comparisons of lattice boltzmann and finite difference methods for a two dimensional viscous burgers equation lattice boltzmann methods have been proposed as a computational means for solving various partial differential equations we look at three lattice boltzmann methods and an analogous finite difference method for solving a two dimensional scalar viscous burgers equation with periodic boundary conditions first using monotonicity arguments we prove that in the ell 1 norm the lattice boltzmann methods converge first order temporally and second order spatially then we provide some computational results which substantiate the theoretical results finally we compare the lattice boltzmann and finite difference methods the basis of comparison includes numerical results with regard to accuracy order of convergence performance and timing measurements memory requirements and conservations laws

simultaneous eigenvalue lower bounds for the lyapunov matrix equation simultaneous eigenvalue lower bounds for the solution of the lyapunov equation are presented the n summation and the n product bounds generalize existing bounds such as those for the trace and the determinant these bounds are stronger than the majority of the relevant bounds shown in the literature

bounds in algebraic riccati and lyapunov equations a survey and some new results this paper summarizes and investigates the relations of a number of bounds for the solutions of the algebraic riccati and lyapunov equations that have been reported during the last two decades also presented are bounds for the unified riccati equation using the delta operator and it is shown that some bounds for the continuous and discrete riccati equations can be unified by them

lower summation bounds for the discrete riccati and lyapunov equations lower eigenvalue summation including trace bounds for the solution of the discrete algebraic riccati and lyapunov matrix equations are presented these are tighter than or supplement existing results

some bounds of the solution of the algebraic riccati equation bounds on extremal eigenvalues and lower and upper bounds of the trace for the solution of the algebraic riccati equation are presented through two examples it is shown that in some special cases the presented bounds can be better than the results recently published

matrix bounds of the discrete are solution this paper provides new lower and upper matrix bounds of the solution to the discrete algebraic riccati equation the lower bound always works if the solution exists the upper bounds are presented in terms of the solution of the discrete lyapunov equation and its upper matrix bound the upper bounds are always calculated if the solution of the lyapunov equation exists a numerical example shows that the new bounds are tighter than previous results in many cases

eigenvalue bounds for algebraic riccati and lyapunov equations the majorization result of wimmer 2 relating the eigenvalues of the matrices involved in the lyapunov equation is extended to the algebraic riccati equation this result coupled with certain results on the eigenvalue bounds for sum and product of matrices yields several lower and upper bounds for the eigenvalues of the solution matrix of the algebraic riccati and lyapunov equations

bounds for the solution of the lyapunov matrix equation a unified approach abstract in recent years several bounds have been reported for the solution of the continuous and the discrete lyapunov equations using the unified lyapunov equation we give in this paper bounds for the solution of this equation in the limiting cases the bounds reduce to existing bounds for both the continuous and discrete lyapunov equations

trace bounds on the covariances of continuous time systems with multiplicative noise matrix equations such as ap pa sup t fpf sup t omega 0 and aq qa sup t qvq fpf sup t omega 0 which arise in the estimation problem of systems with both additive and multiplicative noise are treated trace bounds on the steady state and error covariances p and q are established under complete and incomplete noise information an example illustrates the usefulness of these bounds in determining the size of the estimation error

a note on the discrete lyapunov and riccati matrix equations abstract inequalities which are satisfied by the determinants of the positive definite solutions of the discrete algebraic riccati and lyapunov matrix equations are presented the results give lower bounds for the product of the eigenvalues of the matrix solutions also for a discrete lyapunov equation an algorithm is presented to determine under what conditions a positive diagonal solution will exist if all the conditions are satisfied the algorithm also provides such a diagonal solution

a note on eigenvalue bounds in algebraic riccati equation in this note an upper bound on the maximum eigenvalue of the solution matrix k of the algebraic riccati equation is established the approach outlined also results in several lower bounds which are more general than those derived in 1 for some of the largest eigenvalues of k

lower bounds for the trace of the solution of the discrete algebraic riccati equation two lower bounds for the trace of the solution of the discrete algebraic riccati equation dare are presented it is shown that in many cases these trace bounds are tighter than those in the literature and greater than the trace of the state weighting matrix even when the system matrix is singular the results are illustrated by an example

on bounds of the riccati and lyapunov matrix equations various bounds for the traces of the solutions of the algebraic riccati and lyapunov matrix equations are established in the continuous and discrete domain respectively the presented results give lower bounds for the sum of the eigenvalue of the matrix solutions

improved bounds for the eigenvalues of solutions of lyapunov equations bounds on the solution and on the eigenvalues of the solution of the discrete and of the continuous matrix lyapunov equations are presented these bounds are nontrivial in the case of system matrices having multiple eigenvalues especially in the latter case these bounds are better than bounds found in the literature further these bounds are sharp i e there exist systems such that the lower and upper bounds coincide yielding the solution or its eigenvalues respectively

trace bounds on the solution of the algebraic matrix riccati and lyapunov equation lower and upper bounds on the trace of the positive semidefinite solution of the algebraic matrix riccati and lyapunov equation are derived the upper trace bound obtained in this note in many cases results in a tighter bound as compared to the upper bound for the maximal eigenvalue proposed in 1 and 2

upper matrix bounds for the discrete algebraic riccati matrix equation we propose upper matrix bounds for the discrete algebraic riccati matrix equation we show that our results are less restrictive than the previous bounds finally we give numerical examples in order to verify the effectiveness of our results

a brief summary of the bounds on the solution of the algebraic matrix equations in control theory abstract in recent years several bounds have been reported for different measures of the extent or size of the solution of the algebraic matrix equations arising in control theory such as the riccati equation and the lyapunov equation this paper collects the bounds that have been presented up to now and summarizes them in an unified form this will prove particularly convenient for those wishing to get a ready estimate of the solution while solving the equations numerically or to develop theoretical results that rely on these bounds

bounds on solutions to h algebraic riccati equations and h 2 properties of h central solutions abstract in this paper we give upper bounds on the stabilizing solutions of the algebraic riccati equations arising in standard h estimation and control problems for continuous time finite dimensional linear time invariant systems using these bounds we show that in h estimation or h feedback control via state feedback the so called central solution satisfies an interesting h 2 norm bound this bound may be readily used to find a central solution i e estimator or feedback controller that guarantees an equal comprise between h 2 and h objectives

a cyclic low rank smith method for large sparse lyapunov equations in this paper we present the cyclic low rank smith method which is an iterative method for the computation of low rank approximations to the solution of large sparse stable lyapunov equations it is based on a generalization of the classical smith method and profits by the usual low rank property of the right hand side matrix the requirements of the method are moderate with respect to both computational cost and memory furthermore we propose a heuristic for determining a set of suboptimal alternating direction implicit adi shift parameters this heuristic which is based on a pair of arnoldi processes does not require any a priori knowledge on the spectrum of the coefficient matrix of the lyapunov equation numerical experiments show the efficiency of the iterative scheme combined with the heuristic for the adi parameters

comments on on some bounds in the algebraic riccati and lyapunov equations results presented by moil in the paper which give some bounds for the trace and the determinant of the positive definite solution to the algebraic riccati matrix equation are shown to be erroneous and suggestions to remedy these errors are made

lower bounds on the solution of lyapunov matrix and algebraic riccati equations in this note various lower bounds for all the eigenvalues of the solution matrix k of the lyapunov matrix equation are established a special case of this result is a generalization of that presented in 1 3 where lower bounds for the maximum and minimum eigenvalues of k are given moreover the approach used here enables one to establish various lower bounds for some of the largest eigenvalues of the solution matrix of the algebraic riccati equation

bounds for the eigenvalues of the solution of the discrete riccati and lyapunov equations and the continuous lyapunov equation we present some bounds for the eigenvalues and certain sums and products of the eigenvalues of the solution of the discrete riccati and lyapunov matrix equations and the continuous lyapunov matrix equation nearly all of our bounds for the discrete riccati equation are new the bounds for the discrete and continuous lyapunov equations give a completion of some known bounds for the extremal eigenvalues and the determinant and the trace of the solution of the respective equation