropriate requirement for numerical methods integrating periodic problems as established by lambert and watson 1976 a p stable linear multistep method has order no greater than two in this paper two step multi stage linear methods are proposed for numerical integration of periodic problems and p stable methods of this type obtained in a simple form up to order eight are shown finally those methods are successfully tested

on richardson extrapolation for low dissipation low dispersion diagonally implicit runge kutta schemes abstract in the modeling of wave propagation phenomena it is necessary to use time integration methods which are not only sufficiently accurate but also properly describe the amplitude and phase of the propagating waves it is not clear if amending the developed schemes by extrapolation methods to obtain a high order of accuracy preserves the qualitative properties of these schemes in the perspective of dissipation dispersion and stability analysis it is illustrated that the combination of various optimized schemes with richardson extrapolation is not optimal for minimal dissipation and dispersion errors optimized third order and fourth order methods are obtained and it is shown that the proposed methods combined with richardson extrapolation result in fourth and fifth orders of accuracy correspondingly while preserving optimality and stability the numerical applications include the linear wave equation a stiff system of reaction diffusion equations and the nonlinear euler equations with oscillatory initial conditions it is demonstrated that the extrapolated third order scheme outperforms the recently developed fourth order diagonally implicit runge kutta scheme in terms of accuracy and stability

optimal signal reconstruction in noisy filter bank systems multirate kalman synthesis filtering approach a multirate kalman synthesis filter is proposed in this paper to replace the conventional synthesis filters in a noisy filter bank system to achieve optimal reconstruction of the input signal based on an equivalent block representation of subband signals a state space model is introduced for an m band filter bank system with subband noises the composite effect of the input signal analysis filter bank decimators and interpolators is represented by a multirate state space model the input signal is embedded in the state vector and the corrupting noises in subband paths are generally considered as additive noises hence the signal reconstruction problem in the m band filter bank systems with subband noises becomes a state estimation procedure in the resultant multirate state space model the multirate kalman filtering algorithm is then derived according to the multirate state space model to achieve optimal signal reconstruction in noisy filter bank systems based on the optimal state estimation theory the proposed multirate kalman synthesis filter provides the minimum variance reconstruction of the input signal two numerical examples are also included the simulation results indicate that the performance improvement of signal reconstruction in noisy filter bank systems is remarkable

explicit runge kutta methods for initial value problems with oscillating solutions new pairs of embedded runge kutta methods specially adapted to the numerical solution of first order systems of differential equations which are assumed to possess oscillating solutions are obtained these pairs have been derived taking into account not only the usual properties of accuracy stability and reliability of the local error estimator to adjust the stepsize of the underlying formulas but also the dispersion and dissipation orders of the advancing formula as defined by van der houwen and sommeijer 1989 three nine stage embedded pairs of runge kutta methods with algebraic orders 7 and 5 and higher orders of dispersion and or dissipation are selected among the members of a family of pairs depending on several free parameters some numerical results are presented to show the efficiency of the new methods

second derivative methods applied to implicit first and second order systems a wide variety of methods have been proposed for solving implicit second order systems of differential equations such as arise in structural and mechanical vibration analyses here we discuss a class of methods based on second derivative formulae some of which have previously been used by enright 1 we show them to have desirable analytic and implementation properties we relate these methods to others recently proposed by brusa and nigro and by serbin showing that we obtain algorithms which have all the desirable properties of their methods and which can also be extended to classes of practical problems which were not considered by these other authors since implicit first order systems are also important in practice for example in finite element analyses of heat conduction problems we demonstrate the efficient implementation of the second derivative formulae in this case too

global error estimation based on the tolerance proportionality for some adaptive runge kutta codes modern codes for the numerical solution of initial value problems ivps in odes are based in adaptive methods that for a user supplied tolerance d attempt to advance the integration selecting the size of each step so that some measure of the local error is email protected although this policy does not ensure that the global errors are under the prescribed tolerance after the early studies of stetter considerations concerning a theory for ode solvers in r burlisch r d grigorieff j schroder eds numerical treatment of differential equations proceedings of oberwolfach 1976 lecture notes in mathematics vol 631 springer berlin 1978 pp 188 200 tolerance proportionality in ode codes in r marz ed proceedings of the second conference on numerical treatment of ordinary differential equations humbold university berlin 1980 pp 109 123 and the extensions of higham global error versus tolerance for explicit runge kutta methods ima j numer anal 11 1991 457 480 the tolerance proportionality of adaptive ode solvers j comput appl math 45 1993 227 236 the reliability of standard local error control algorithms for initial value ordinary differential equations in proceedings the quality of numerical software assessment and enhancement ifip series springer berlin 1997 it has been proved that in many existing explicit runge kutta codes the global errors behave asymptotically as some rational power of d this step size policy for a given ivp determines at each grid point t n a new step size h n 1 h t n d so that h t d is a continuous function of t in this paper a study of the tolerance proportionality property under a discontinuous step size policy that does not allow to change the size of the step if the step size ratio between two consecutive steps is close to unity is carried out this theory is applied to obtain global error estimations in a few problems that have been solved with the code gauss2 s gonzalez pinto r rojas bello gauss2 a fortran 90 code for second order initial value problems based on an adaptive two stage runge kutta gauss method with this discontinuous step size policy

generalized laguerre pseudospectral method based laguerre interpolation abstract in this paper we develop the generalized laguerre pseudospectral method for the second order ordinary differential equation based on the modified generalized laguerre approximation which is very efficient for the long time numerical simulations of dynamical systems the numerical solution possesses the spectral accuracy the global convergence of the proposed algorithm is proved numerical results demonstrate the effectiveness of the proposed method and coincide well with theoretical analysis

block runge kutta methods for periodic initial value problems abstract block runge kutta methods with minimal phase lag for first order periodic initialvalue problems are developed it should be noted that the new methods are based on the runge kutta methods of algebraic order three and on a new error estimate introduced in this paper the numerical results indicate that these new methods are efficient for the numerical solution of differential equations with periodic solutions using variable step size

convergence results on the design of qmf banks this correspondence considers the design of quadrature mirror filter qmf banks whose analysis and synthesis filters are fir and have linear phase the design criterion is of least squares type an iterative computation algorithm is proposed and its convergence is proven that offers new insight to the design of qmf banks and relates it to a more general nonlinear optimization problem

damping and phase analysis for some methods for solving second order ordinary differential equations we consider numerical methods for initial value problems for second order systems of ordinary differential equations analysing them by applying them to the test equation we discuss conditions which ensure an oscillatory numerical solution and the desirability of such a property we also use a slightly more general test equation to derive conditions which ensure that the numerical forced oscillation is in phase with the true forced oscillation to illustrate the theory we consider the damping and phase properties of some frequently used methods

variable order variable step algorithms for second order systems part 2 the codes we develop a new class of multistage algorithms for second order systems of ordinary differential equations ode s based on methods by zienkiewicz et al 13 our new methods are designed to provide a set of efficient methods which can be used with local error estimators based on embedding techniques in a companion paper we provide a set of subroutines implementing the algorithms derived and a discussion of numerical experience with the resulting codes

perfect reconstruction filter banks having linear phase fir filters with equiripple response the design of equiripple linear phase analysis and synthesis fir filters of two channel perfect reconstruction pr filter banks is formulated as the minimization of a weighted peak error under both linear inequality arising from the desired responses of the analysis filters and nonlinear equality pr constraints the effectiveness of a proposed method to solve the design problem a modified dual affine scaling variant of karmarkar s 1989 algorithm and an approximation scheme is illustrated through several design examples

families of methods for ordinary differential equations based on trigonometric polynomials abstract we consider the construction of methods based on trigonometric polynomials for the initial value problems whose solutions are known to be periodic it is assumed that the frequency w can be estimated in advance the resulting methods depend on a parameter hw where h is the step size and reduce to classical multistep methods if 0 gautschi 4 developed adams and stormer type methods in our paper we construct nystrom s and milne simpson s type methods numerical experiments show that these methods are not sensitive to changes in w but require the jacobian matrix to have purely imaginary eigenvalues

stability analysis of theta methods for neutral functional differential equations this paper deals with the subject of numerical stability for the neutral functional differential equation y t ay t by qt cy pt qquad t 0 it is proved that numerical solutions generated by theta methods are convergent if c b if and only if theta geq 1 2 and that the one leg theta method is lambda stable if theta 1 we also find out that inappropriate stepsize causes spurious solution in the marginal case where rm re a 0 and a b

polyphase networks block digital filtering lptv systems and alias free qmf banks a unified approach based on pseudocirculants the relationship between block digital filtering and quadrature mirror filter qmf banks is explored necessary and sufficient conditions for alias cancellation in qmf banks are expressed in terms of an associated matrix derived from the polyphase components of the analysis and synthesis filters these conditions called the pseudocirculant conditions make it possible to unite qmf banks with the framework of block digital filtering directly absence of amplitude distortion in an alias free qmf bank translates into the losslessness property of the pseudocirculant matrix involved

a noumerov type method with minimal phase lag for the integration of second order periodic initial value abstract in a recent paper 2 we gave a noumerov type method with minimal phase lag for the integration of second order initial value problems y f t y y t0 y0 y t0 y 0 however the method given there is implicit we show here the interesting result that if the noumerov type methods of 2 are made explicit with the help of the classical second order method then there exists a selection of the free parameter for which the resulting method has a considerably small frequency distortion of size 1 40320 h 6 and also a slightly larger interval of periodicity of size 2 75 than the phase lag of size 1 12096 h 6 and interval of periodicity of size 2 71 for the implicit method 2 more interestingly it turns out that noumerov made explicit of chawla 3 also has less frequency distortion than the implicit noumerov method we shall assume a familiarity with the notation and discussion given in 2

the design of arbitrary fir digital filters using the eigenfilter method the design of finite impulse response fir digital filters for approximating an arbitrary function in both magnitude and phase in the least square sense is studied the design method is based on the computation of an eigenvector of an appropriate real symmetric and positive definite matrix the design of the complex coefficient filter is shown to be an extension of the design of the real coefficient filter several design examples including the constant group delay filters and digital phase all pass filters are presented comparisons to existing methods are made

two step fourth order p stable methods with phase lag of order six for y t y abstract we present a new family of two step fourth order methods which when applied to the test equation y 2y 0 are at once p stable and have a phase lag of order h6 h h h is the step size

signal adapted multiresolution transform for image coding the authors consider the problem of designing multiresolution transforms that are adapted to the given image signal in the sense that they maximize the coding gain at each resolution level a simple alternating optimization algorithm is derived for solving this problem in the framework of the lattice realization of para unitary quadrature mirror filters the resulting large coding scheme is discussed in some detail and its performance is compared with that of the discrete cosine transform jpeg technique and with that of some nonadapted multiresolution transforms

an easily implementable fourth order method for the time integration of wave problems abstract we are concerned with the time integration of systems of ordinary differential equations arising from the space discretization of partial differential wave equations with smooth solutions a method is suggested that while being as easily implementable as the standard implicit mid point rule is fourth order accurate the new method is symplectic so that it is very well suited for long time integrations of problems with a hamiltonian structure numerical experiments are reported that refer to a fourth order galerkin space discretization of the korteweg de vries equation and to a pseudospectral space discretization of the same equation

theory and design of m channel maximally decimated quadrature mirror filters with arbitrary m having the perfect reconstruction property based on the concept of losslessness in digital filter structures this paper derives a general class of maximally decimated m channel quadrature mirror filter banks that lead to perfect reconstruction the perfect reconstruction property guarantees that the reconstructed signal hat x n is a delayed version of the input signal x n i e hat x n x n n 0 it is shown that such a property can be satisfied if the alias component matrix ac matrix for short is unitary on the unit circle of the z plane the number of channels m is arbitrary and when m is two the results reduce to certain recently reported 2 channel perfect reconstruction qmf structures a procedure based on recently reported fir cascaded lattice structures is presented for optimal design of such fir m channel filter banks design examples are included

a runge kutta fehlberg method with phase lag of order infinity for initial value problems with oscillating solution abstract a runge kutta method is developed for the numerical solution of initial value problems with oscillating solution based on the runge kutta fehlberg 2 3 method a runge kutta method with phase lag of order infinity is developed based on these methods we produce a new embedded runge kutta fehlberg 2 3 method with phase lag of order infinity this method is called as runge kutta fehlberg phase fitted method rkfpf the numerical results indicate that this new method is much more efficient compared with other well known runge kutta methods for the numerical solution of differential equations with oscillating solution using variable step size

least squares approximation of perfect reconstruction filter banks designing good causal filters for perfect reconstruction pr filter banks is a challenging task due to the unusual nature of the design constraints we present a new least squares ls design methodology for approximating prfbs that avoids most of these difficult constraints the designer first selects a set of subband analysis filters from an almost unrestricted class of rational filters then given some desired reconstruction delay this design procedure produces the causal and rational synthesis filters that result in the best least squares approximation to a prfb this technique is built on a multi input multi output mimo system model for filter banks derived from the filter bank polyphase representation using this model we frame the ls approximation problem for prfbs as a causal ls equalization problem for mimo systems we derive the causal ls solution to this design problem and present an algorithm for computing this solution the resulting algorithm includes a mimo spectral factorization that accounts for most of the complexity and computational cost for this design technique finally we consider some design examples and evaluate their performance

explicit runge kutta nystro m methods with reduced phase errors for computing oscillating solutions we construct explicit runge kutta nystrom methods for the integration of first and second order differential equations having an oscillatory solution special attention is paid to the phase errors or dispersion of the dominant components in the numerical oscillations when these methods are applied to a linear homogeneous test model rk n methods are constructed which are dispersive of orders up to 10 whereas the algebraic order of accuracy is only 2 or 3 application of these methods to equations describing free and weakly forced oscillations and to semidiscretized hyperbolic equations reveals that the phase errors can significantly be reduced

the design of nonuniform modulated filterbanks we present an approach to nonuniform filterbank design based on modulated filters and the principle of adjacent channel aliasing cancellation the approach is a generalization of pseudo qmf designs to nonuniform channel arrangements and the essential idea is to form nonuniform filterbanks from uniform sections joined by transition filters all channel filters are formed by modulating lowpass prototypes to meet the aliasing cancellation conditions the transition filters are modulated from complex lowpass prototypes although the resulting channel filters are still real the approach provides a computationally simple method for designing filterbanks with large numbers of channels because of the fact that channel filters are modulated from a few lowpass prototypes and because the lowpass prototype designs can be scaled that is designs for narrow channel spacing can be produced from designs for wider channel spacing by interpolating the lowpass prototype impulse response the development is limited to the design of nonuniform filterbanks with integer decimation factors although we demonstrate that the technique can be used to produce excellent rational decimation factor designs by combining channel outputs using an inverse polyphase transform

information theoretic performance of quadrature mirror filters most existing quadrature mirror filters qmfs closely match the derived closed form expression for an efficient class of qmfs we use the closed form expressions to derive the relationship between information theoretic loss and the frequency selectivity of the qmf by calculating first order entropy as well as rate distortion theoretic performance of a two band qmf system we find that practical qmfs do not suffer a significant information theoretic loss with first order autoregressive gaussian sources with second order autoregressive sources we find that practical qmfs suffer a notable information theoretic loss when the bandwidth of the source is extremely narrow but incur a small loss when the bandwidth is wider we suggest that our results broadly apply to higher order autoregressive sources as well

phase lag analysis of implicit runge kutta methods we analyse the phase errors introduced by implicit runge kutta methods when a linear inhomogeneous test equation is integrated it is shown that the homogeneous phase errors dominate if long interval integrations are performed homogeneous dispersion relations for the special class of dirk methods are derived and a few high order dispersive dirk methods are constructed these methods are applied to systems of linear differential equations with oscillating solutions and compared with the conventional dirk methods of norsett and crouzeix

trigonometrically fitted multi step rkn methods for second order oscillatory initial value problems abstract trigonometrically fitted multi step runge kutta nystrom tfmsrkn methods for solving numerically oscillatory special second order initial value problems are introduced tfmsrkn methods integrate exactly the differential system whose solutions can be expressed as the linear combinations of functions from the set exp i w t exp i w t or equivalently the set cos wt sin wt where w represents an approximation of the main frequency of the problem the corresponding order conditions are given and two explicit four stage tfmsrkn methods with order five are constructed stability of the new methods is examined and the regions of stability are depicted numerical results show that our new methods are more efficient in comparison with other well known high quality methods proposed in the scientific literature

embedded diagonally implicit runge kutta nystrom 4 3 pair for solving special second order ivps abstract in this paper third order 3 stage diagonally implicit runge kutta nystrom method embedded in fourth order 4 stage for solving special second order initial value problems is constructed the method has the property of minimized local truncation error as well as the last row of the coefficient matrix is equal to the vector output the stability of the method is investigated and a standard set of test problems are tested upon and comparisons on the numerical results are made when the same set of test problems are reduced to first order system and solved using existing runge kutta method the results clearly shown the advantage and the efficiency of the new method

a note on a diagonally implicit runge kutta nystro m method abstract it is shown that it is possible to obtain fourth order accurate diagonally implicit runge kutta nystrom methods with only 2 stages the scheme with the largest interval of periodicity i e 0 12 is given furthermore the requirement of p stability decreases the order to 2

structures for m channel perfect reconstruction fir qmf banks which yield linear phase analysis filters the authors develop structures for fir finite impulse response perfect reconstruction qmf quadrature mirror filter banks which cover a subclass of systems that yield linear phase analysis filters for arbitrary m the parameters of these structures can be optimized to design analysis filters with minimum stopband energy which at the same time have linear phase and satisfy the perfect reconstruction property if there are m subbands then depending on whether the coefficients h sub k n of each analysis filter are symmetric or antisymmetric several combinations of filter banks are possible some of these permit perfect reconstruction and some do not for a given m the authors develop a formula for the number of combinations for a subclass of linear phase perfect reconstruction structures as an example they elaborate on a perfect reconstruction linear phase lattice structure for three channels the lattice structure is such that regardless of the parameter values the qmf bank enjoys the perfect reconstruction property while at the same time the analysis filters have linear phase a design example is presented along with tables of impulse response coefficients

improved technique for design of perfect reconstruction fir qmf banks with lossless polyphase matrices a technique is developed for the design of analysis filters in an m channel maximally decimated perfect reconstruction finite impulse response quadrature mirror filter fir qmf bank that has a lossless polyphase component matrix e z the aim is to optimize the parameters characterizing e z until the sum of the stopband energies of the analysis filters is minimized there are four novel elements in the procedure reported here the first is a technique for efficient initialization of one of the m analysis filters as a spectral factor of an mth band filter the factorization itself is done in an efficient manner using the eigenfilters approach without the need for root finding techniques the second element is the initialization of the internal parameters which characterize e z based on the above spectral factor the third element is a modified characterization mostly free from rotation angles of the fir e z the fourth is the incorporation of symmetry among the analysis filters so as to minimize the number of unknown parameters being optimized the resulting design procedure always gives better filter responses than earlier ones for a given filter length and converges much faster

diagonally implicit runge kutta nystro m methods for oscillatory problems implicit runge kutta nystrom rkn methods are constructed for the integration of second order differential equations possessing an oscillatory solution based on a linear homogeneous test model we analyse the phase errors or dispersion introduced by these methods and derive so called dispersion relations diagonally implicit rkn methods of relatively low algebraic order are constructed which have a high order of dispersion up to 10 application of these methods to a number of test examples linear as well as nonlinear yields a greatly reduced phase error when compared with conventional dirkn methods

stability of runge kutta methods for the generalized pantograph equation this paper deals with stability properties of runge kutta rk methods applied to a non autonomous delay differential equation dde with a constant delay which is obtained from the so called generalized pantograph equation an autonomous dde with a variable delay by a change of the independent variable it is shown that in the case where the rk matrix is regular stability properties of the rk method for the dde are derived from those for a difference equation which are examined by similar techniques to those in the case of autonomous ddes with a constant delay as a result it is shown that some rk methods based on classical quadrature have a superior stability property with respect to the generalized pantograph equation stability of algebraically stable natural rk methods is also considered

lattice structures for optimal design and robust implementation of two channel perfect reconstruction qmf banks a lattice structure and an algorithm are presented for the design of two channel qmf quadrature mirror filter banks satisfying a sufficient condition for perfect reconstruction the structure inherently has the perfect reconstruction property while the algorithm ensures a good stopband attenuation for each of the analysis filters implementations of such lattice structures are robust in the sense that the perfect reconstruction property is preserved in spite of coefficient quantization the lattice structure has the hierarchical property that a higher order perfect reconstruction qmf bank can be obtained from a lower order perfect reconstruction qmf bank simply by adding more lattice sections several numerical examples are provided in the form of design tables

h stability of runge kutta methods with general variable stepsize for pantograph equation this paper deals with h stability of runge kutta methods with variable stepsize for pantograph equations it is shown that the runge kutta methods with a regular matrix a are h stable if and only if the modulus of stability function at infinity is less than 1 furthermore the same conclusion can be obtained if the runge kutta methods are stiffly accurate

on using explicit runge kutta nystrom methods for the treatment of retarded differential equations with periodic solutions in this work we dial with the treatment of second order retarded differential equations with periodic solutions by explicit runge kutta nystrom methods in the past such methods have not been studied for this class of problems we refer to the underline theory and study the behavior of various methods proposed in the literature when coupled with hermite interpolants among them we consider methods having the characteristic of phase lag order then we consider continuous extensions of the methods to treat the retarded part of the problem finally we construct scaled extensions and high order interpolants for rkn pairs which have better characteristics compared to analogous methods proposed in the literature in all cases numerical tests and comparisons are done over various test problems

multiderivative methods for periodic initial value problems a family of two step multiderivative methods is developed for the numerical solution of periodic initial value problems of the form bf y t bf f t bf y with initial conditions bf y t 0 bf y 0 bf y t 0 bf y 0 given the methods are developed by showing that the solution of the single test equation y t lambda 2 y t lambda real satisfies a recurrence relation in which exponential functions are involved the methods of the family are then derived by replacing the exponential terms by pade approximants the methods are analyzed for accuracy and periodicity properties the error constants and intervals of periodicity are contained in an appendix those pade approximants for which the degree of the numerator do not exceed the degree of the denominator are seen to be p stable suitable predictor corrector combinations of the methods are analyzed for use in pece mode the methods are compared with existing methods and they are tested on four problems from the

a low order embedded runge kutta method for periodic initial value problems sideridis a b and t e simos a low order embedded runge kutta method for periodic initial value problems journal of computational and applied mathematics 44 1992 23 5 244 an embedded runge kutta fehlberg method is developed it should be noted that this embedded method is procuced using the runge kutta fehlberg method with algebraic order four to estimate a truncation phase lag error of algebraic order three the numerical results indicate that this new method is efficient for the numerical solution of differential equations with periodic solution using variable stepsize

on neutral functional differential equations with proportional delays abstract in this paper we develop a comprehensive theory on the well posedness of the initial value problem for the neutral functional differential equation y t ay t i 1 b i y q i t i 1 c i y p i t t 0 y 0 y 0 and the asymptotic behaviour of its solutions we prove that the existence and uniqueness of solutions depend mainly on the coefficientsci i 1 2 and on the smoothness of functions in the solution space as far as the asymptotic behaviour of analytic solutions is concerned thecihave little effect we prove that if re a 0 then the solutiony t either grows exponentially or is polynomial the most interesting result is that if re a 0 anda 0 then the asymptotic behaviour of the solution depends mainly on the characteristic equation a i 1 b i q i 0 these results can be generalized to systems of equations finally we present some examples to illustrate the change of asymptotic behaviour in response to the variation of some parameters the main idea used in this paper is to express the solution in either dirichlet or dirichlet taylor series form

adaptive two step methods for semidiscretized wave equations adaptive two step direct integration methods are constructed for the integration of second order semidiscrete evolution equations possessing oscillatory solutions the methods are based on a class of adaptive multistep methods for a semilinear test model whose frequency is known they are constructed following the notion of diagonally implicit rk methods by using efficient rational approximations to cos v v 0 our interest is centered on the dispersion or phase errors of the dominant components in the numerical oscillations when these methods are applied to a linear homogeneous test model two step methods which have high order of dispersion up to 12 whereas the algebraic order is relatively low 2 or 4 are derived applications of these methods to linear as well as non linear test models and to semidiscretized hyperbolic equations reveal a good behaviour with regard to error propagation when they are compared with other conventional methods

quadrature mirror filter design in the time domain a new technique for designing quadrature mirror filters is described the formulation carried out in the time domain is shown to result in an optimization problem requiring minimization of a quartic multinomial an iterative solution is suggested which involves computation of the eigenvector of a matrix with a dimensionality equal to one half the number of filter taps our experiments show that convergence to the optimum tap weights is stable and the accuracy of the final solution is limited only by the accuracy of the eigenvalue eigenvector routine as in an earlier technique the user can specify the stop band frequency the relative weights of the passband ripple energy and the stopband residual energy and of course the number of filter taps

on the digital filter associated with daubechies wavelets the magnitude of the filters associated with daubechies wavelets sub n spl psi is shown to monotonically converge to an ideal highpass filter when n spl rarr spl infin the rate of the convergence is also given the magnitude of each filter is shown to be monotonically increasing from 0 spl pi the convergence does not have gibbs phenomenon

stability of a class of runge kutta methods for a family of pantograph equations of neutral type this paper deals with the stability of runge kutta methods for a class of neutral infinite delay differential equations with different proportional delays under suitable conditions the asymptotic stability of some runge kutta methods with variable stepsize are considered by the stability function at infinity it is proved that the even stage gauss legendre methods are not asymptotically stable but the radau ia methods radau iia methods and lobatto iiic methods are all asymptotically stable furthermore some numerical experiments are given to demonstrate the main conclusions

efficient iterative design method for cosine modulated qmf banks an iterative algorithm for the design of multichannel cosine modulated quadrature mirror image filter qmf banks with near perfect reconstruction is proposed the objective function is formulated as a quadratic function in each step whose minimum point can be obtained using a closed form solution this approach has high design efficiency and leads to filter banks with high stopband attenuation and low aliasing and amplitude distortions the proposed approach is then extended to the design of multichannel cosine modulated qmf banks with low reconstruction delays which are often required especially in real time applications several design examples are included to demonstrate the proposed algorithms and some comparisons are made with existing designs

continuous runge kutta nystrom methods for initial value problems with periodic solutions abstract in the present work we are concerned with the derivation of continuous runge kutta nystrom methods for the numerical treatment of second order ordinary differential equations with nystrom methods for the numerical treatment of second order ordinary differential equations with periodic solutions numerical methods used for solving such problems are better to have the characteristics of high phase lag order first we analyse the construction algorithm for a high phase lag order scaled extension of an explicit runge kutta nystrom method using this procedure we manage to construct a phase lag order 14 continuous extension of a popular nine stages 8 6 order erkn pair in the literature only phase lag order 12 continuous extension of nine stage 8 6 erkn pairs can be found so the proposed scaling method has the higher until now dispersion order numerical tests for the proposed methods are done over various test problems

explicit two step methods with minimal phase lag for the numerical integration of special second order initial value problems and their application to the one dimensional schro dinger equation abstract two step sixth order methods with phase lag of order eight ten and twelve are developed for the numerical integration of the special second order initial value sproblem an application to the one dimensional schrodinger equation on the resonance problem indicates that these new methods are generally more accurate than methods developed by chawla rao and neta this journal 1986 1987

efficient iterations for gauss methods on second order problems we consider some important aspects about the implementation of high order implicit formulas specially the gauss methods for solving second order differential systems having high frequencies and small amplitudes superimposed the choice of an appropriate iterative scheme is discussed in detail important topics about the predictors initial guesses are analyzed and a variable order strategy to select the best predictor at each integration step is supplied a few numerical experiments on some standard test problems confirm the theory presented

on the generation of p stable exponentially fitted runge kutta nystrom methods by exponentially fitted runge kutta methods this paper provides an investigation of the stability properties of a family of exponentially fitted runge kutta nystrom efrkn methods p stability is a very important property usually demanded for the numerical solution of stiff oscillatory second order initial value problems p stable efrkn methods with arbitrary high order are presented in this work we have proved our results based on a symmetry argument