explicit solutions of the discrete time lyapunov matrix equation and kalman yakubovich equations a general solution for the nonsquare nonsymmetric lyapunov matrix equation in a canonical form is presented the solution is shown to be a toeplitz matrix which may be calculated using the backwards levinson algorithm this solution is then applied to the kalman yakubovich equations to derive a method for generating strictly positive real functions via the positive real lemma this latter result has an application in system identification

modeling and identification of symmetric noncausal impulse responces spectrally equivalent causal sec models for single input single output noncausal systems are derived with the objective of estimating the noncausal impulse response attention is restricted to symmetric noncausal impulse responses and the measurements of the system output are assumed to be noisy it is shown that by using structured sec models either state space or arma plus noise consistent estimates of the noncausal impulse response can be obtained by exploiting only the second order statistics of the measurements a new version of the correlation technique particularly suited to structured models is presented and analyzed computer simulation results are also presented to illustrate the performance of the proposed approach

closed form solutions to the nonhomogeneous yakubovich conjugate matrix equation some closed form solutions are provided for the nonhomogeneous yakubovich conjugate matrix equation email protected f by r with x and y being unknown matrices the presented solutions can offer all the degrees of freedom which is represented by an arbitrarily chosen parameter matrix the primary feature of the solutions is that the matrices f and r are not restricted to be in any canonical form or may be even unknown a priori one of the solutions is neatly expressed in terms of controllability matrices and observability matrices

efficient iterative methods applied to the solution of transonic flows we investigate the use of an inexact newton s method to solve the potential equations in the transonic regime as a test case we solve the two dimensional steady transonic small disturbance equation approximate factorization adi techniques have traditionally been employed for implicit solutions of this nonlinear equation instead we apply newton s method using an exact analytical determination of the jacobian with preconditioned conjugate gradient like iterative solvers for solution of the linear systems in each newton iteration two iterative solvers are tested a block s step version of the classical orthomin k algorithm called orthogonal s step orthomin osomin and the well known gmres method the preconditioner is a vectorizable and parallelizable version of incomplete lu ilu factorization efficiency of the newton iterative method on vector and parallel computer architectures is the main issue addressed in vectorized tests on a single processor of the cray c 90 the performance of newton osomin is superior to newton gmres and a more traditional monotone af adi method maf for a variety of transonic mach numbers and mesh sizes newton gmres is superior to maf for some cases the parallel performance of the newton method is also found to be very good on multiple processors of the cray c 90 and on the massively parallel thinking machine cm 5 where very fast execution rates up to 9 gflops are found for large problems

diagonal matrix solutions of a discrete time lyapunov inequality diagonal solutions of a lyapunov inequality for companion matrices are studied such solutions are required if states of a discrete time linear system are computed with a finite precision arithmetic

on iterative solutions of general coupled matrix equations in this paper we study coupled matrix equations which are encountered in many systems and control applications first we extend the well known jacobi and gauss seidel iterations and present a large family of iterative methods which are then applied to develop iterative solutions to coupled sylvester matrix equations the basic idea is to regard the unknown matrices to be solved as parameters of a system to be identified and to obtain the iterative solutions by applying a hierarchical identification principle next we generalize the sylvester equations to general coupled matrix equations and propose a gradient based iterative algorithm for the solutions using a block matrix inner product the star star product we prove that the iterative algorithm always converges to the unique solutions for any initial values one advantage of the algorithms proposed is that they require less storage space in implementation than existing numerical methods finally we test the algorithms and show their effectiveness using numerical examples

oscillatory and asymptotic behavior of delay differential equations with a nonlinear damping term in this paper we study the oscillatory and asymptotic behavior of the solutions of delay differential equations of the form x n t f t x t x t 0 n even by comparing with certain differential equations of the same or lower order whose behavior is known the obtained criteria can be applied to the delay differential equation x n t q t x t x t sign x t 0 n even where and are positive constants

identification of linear stochastic systems via second and fourth order cumulant matching the identification problem for time invariant single input single output linear stochastic systems driven by non gaussian white noise is considered the system is not restricted to be minimum phase and it is allowed to contain all pass components a least squares criterion that involves matching the second and the fourth order cumulant functions of the noisy observations is proposed knowledge of the probability distribution of the driving noise is not required an order determination criterion that is a modification of the akaike information criterion is also proposed strong consistency of the proposed estimator is proved under certain sufficient conditions simulation results are presented to illustrate the method

the numerical stability analysis of pipelined conjugate gradient methods historical context and methodology algebraic solvers based on preconditioned krylov subspace methods are among the most powerful tools for large scale numerical computations in applied mathematics sciences technology as well as i

oscillation theorems for second order nonlinear functional differential equations with damping abstract new oscillation criteria are given for a second order nonlinear functional differential equation t t p t t q t g t sgn g t 0 where 1 and 0

a shift splitting hierarchical identification method for solving lyapunov matrix equations in the present paper we propose a hierarchical identification method sshi for solving lyapunov matrix equations which is based on the symmetry and skew symmetry splitting of the coefficient matrix we prove that the iterative algorithm consistently converges to the true solution for any initial values with some conditions and illustrate that the rate of convergence of the iterative solution can be enhanced by choosing the convergence factors appropriately furthermore we show that the method adopted can be easily extended to study iterative solutions of other matrix equations such as sylvester matrix equations finally we test the algorithms and show their effectiveness using numerical examples

order reduction of siso nonminimum phase stochastic systems we consider the order reduction problem for single input single output nonminimum phase linear stochastic systems the problem is formulated as that of approximating the original system output process by another stochastic process generated via a reduced order model it is shown that in order to obtain a correct phase approximation it is necessary to consider some higher order statistics of the original process in addition to the usual second order statistics a computationally feasible approximation criterion is proposed and analyzed an example is also presented to illustrate the proposed method

lyapunov bezout and hankel abstract given a polynomial f of degree n we denote by c its companion matrix and by s the truncated shift operator of order n we consider lyapunov type equations of the form x sxc w and x cxs w we derive some properties of these equations which make it possible to characterize bezoutian matrices as solutions of the first equation with suitable right hand sides w similarly for hankel and the second equation and to write down explicit expressions for these solutions this yields explicit factorization formulae for polynomials in c for the schur cohn matrix and for matrices satisfying certain intertwining relations as well as for bezoutian matrices

a generalization of s step variants of gradient methods the s step methods were proposed by chronopoulos to gain efficiency in parallel programming of iterative methods for linear systems they are variants of classical iterative methods based on the construction of a krylov subspace basis on each iteration these s step methods were inferred from algorithms like the conjugate gradient generalized conjugate residual or the minimal residual they converge for all symmetric nonsymmetric definite and some nonsymmetric indefinite matrices in this paper we introduce an s step variant of a general orthogonalization algorithm that is a generalization of s step variants of gradient methods we prove convergence and obtain error estimates we also describe an orthomin variant together with a convergence theorem from this we derive the well known s step methods as particular cases and some which are newfound to our knowledge this provides a unified framework to derive and study s step methods some of the methods obtained are convergent for every nonsingular matrix finally we give some numerical results for the new proposed methods showing that the parallel implementation of these overcomes the original ones

a parallel alternating direction implicit preconditioning method abstract the alternating direction implicit adi iterative method is an efficient iterative method to solve systems of linear equations due to its extremely fast convergence the adi method has also been used successfully as a preconditioner in some other iterative methods such as the preconditioned conjugate gradient in this paper a parallel algorithm for the adi preconditioning is proposed in this algorithm several steps of the adi iteration are computed simultaneously this means that several tridiagonal systems that are traditionally solved sequentially are now solved concurrently the high performance of this algorithm is achieved by increasing the degree of parallelism and reducing memory contention the algorithm can easily be implemented in a multiprocessor architecture experiments have been conducted on the myrias sps 2 computer with 64 processors and good performance of this algorithm is observed

on the real convergence rate of the conjugate gradient method abstract we present a parametrized class of matrices for which the rate of convergence of the conjugate gradient method varies greatly with the parameter and does not appreciably depend on the algorithm implementation a small change in the eigenvalue distribution can lead to a large change in the sensitivity of cg to rounding errors a theorem is proved which gives a necessary and sufficient condition for ordering exact arithmetic cg processes for systems with different spectra according to the energy norm of the error theorems 4 1 and 4 2 continue paige s and greenbaum s work

oscillations caused by several retarded and advanced arguments in this paper we study the oscillatory behavior of equations of the forms y t i 1npiy t i 0 and y t i 1npiy t i 0 where piand i i 1 2 n are positive constants we prove that each one of the following conditions 1 pi i1 e for some i i 1 2 n 2 i 1npi 1 e where min 1 2 n 3 i 1npi 1 n i 1n i 1 e or 4 1 n i 1n pi i 1 2 21 e implies that every solution of and oscillates a generalization in the case where the coefficientspi i 1 2 n are positive and continuous functions is also presented

s step iterative methods for symmetric linear systems abstract in this paper we introduce s step conjugate gradient method for symmetric and positive definite spd linear systems of equations and discuss its convergence in the s step conjugate gradient method iteration s new directions are formed simultaneously from r i ar i a s 1 r i and the preceding s directions all s directions are chosen to be a orthogonal to the preceding s directions the approximation to the solution is then advanced by minimizing an error functional simultaneously in all s directions this intuitively means that the progress towards the solution in one iteration of the s step method equals the progress made over s consecutive steps of the one step method this is proven to be true

on the theory of state covariance assignment for single input linear discrete systems an alternative method for state covariance assignment for discrete single input linear systems is proposed this method is based on the inverse solution of the lyapunov equation and the resulting formulas are similar in structure to the formulas for pole placement further the set of all covariance matrices assignable to a single input discrete system is also characterized the proposed method is illustrated by a numerical example

scalable asynchronous domain decomposition solvers parallel implementations of linear iterative solvers generally alternate between phases of data exchange and phases of local computation increasingly large problem sizes and more heterogeneous com

strong and weak oscillatory behavior in time varying linear difference equations of arbitrary order in the presence of unmodeled dynamics and nominal parametrical errors abstract this paper is devoted to investigate the presence of oscillating solutions in time varying difference equations of arbitrary n th order even in the case when there are parametrical errors i e errors in the sequences defining their coefficients or unmodeled dynamics i e the current order n is unknown with n n 0 the nominal known order the problem is formulated linked to the concepts of conjugacy disconjugacy positivity and generalized zeros and very general conditions of oscillation are obtained both over particular intervals and for the whole solution some results concerned with the presence of stable asymptotically stable oscillatory solutions are also presented

the minkowski lyapunov equation for linear dynamics theoretical foundations we consider the lyapunov equation for the linear dynamics which arises naturally when one seeks for a lyapunov function with a uniform exact decrease in this setting a solution to the lyapunov equation has been characterized only for quadratic lyapunov functions we demonstrate that the lyapunov equation is a well posed equation for strictly stable dynamics and a much more general class of lyapunov functions specified via minkowski functions of proper c sets which include euclidean and weighted euclidean vector norms polytopic and weighted polytopic 1 vector norms as well as vector semi norms induced by the minkowski functions of proper c sets furthermore we establish that the lyapunov equation admits a basic solution i e the unique solution within the class of minkowski functions associated with proper c sets finally we provide a characterization of the lower and upper approximations of the basic solution that converge pointwise and compactly to it while in addition the upper approximations satisfy the classical lyapunov inequality

an efficient nonsymmetric lanczos method on parallel vector computers abstract in this paper we introduce the s step biorthogonal lanczos method for finding a few eigenvalues of a large sparse nonsymmetric matrix and we prove that the s step method generates reduction matrices which are similar to reduction matrices generated by the standard method we prove that the breakdown conditions of the s step method are less stringent than the standard one one iteration of the s step biorthogonal lanczos algorithm corresponds to s iterations of the standard biorthogonal lanczos algorithm and the s step method has improved data locality and minimized global communication and superior parallel properties to the standard one on parallel machines chronopoulos and gear 1989 and kim and chronopoulos 1991 we implement the s step biorthogonal lanczos method on the cray 2 super computer and discuss the breakdown conditions and demonstrate the superior performance of the s step method to the standard one

a numerical algorithm for lyapunov equations in this paper we give a new numerical iterative method for lyapunov matrix equations which is called kronecker products iterative method kpim by means of the kronecker products we get an iterative convergent sequence from the lyapunov matrix equations at the same time we give its algorithms and corresponding improved algorithm in order to reduce the computation cost finally we discuss how to choose the optimal parameter of kpim for the continuous lyapunov matrix equations in the case of symmetry

oscillation criteria of a certain class of third order nonlinear delay differential equations with damping abstract in this paper we are concerned with the oscillation of third order nonlinear delay differential equations of the form r 2 t r 1 t y p t y q t f y g t 0 by using a generalized riccati transformation and integral averaging technique we establish some new sufficient conditions which insure that any solution of this equation oscillates or converges to zero in particular several examples are given to illustrate the importance of our results

projection minimization methods for nonsymmetric linear systems abstract we consider projection minimization methods for solving systems of linear equations we transform these methods to new ones and show that they converge faster in some sense we present in particular the transform of the norm decomposition method of gastinel the new algorithm presents advantages over some known conjugate gradient like methods this new method is studied and numerical examples are given

iterative solutions to the kalman yakubovich conjugate matrix equation abstract two operations are introduced for complex matrices in terms of these two operations an infinite series expression is obtained for the unique solution of the kalman yakubovich conjugate matrix equation based on the obtained explicit solution some iterative algorithms are given for solving this class of matrix equations convergence properties of the proposed algorithms are also analyzed by using some properties of the proposed operations for complex matrices

gradient based iterative algorithms for solving a class of matrix equations in this note we apply a hierarchical identification principle to study solving the sylvester and lyapunov matrix equations in our approach we regard the unknown matrix to be solved as system parameters to be identified and present a gradient iterative algorithm for solving the equations by minimizing certain criterion functions we prove that the iterative solution consistently converges to the true solution for any initial value and illustrate that the rate of convergence of the iterative solution can be enhanced by choosing the convergence factor or step size appropriately furthermore the iterative method is extended to solve general linear matrix equations the algorithms proposed require less storage capacity than the existing numerical ones finally the algorithms are tested on computer and the results verify the theoretical findings

iterative least squares solutions of coupled sylvester matrix equations abstract in this paper we present a general family of iterative methods to solve linear equations which includes the well known jacobi and gauss seidel iterations as its special cases the methods are extended to solve coupled sylvester matrix equations in our approach we regard the unknown matrices to be solved as the system parameters to be identified and propose a least squares iterative algorithm by applying a hierarchical identification principle and by introducing the block matrix inner product the star product for short we prove that the iterative solution consistently converges to the exact solution for any initial value the algorithms proposed require less storage capacity than the existing numerical ones finally the algorithms are tested on computer and the results verify the theoretical findings

oscillation theorems for nth order delay differential equations abstract oscillation criteria are obtained for a class of functional differential equations including the generalized emden fowler equation x n q t x g t sgn x g t 0 0 as a special case the purpose of this paper is to extend and improve known fundamental oscillation criteria for superlinear sublinear and linear differential equations with retarded arguments

oscillation criteria for second order nonlinear differential equations with nonlinear damping this paper is concerned with the oscillation of a class of general type second order differential equations with nonlinear damping several new oscillation criteria are established under quite general assumptions which improve and extend the known results in the literature

on finite precision lyapunov functions for companion matrices the finite precision stability of a discrete time system realized in companion form is shown to be closely connected to a positive real character of a related all pole minus one half transfer function this is shown as a consequence of some novel characterizations of solutions to a lyapunov equation for companion matrices the results are of practical significance as companion matrix state recursions are cheap to implement

residual replacement strategies for krylov subspace iterative methods for the convergence of true residuals in this paper a strategy is proposed for alternative computations of the residual vectors in krylov subspace methods which improves the agreement of the computed residuals and the true residuals to the level of o u a x building on earlier ideas on residual replacement and on insights in the finite precision behavior of the krylov subspace methods computable error bounds are derived for iterations that involve occasionally replacing the computed residuals by the true residuals and they are used to monitor the deviation of the two residuals and hence to select residual replacement steps so that the recurrence relations for the computed residuals which control the convergence of the method are perturbed within safe bounds numerical examples are presented to demonstrate the effectiveness of this new residual replacement scheme

kronecker maps and sylvester polynomial matrix equations the concepts of kronecker maps and sylvester polynomial matrix equations are proposed some properties of the kronecker maps are deduced it is shown that many kinds of matrix equations can be formulated into the so called sylvester polynomial matrix equations with the help of the kronecker maps general explicit solutions of the sylvester polynomial matrix equations are established when the polynomial matrices possess f primeness for the nonsquare case the proposed solutions can offer all the degrees of freedom represented by a free parameter matrix

oscillation of even order linear functional differential equations with deviating arguments of mixed type where n is even p 0 co r and g o 00 r are continuous p t 0 g t is nondecreasing and cm g t 03 by a proper solution of eq 1 we mean a function x tx 00 r which satisfies 1 for all sufficiently large t and sup lx t j t t 0 for any t tx we make the standing hypothesis that 1 does possess proper solutions a proper solution is called oscillatory if it has arbitrarily large zeros otherwise it is called nonoscillatory if x t is a nonoscillatory solution of l then there exist an even integer 1 e 0 2 n and a t tx such that

implementation of an adaptive algorithm for richardson s method abstract we discuss the implementation of an adaptive algorithm proposed by one of us the algorithm is a hybrid of the gmres method and richardson s method richardson s method rm depends on a set of parameters that are computed by minimizing the l 2 norm of a polynomial over the convex hull of eigenvalues execution of gmres yields not only an approximate solution but also the approximate convex hull rm is used to avoid storing and working with a large number of vectors as gmres often requires this method is also advantageous for the solution of large problems we consider several test problems and compare our algorithm primarily with the conjugate gradient squared algorithm but also with gmres and to cg applied to the normal equations for many test problems our algorithm takes roughly 50 percent more work than the conjugate gradient squared algorithm although if the matrix is either preconditioned or indefinite our algorithm is more efficient however our algorithm currently imposes an undesirable burden on the user who is invited to consider a variety of numerical parameters to manipulate such as the number of steps of rm in order to enhance performance the values we suggest are only empirical

an output error method for reduced order controller design an efficient computational technique is presented for the design of reduced order controllers for linear discrete time systems the technique is based on the minimization of the output error between the closed loop system and a specified reference model

the oscillation of higher order differential equations with deviating arguments in this paper some new criteria for the oscillation of higher order functional differential equations of the form lnx t f t x g t 0 n is even are established some of our results are obtained via comparing it with second order ordinary linear and first order delay differential equations the oscillation of this equation when n is odd is also considered then we shall use the obtained results to study the oscillatory behavior of the neutral functional differential equations of the form ln x t c t x t f t x g t 0 n is even and the damped functional differential equations of the type lnx t h t x g t ddtx h t 0 n is even the obtained results extend improve and correlate a number of existing criteria

on the lanczos method for solving symmetric linear systems with several right hand sides this paper analyzes a few methods based on the lanczos algorithm for solving large sparse symmetric linear systems with several right hand sides the methods examined are suitable for the situation when the right sides are not too different from one another as is often the case in time dependent or parameter dependent problems we propose a theoretical error bound for the approximation obtained from a projection process onto a krylov subspace generated from processing a previous right hand side

the least squares anti bisymmetric solution and the optimal approximation solution for sylvester equation in this paper a modified conjugate gradient iterative method for solving sylvester equation is presented by using this iterative method the least squares anti bisymmetric solution and the optimal approximation solution can be obtained here we present the derivation and theoretical analysis of our iterative method numerical results illustrate the feasibility and effectiveness of the proposed iterative method

integral criteria for oscillation of third order nonlinear differential equations abstract in this paper we are concerned with the oscillation of third order nonlinear differential equations of the form r 2 t r 1 t y p t y q t f y 0 t t 0 where t 0 0 is a fixed real number r 1 r 2 p q i t 0 r are nonnegative continuous functions with r 1 0 r 2 0 f r r is a continuous function by making use of a generalized riccati transformation we establish some new sufficient integral conditions under which the equation has at least one oscillatory solution examples are given to illustrate the importance of our results