chebyshev hankel matrices and the splitting approach for centrosymmetric toeplitz plus hankel matrices abstract centrosymmetric toeplitz plus hankel matrices are investigated on the basis of their splitting property which is their similarity to the direct sum of two special toeplitz plus hankel matrices these matrices can be considered as hankel matrices moment matrices in bases of chebyshev polynomials and are called chebyshev hankel matrices chebyshev hankel matrices have similar properties like hankel matrices this concerns inversion formulas and fast algorithms a superfast algorithm for solving chebyshev hankel and centrosymmetric toeplitz plus hankel systems is presented that is based on real trigonometric transforms the main tool of investigation is the interpretation of chebyshev hankel matrices as matrices of restricted multiplication operators with respect to chebyshev bases

fast exact multiplication by the hessian just storing the hessian h the matrix of second derivatives 2e wi wj of the error e with respect to each pair of weights of a large neural network is difficult since a common use of a large matrix like h is to compute its product with various vectors we derive a technique that directly calculates hv where v is an arbitrary vector to calculate hv we first define a differential operator rv f w r f w rv r 0 note that rv w hv and rv w v and then apply rv to the equations used to compute w the result is an exact and numerically stable procedure for computing hv which takes about as much computation and is about as local as a gradient evaluation we then apply the technique to a one pass gradient calculation algorithm backpropagation a relaxation gradient calculation algorithm recurrent backpropagation and two stochastic gradient calculation algorithms boltzmann machines and weight perturbation finally we show that this technique can be used at the heart of many iterative techniques for computing various properties of h obviating any need to calculate the full hessian

differential and integral methods for multidimensional inverse scattering problems a layer stripping procedure for solving three dimensional schrodinger equation inverse scattering problems is developed this procedure operates by recursively reconstructing the radon transform of the potential from the jump in the radon transform of the scattered field at the wave front this reconstructed potential is then used to propagate the wave front and scattered field differentially further into the support of the potential the connections between this differential procedure and integral equation procedures are then illustrated by the derivations of two well known exact integral equation procedures using the radon transform and a generalized radon transform these procedures as well as the layer stripping procedure are then reduced to the familiar born approximation result for this problem by neglecting multiple scattering events this illustrates the central role of the radon transform in both exact and approximate inversion procedures

scattering by a potential using hyperbolic methods etude du probleme de diffusion inverse developpement d un algorithme correspondant pour deux variables

nearly optimal control laws for nonlinear systems with saturating actuators using a neural network hjb approach the hamilton jacobi bellman hjb equation corresponding to constrained control is formulated using a suitable nonquadratic functional it is shown that the constrained optimal control law has the largest region of asymptotic stability ras the value function of this hjb equation is solved for by solving for a sequence of cost functions satisfying a sequence of lyapunov equations le a neural network is used to approximate the cost function associated with each le using the method of least squares on a well defined region of attraction of an initial stabilizing controller as the order of the neural network is increased the least squares solution of the hjb equation converges uniformly to the exact solution of the inherently nonlinear hjb equation associated with the saturating control inputs the result is a nearly optimal constrained state feedback controller that has been tuned a priori off line

reinforcement learning in continuous time and space this article presents a reinforcement learning framework for continuous time dynamical systems without a priori discretization of time state and action based on the hamilton jacobi bellman hjb equation for infinite horizon discounted reward problems we derive algorithms for estimating value functions and improving policies with the use of function approximators the processof value function estimation is formulated asthe minimization of a continuous time form of the temporal difference td error update methods based on backward euler approximation and exponential eligibility traces are derived and their correspondences with the conventional residual gradient td 0 and td lambda algorithms are shown for policy improvement two methods a continuous actor critic method and a value gradient based greedy policy are formulated as a special case of the latter a nonlinear feedback control law using the value gradient and the model of the input gain is derived the advantage updating a model free algorithm derived previously is also formulated in the hjbbased framework the performance of the proposed algorithms is first tested in a nonlinear control task of swinging a pendulum up with limited torque it is shown in the simulations that 1 the task is accomplished by the continuous actor critic method in a number of trials several times fewer than by the conventional discrete actor critic method 2 among the continuous policy update methods the value gradient based policy with a known or learned dynamic model performs several times better than the actor critic method and 3 a value function update using exponential eligibility traces is more efficient and stable than that based on euler approximation the algorithms are then tested in a higher dimensional task cartpole swing up this task is accomplished in several hundred trials using the value gradient based policy with a learned dynamic model

a kalman filter solution of the inverse scattering problem with a rational reflection coefficient abstract this paper presents a new inverse scattering method for reconstructing the reflectivity function of symmetric two component wave equations or the potential of a schrodinger equation when the reflection coefficient is rational this method relies on the so called chandrasekhar equations which implement the kalman filter associated to a stationary state space model these equations are derived by using first a general layer stripping principle to obtain some differential equations for reconstructing a general scattering medium and by specializing these recursions to the case when the probing waves have a state space model

goupillaud inverse problem with arbitrary input a new impulse inverse formula and it s generalization including arbitrary input are proposed the new generalized inverse formula has the advantage of not being necessary to specify the first digital value of the input numerical instability of the inverse formula for arbitrary input is investigated carefully an analysis of the perturbed stability and lyapunov exponent are carried out as the result of this analysis it is shown that the estimated error is sensitive to the experimental error and previously estimated error as a sequential algorithm simulation using experimental data is also shown in the paper

the stabilized probability plot summary the stabilized probability plot is introduced an attractive feature of the plot that enhances its interpretability is that the variances of the plotted points are approximately equal this prompts the definition of a new and powerful goodness of fit statistic dsp which analogous to the standard kolmogorov smirnov statistic d is defined to be the maximum deviation of the plotted points from their theoretical values using either d or dsp it is shown how to construct acceptance regions for qq pp and the new plots acceptance regions can help remove much of the subjectivity from the interpretation of these probability plots

fast matrix factorizations via discrete transmission lines abstract fast algorithms associated with the names of schur and levinson are known for the triangular factorization of symmetric positive definite toeplitz matrices and their inverses in this paper we show that these algorithms can be derived from simple arguments involving causality symmetry and energy conservation in discrete lossless transmission lines the results not only provide a nice interpretation of the classical schur and levinson algorithms and a certain toeplitz inversion formula of gohberg and semencul but they also show immediately that the same fast algorithms apply not only to toeplitz matrices but to all matrices with so called displacement inertia 1 1 the results have been helpful in suggesting new digital filter structures and in the study of nonstationary second order processes

the one dimensional inverse problem of reflection seismology an impulsive normal traction is applied uniformly over the surface of a perfectly stratified plane layered earth the ensuing particle velocity at the surface is assumed to be measured the problem of calculating the subsurface characteristic impedance from knowledge of the input pulse and the measured data is the one dimensional inverse problem of reflection seismology the problem is set up mathematically as an inverse problem for a first order 2 times 2 hyperbolic system the role of propagation of singularities is explained and the relation between the solution of the inverse problem and the cholesky factorization of certain matrices constructed from the data is elaborated for both the continuum problem and the related discrete problem it is found that recursive numerical techniques such as the levinson algorithm and certain other fast methods for cholesky factorization are intimately related to explicit finite difference schemes for solving hyperbolic systems the order of approximation between the

source wavelet and its angular spectrum from plane wave seismograms in a typical seismic experiment land or marine the generated source wavelet is either unknown or known only approximately in order to solve many geophysical problems an accurate wavelet estimation is crucial this paper demonstrates a fast numerical algorithm which allows not only estimating the source wavelet but also its angular spectrum radiation pattern from plane wave decomposed seismograms surprisingly literature on this subject is virtually missing although a nonuniform angular spectrum of the generated wavelet can substantially affect the recorded data the technique presented here is based on the downward continuation dc and reverse time extrapolation rte of the recorded data both pressure and its vertical derivative the proposed method is iterative the rate of convergence does not depend on phase characteristics of the generated wavelet we demonstrate that both the wavelet signature and its angular spectrum can be estimated accurately from plane wave seismograms we also illu

variable background born inversion by wavefield backpropagation the inverse scattering problem for an acoustic medium is formulated by using the variable background born approximation a constant density acoustic medium is probed by a wide band point source and the scattered field is observed along a curved receiver array located outside the region where the medium velocity is different from the assumed background velocity function the solution proposed relies on the introduction of a backpropagated field this field is obtained by using a finite difference scheme backward in time to backpropagate into the medium the scattered field observed along the receiver array the backpropagated field is imaged at the source travel times giving an image of the same type as obtained by reverse time finite difference migration techniques the gradient of this image is then taken along rays linking the source to points in the medium and after scaling this gives the reconstructed potential to relate the reconstructed potential to the true scattering potential high frequency

rotation number associated with difference equations satisfied by polynomials orthogonal on the unit circle abstract the techniques of topological dynamics and differential dynamical systems are used to study polynomials orthogonal with respect to a measure supported on the unit circle it is assumed that the reflection coefficients associated with these polynomials form a stationary stochastic ergodic process in particular the techniques mentioned above are used to prove a gap labelling result

n player statistical nash game control m th cost cumulant optimization in this note we study a statistical nash game control problem where multiple players optimize system performance by shaping the distribution of their cost functions through cost cumulants our focus is on the nonzero sum dynamic statistical nash game which generalizes the risk sensitive nash game the necessary and sufficient conditions for the nash equilibrium solution are given for the n player m th cost cumulant optimization using the hamilton jacobi bellman hjb equations in addition we derive the optimal control strategies and provide simulation results to demonstrate the performance of our method

acoustic impulse responses for nonuniform media abstract the acoustic impulse responses of a family of nonuniform lossless one dimensional media are determined the nonuniform portions of the media are confined to layers of either finite or infinite thicknesses the acoustic impedances of the nonuniform layers as functions of travel time are assumed known the impulse responses are determined by evaluating the fourier transforms of their reflection coefficients using infinite series expansions and contour integrations

fast algorithms for estimation and signal processing an inverse scattering framework the inverse scattering problem for a lossless transmission line has been used as a framework for deriving the levinson and schur fast algorithms for toeplitz matrices here the same framework is used to derive and attach novel simple scattering interpretations to the lattice algorithm and the split algorithms this demonstrates that the scattering and interpretation is more fundamental to fast signal processing algorithms than previously noted

exact analytical solutions for elastodynamic impact abstract we consider the one dimensional impact problem in which a semi infinite flyer collides with and adheres to the front face of a stationary target plate of finite thickness with the back face of the target bonded to another semi infinite medium all three bodies are assumed to be linear elastic and homogeneous our interest is in explicit expressions for the stress and velocity in the target at all times after impact the analysis of this problem is simplified by reducing it to an initial boundary value problem for the target only which is solved by combining the d alembert solution of the wave equation with the laplace transform method an appropriate impact boundary condition is required on the front face of the target in the literature this is usually taken as a prescribed step in stress or velocity but the correct boundary condition involves a linear combination of the unknown stress and velocity at the impact face our solutions are expressed in an apparently new compact form involving the floor or greatest integer function the results are amenable to asymptotic analysis in particular solutions for stress free or rigid back faces follow easily as limiting cases of the backing impedance and the long time asymptotes of stress and velocity in the target are seen to be independent of the target s elastic properties all of our results are corroborated by derivation of exact discrete solutions from recursive equations for the impact problems

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

tests for the exponential weibull and gumbel distributions based on the stabilized probability plot goodness of fit procedures for composite testing of the exponential weibull and gumbel distributions are investigated these procedures are such that they can be used graphically with and to aid interpretation of probability plots in the situations of interest here the new procedure is generally more powerful than the kolmogorovsmirnov test which appears to be the only other readily available goodness of fit procedure that can be used graphically with probability plots

an iterative method for the inversion of the two dimensional wave equation with a potential a numerical method for a nonlinear inversion problem for the 2d wave equation with a potential is discussed in order to avoid the ill posedness we substitute a coupled system of one way wave equations for the original wave equation an iterative algorithm is constructed to improve the accuracy of the inversion numerical experiments are performed on several examples to examine the effectiveness of this method

layer stripping solutions of multidimensional inverse scattering problems a layer stripping procedure for solving three dimensional schrodinger equation inverse scattering problems is developed this method operates by recursively reconstructing the potential from the jump in the scattered field at the wave front and then using the reconstructed potential to propagate the wave front and the scattered field further into the inhomogeneous region it is thus a generalization of algorithms that have been developed for one dimensional inverse scattering problems although the procedure has not yet been numerically tested the corresponding one dimensional algorithms have performed well on synthetic data the procedure is applied to a two dimensional inverse seismic problem connections between simplifications of this method and born approximation inverse scattering methods are also noted

reconstruction of a spherically symmetric speed of sound consider the inverse acoustic scattering problem for a spherically symmetric inhomogeneity of compact support that arises among other places in nondestructive testing define the corresponding homogeneous and inhomogeneous interior transmission problems see e g d colton and p monk quart j mech math 41 1988 pp 97 125 here the authors study the subset of transmission eigenvalues corresponding to spherically symmetric eigenfunctions of the homogeneous interior transmission problem it is shown in mclaughlin and polyakov j differential equations to appear that these eigenvalues are the zeros of an average of the scattering amplitude and a uniqueness theorem for the inverse acoustic scattering problem is presented where these eigenvalues are the given data in the present paper an algorithm for finding the solution of the inverse acoustic scattering problem from this subset of transmission eigenvalues is developed and implemented the method given here completely determines the sound

successive galerkin approximation algorithms for nonlinear optimal and robust control nonlinear optimal control and nonlinear h infinity control are two of the most significant paradigms in nonlinear systems theory unfortunately these problems require the solution of hamilton jacobi equations which are extremely difficult to solve in practice to make matters worse approximation techniques for these equations are inherently prone to the so called curse of dimensionality while there have been many attempts to approximate these equations solutions resulting in closed loop control with well defined stability and robustness have remained elusive this paper describes a recent breakthrough in approximating the hamilton jacobi bellman and hamilton jacobi isaacs equations successive approximation and galerkin approximation methods are combined to derive a novel algorithm that produces stabilizing closed loop control laws with well defined stability regions in addition we show how the structure of the algorithm can be exploited to reduce the amount of computation from exponential to po

on goodness of fit tests for the two parameter weibull distribution derived from the stabilized probability plot summary michael s statistic dsp based on the stabilized probability plot is considered for the assessment of goodness of fit of a two parameter weibull distribution this statistic is shown to have high power when the parameters are estimated using a procedure due to blom 1958 the correlation coefficient of the stabilized plot rsp is also proposed as a test statistic for this situation finally a family of transformations is obtained to reduce the null distribution of rsp to standard normality

fast algorithms for toeplitz and hankel matrices abstract the paper gives a self contained survey of fast algorithms for solving linear systems of equations with toeplitz or hankel coefficient matrices it is written in the style of a textbook algorithms of levinson type and schur type are discussed their connections with triangular factorizations pade recursions and lanczos methods are demonstrated in the case in which the matrices possess additional symmetry properties split algorithms are designed and their relations to butterfly factorizations are developed

on a neural approximator to odes the object of this brief is to present and analyze the training of a single layer neural network in order to solve ordinary differential equations odes properties of the approximator are derived and some examples of its application are shown

biometrika centenary theory and general methodology note reprinted in biometrika one hundred years edited by d m titterington and d r cox oxford university press 11 50 reference stat article 2001 002doi 10 1093 biomet 88 1 13view record in web of science record created on 2006 04 21 modified on 2017 05 12

inverse problem in nonstationary multidimensional medium the problem of a scalar wave propagation from the point impulsive source in the layer of a nonstationary multidimensional medium is considered the boundary problem for the wave equation is reformulated in the problem with the initial condition using the invariant imbedding method the integral differential inverse procedures of the various orders were obtained from the imbedding equations using the singularities method the order of inverse procedure is defined by the degree of a polynomial in the analytical representation of the medium characteristic near the layer boundary it was shown that the coefficients of the polynomial are calculated with the help of the differential characteristics of the point impulsive source in the inhomogeneous medium the cause and character of the multidimensional inverse problem overdefiniteness are considered the application of the proposed procedure for a statistical problem is discussed

online optimal control of affine nonlinear discrete time systems with unknown internal dynamics by using time based policy update in this paper the hamilton jacobi bellman equation is solved forward in time for the optimal control of a class of general affine nonlinear discrete time systems without using value and policy iterations the proposed approach referred to as adaptive dynamic programming uses two neural networks nns to solve the infinite horizon optimal regulation control of affine nonlinear discrete time systems in the presence of unknown internal dynamics and a known control coefficient matrix one nn approximates the cost function and is referred to as the critic nn while the second nn generates the control input and is referred to as the action nn the cost function and policy are updated once at the sampling instant and thus the proposed approach can be referred to as time based adp novel update laws for tuning the unknown weights of the nns online are derived lyapunov techniques are used to show that all signals are uniformly ultimately bounded and that the approximated control signal approaches the optimal control input with small bounded error over time in the absence of disturbances an optimal control is demonstrated simulation results are included to show the effectiveness of the approach the end result is the systematic design of an optimal controller with guaranteed convergence that is suitable for hardware implementation

matrix representations of split bezoutians abstract inverses of symmetric or skewsymmetric toeplitz matrices as well as of centrosymmetric or centro skewsymmetric toeplitz plus hankel matrices can be represented as sums of two split bezoutians which are highly structured matrices since all of their rows and columns are symmetric or skewsymmetric vectors thus it is desirable to find matrix representations for split bezoutians b this is the main aim of the present paper recursion formulas for the entries of b are presented bases of very simple split bezoutians or of sparse matrices are constructed and b is represented as a corresponding linear combination moreover matrix representations of gohberg semencul type are established

least squares solutions of the hjb equation with neural network value function approximators in this paper we present an empirical study of iterative least squares minimization of the hamilton jacobi bellman hjb residual with a neural network nn approximation of the value function although the nonlinearities in the optimal control problem and nn approximator preclude theoretical guarantees and raise concerns of numerical instabilities we present two simple methods for promoting convergence the effectiveness of which is presented in a series of experiments the first method involves the gradual increase of the horizon time scale with a corresponding gradual increase in value function complexity the second method involves the assumption of stochastic dynamics which introduces a regularizing second derivative term to the hjb equation a gradual reduction of this term provides further stabilization of the convergence we demonstrate the solution of several problems including the 4d inverted pendulum system with bounded control our approach requires no initial stabilizing policy or any restrictive assumptions on the plant or cost function only knowledge of the plant dynamics in the appendix we provide the equations for first and second order differential backpropagation

estimation of coefficients and boundary parameters in hyperbolic systems we consider semi discrete galerkin approximation schemes in connection with inverse problems for the estimation of spatially varying coefficients and boundary condition parameters in second order hyperbolic systems typical of those arising in 1 d surface seismic problems spline based algorithms are proposed for which theoretical convergence results along with a representative sample of numerical findings are glven

inversion of p sv seismic data in a layered elastic material density shear velocity and compressional velocity can be found at any depth from broadband surface measurements at two distinct nonzero precritical values of plane wave incidence angle layer stripping inversion uses three component surface velocity measurements generated by a polarized surface source to determine subsurface properties incrementally the surface velocity measurements initialize a first order nonlinear matrix riccati equation derived from the elastic wave equation which takes advantage of an attractive fixed point condition in the complex frequency plane to extract subsurface mechanical impedances subsurface density and velocities are recovered from the inverted impedances at two or more plane wave incidence angles general properties of the matrix riccati equation in the complex frequency plane aid in incorporating bandwidth constraints inversion of synthetic plane wave data from a piece wise continuous model illustrates inversion effects when only

analytical methods for recovering coefficients in differential equations from spectral data in this paper analytical methods for recovering coefficients from spectral data are discussed properties of these methods are given and mathematical models for which these methods might apply are presented

zhang neural network for online solution of time varying linear matrix inequality aided with an equality conversion in this paper for online solution of time varying linear matrix inequality lmi such an lmi is first converted to a time varying matrix equation by introducing a time varying matrix of which each element is greater than or equal to zero then by employing zhang s neural dynamic method a special recurrent neural network termed zhang neural network znn is proposed and investigated for solving online the converted time varying matrix equation as well as the time varying lmi such a znn model showed in an explicit dynamics exploits the time derivative information of time varying coefficients in addition theoretical analysis and results of the proposed znn model are discussed and presented to show its excellent performance on solving the time varying lmi computer simulation results further demonstrate the efficacy of the proposed znn model for online solution of the time varying lmi and the converted time varying matrix equation

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

iterative trace deconvolution and noncausal transform for processing band limited data in 1983 carrion and patton showed that if recorded seismic data do not have low frequencies in their spectrum inversion for acoustic impedance is unstable this concept is commonly accepted here we infer acoustic impedance from band limited data without low frequencies using iterative trace deconvolution with the so called noncausal projection which moves initial data samples corresponding to small eigenvalues to negative times we show that this procedure can solve ill posed problems by migrating large residuals uncertainties in the trend of the recovered impedance to the bottom of the model an important property of the proposed method is that it converges to the true solution independently of the chosen initial model another advantage of the proposed algorithm is that unlike conventional dynamic deconvolution characteristic integration schemes it increases neither errors nor the condition number with depth the method is illustrated on synthetic and real data

convergence of numerical inversion methods for discontinuous impedance profiles we study numerical methods for the one dimensional inverse problem of reflection seismology an impulsive plane wave in pressure is applied to the surface of a piecewise smooth stratified elastic half space and the resulting particle velocity is measured at the surface the characteristic impedance of the medium is to be recovered as a function of travel time we consider two methods for the numerical solution of this inverse problem a discretization of the gopinath sondhi integral equation and a downward continuation method both methods are exact for so called goupillaud layered media both use second order difference schemes to reconstruct a solution of the system of partial differential equations governing motion in the medium a formally second order accurate equation based on the theory of geometrical acoustics is used to recover the impedance by analyzing the discretization of the gopinath sondhi integral equation we prove that if the impulse response is measured by sampling the step response or

z transform methods for the optimal design of one dimensional layered elastic media in this work we develop a finite trigonometric series representation for the stress in a multilayered goupillaud type elastic strip subjected to transient heaviside loading on one end while the other end is held fixed this representation is achieved by means of the z transform method and involves the so called base angles generally different layered designs could share the same set of base angles and the more layers the design has the more base angles are expected necessary conditions for the base angles and design parameters for any given design are described as a result of the stress representation we are able to identify optimal layered designs which provide the smallest stress amplitude for two and three layered designs for which the coefficients of the stress representation are easy to find the optimization results are achieved using a custom made discrete optimization technique applied in a p velo and g a gazonas int j solids structures 40 2003 pp 6417 6428 for other laye

some nniqueness results related to soil and building structural identification the need for accurate predictions of the dynamic responses of soil and structural systems has led to the widespread use of forced vibration testing for obtaining knowledge of the dynamic properties of the systems under consideration this paper is concerned with the uniqueness of the results in the identification of such properties more specifically the damping and stiffness distributions which are of importance in the linear range of response have been investigated an n storied structure or an n layered soil system is modeled as an n degree of freedom lumped mass system consisting of masses springs and dampers assuming that the mass distribution is known the conditions under which unique identification of the stiffness and damping distributions can be obtained are established several useful results on the ability to identify uniquely some but not all of the properties of the system have also been obtained the paper extends some of the known results of one dimensional reflection seismology to

asymptotic impact behavior of goupillaud type layered elastic media abstract in this paper we use the method of riemann invariants to develop a system of recurrence relations for the one dimensional elastic impact problem in which a semi infinite flyer collides with and adheres to a stationary goupillaud type layered elastic target plate of finite thickness that is bonded to a semi infinite half space we derive explicit expressions for the long time asymptotes of stress and particle velocity that are independent of the layered target s elastic properties this result is verified by numerical simulations of impact into layered targets with periodic or randomly distributed elastic properties and holds when the half space backing the target is replaced by either free or fixed boundary conditions our discrete recurrence relation solutions for impact into layered media generalize our recent analytical results concerning impact into homogeneous single layer targets as well as prior discrete recurrence relation solutions for stress wave propagation in goupillaud type layered elastic media