new upper solution bounds of the continuous algebraic riccati matrix equation by utilizing some linear algebraic techniques new upper bounds of the solution of the continuous algebraic riccati equation care are derived according to the present bounds iterative procedures are also developed for obtaining more precise estimations comparing with existing results the obtained bounds are less restrictive

liapunov schmidt reduction and continuation for nonlinear schro dinger equations we study the bifurcation scenario of nonlinear schro dinger equations nls the liapunov schmidt reduction is applied to show that the simple bifurcations of a single nls are pitchfork the pitchfork bifurcation can be subcritical or supercritical depending on the coefficient of the cubic term we choose we also describe numerical methods so that the liapunov schmidt reduction can effectively handle a corank 2 bifurcation point next we apply numerical continuation methods to trace solution curves and surfaces of the nls where the system is discretized by the centered difference approximations numerical results on two and three dimensional m coupled nls are reported where the physical properties such as the effect of trapping potentials isotropic and nonisotropic trapping potentials mass conservation constraints and strong and weak repulsive interactions are considered in our numerical experiments

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

a spectral galerkin continuation method using chebyshev polynomials for the numerical solutions of the gross pitaevskii equation we study an efficient spectral galerkin continuation method sgcm and two grid centered difference approximations for the numerical solutions of the gross pitaevskii equation gpe where the second kind chebyshev polynomials are used as the basis functions for the trial function space some basic formulae for the sgcm are derived so that the eigenvalues of the associated linear eigenvalue problems can be easily computed the sgcm is implemented to investigate the ground and first excited state solutions of the gpe both the parabolic and quadruple well trapping potentials are considered we also study bose einstein condensates bec in optical lattices where the periodic potential described by the sine or cosine functions is imposed on the gpe of particular interest here is the investigation of symmetry breaking solutions sample numerical results are reported

upper matrix bound of the solution for the discrete riccati equation a new upper matrix bound of the solution for the discrete algebraic matrix riccati equation is developed this matrix bound is then used to derive bounds on the eigenvalues trace and determinant of the same solution it is shown that these eigenvalue bounds are less restrictive than previous results

mode jumping in the von karman equations we study numerical approximations of bifurcating solution curves of the von karman equations with simply supported and clamped boundary conditions respectively of special interest here is the splitting of a double bifurcation point into two simple bifurcation points and the tracing of the associated secondary solution branches which corresponds to the phenomenon of mode jumping in the buckling of a rectangular plate a continuation unsymmetric lanczos algorithm is utilized for curve tracking our numerical results verify the theoretical prediction of schaeffer and golubitsky namely mode jumping occurs when the boundary is partially clamped but not if it is merely simply supported

brief lower matrix bounds for the continuous algebraic riccati and lyapunov matrix equations in this paper we propose lower matrix bounds for the continuous algebraic riccati and lyapunov matrix equations we give comparisons between the parallel estimates finally we give examples showing that our bounds can be better than the previous results for some cases

numerical study of time splitting spectral discretizations of nonlinear schrodinger equations in the semiclassical regimes in this paper we study the performance of time splitting spectral approximations for general nonlinear schrodinger equations nls in the semiclassical regimes where the planck constant varepsilon is small the time splitting spectral approximation under study is explicit unconditionally stable and conserves the position density in l 1 moreover it is time transverse invariant and time reversible when the corresponding nls is extensive numerical tests are presented for weak strong focusing defocusing nonlinearities for the gross pitaevskii equation and for current relaxed quantum hydrodynamics the tests are geared towards the understanding of admissible meshing strategies for obtaining correct physical observables in the semiclassical regimes furthermore comparisons between the solutions of the nls and its hydrodynamic semiclassical limit are presented

iterative matrix bounds and computational solutions to the discrete algebraic riccati equation bilateral matrix bounds for the solution of the discrete algebraic riccati equation dare are presented they are new or tighter than the existing bound computational algorithms to solve the dare follow

new matrix bounds and iterative algorithms for the discrete coupled algebraic riccati equation abstractthe discrete coupled algebraic riccati equation dcare has wide applications in control theory and linear system in general for the dcare one discusses every term of the coupled term respectively in this paper we consider the coupled term as a whole which is different from the recent results when applying eigenvalue inequalities to discuss the coupled term our method has less error in terms of the properties of special matrices and eigenvalue inequalities we propose several upper and lower matrix bounds for the solution of dcare further we discuss the iterative algorithms for the solution of the dcare in the fixed point iterative algorithms the scope of lipschitz factor is wider than the recent results finally we offer corresponding numerical examples to illustrate the effectiveness of the derived results

all multipliers for repeated monotone nonlinearities we have derived the largest class of bounded linear time invariant operators that preserve positivity of repeated monotone nonlinearities it reduces conservativeness of integral quadratic constraint iqc stability analysis of systems having repeated monotone nonlinearities to the minimum possible utility of this characterization stems from the fact that such nonlinearities are typically encountered in stability analysis of anti windup schemes among other applications we have also derived the time varying counterpart of this class of operators

numerical studies on the split step finite difference method for nonlinear schrodinger equations in this study we use the split step finite difference method to solve various nonlinear schrodinger equations including coupled ones we study the numerical accuracy of the method detailed numerical results including higher dimensions show that the method provides accurate and stable solutions for nonlinear schrodinger equations

optimizing schrodinger functionals using sobolev gradients applications to quantum mechanics and nonlinear optics in this paper we study the application of the sobolev gradients technique to the problem of minimizing several schrodinger functionals related to timely and difficult nonlinear problems in quantum mechanics and nonlinear optics we show that these gradients act as preconditioners over traditional choices of descent directions in minimization methods and show a computationally inexpensive way to obtain them using a discrete fourier basis and a fast fourier transform we show that the sobolev preconditioning provides a great convergence improvement over traditional techniques for finding solutions with minimal energy as well as stationary states and suggest a generalization of the method using arbitrary linear operators

a new approach for upper bound estimations of the solution of the continuous riccati equation by utilizing some linear algebraic techniques a new approach for measuring upper bounds of the solution of the continuous algebraic riccati equation care is developed two upper matrix bounds are developed comparing with those appeared in the literature the obtained bounds are less restrictive or more concise therefore they can be applied easily to deal with control problems

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

new upper solution bounds of the discrete algebraic riccati matrix equation in this note we present upper matrix bounds for the solution of the discrete algebraic riccati equation dare using the matrix bound of theorem 2 2 we then give several eigenvalue upper bounds for the solution of the dare and make comparisons with existing results the advantage of our results over existing upper bounds is that the new upper bounds of theorem 2 2 and corollary 2 1 are always calculated if the stabilizing solution of the dare exists whilst all existing upper matrix bounds might not be calculated because they have been derived under stronger conditions finally we give numerical examples to demonstrate the effectiveness of the derived results

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

upper bounds for the solution of the discrete riccati equation upper bounds for individual eigenvalues and for summations of eigenvalues including the trace of the solution of the discrete algebraic riccati equation are presented some are new and some supplement bounds in the literature

computing the ground state solution of bose einstein condensates by a normalized gradient flow in this paper we present a continuous normalized gradient flow cngf and prove its energy diminishing property which provides a mathematical justification of the imaginary time method used in the physics literature to compute the ground state solution of bose einstein condensates bec we also investigate the energy diminishing property for the discretization of the cngf two numerical methods are proposed for such discretizations one is the backward euler centered finite difference befd method the other is an explicit time splitting sine spectral tssp method energy diminishing for befd and tssp for the linear case and monotonicity for befd for both linear and nonlinear cases are proven comparison between the two methods and existing methods e g crank nicolson finite difference cnfd or forward euler finite difference fefd shows that befd and tssp are much better in terms of preserving the energy diminishing property of the cngf numerical results in one two and three dimensions with magnetic trap confinement potential as well as a potential of a stirrer corresponding to a far blue detuned gaussian laser beam are reported to demonstrate the effectiveness of befd and tssp methods furthermore we observe that the cngf and its befd discretization can also be applied directly to compute the first excited state solution in bec when the initial data is chosen as an odd function

on time splitting spectral approximations for the schrodinger equation in the semiclassical regime in this paper we study time splitting spectral approximations for the linear schrodinger equation in the semiclassical regime where the planck constant e is small in this regime the equation propagates oscillations with a wavelength of o e and finite difference approximations require the spatial mesh size h o e and the time step k o e in order to obtain physically correct observables much sharper mesh size constraints are necessary for a uniform l2 approximation of the wave function the spectral time splitting approximation under study will be proved to be unconditionally stable time reversible and gauge invariant it conserves the position density and gives uniform l2 approximation of the wave function for k o e and h o e extensive numerical examples in both one and two space dimensions and analytical considerations based on the wigner transform even show that weaker constraints e g k independent of e and h o e are admissible for obtaining correct observables finally we address the application to nonlinear schrodinger equations and conduct some numerical experiments to predict the corresponding admissible meshing strategies

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

a two grid discretization scheme for semilinear elliptic eigenvalue problems we present a two grid finite element discretization scheme with a two loop continuation algorithm for tracing solution branches of semilinear elliptic eigenvalue problems first we use the predictor corrector continuation method to compute an approximating point for the solution curve on the coarse grid then we use this approximating point as a predicted point for the solution curve on the fine grid in the corrector step we solve the first and the second order approximations of the nonlinear pde to obtain corrections for the state variable on the fine grid and the coarse grid respectively the continuation parameter is updated by computing the rayleigh quotient on the fine space to guarantee the approximating point we just obtained lies on the solution curve we perform newton s method we repeat the process described above until the solution curve on the fine space is obtained we show how the singular points such as folds and bifurcation points can be well approximated comprehensive numerical experiments show that the two grid finite element discretization scheme with a two loop continuation algorithm is efficient and robust for solving second order semilinear elliptic eigenvalue problems

upper solution bounds of the continuous and discrete coupled algebraic riccati equations in this paper we propose upper bounds for the sum of the maximal eigenvalues of the solutions of the continuous coupled algebraic riccati equation ccare and the discrete coupled algebraic riccati equation dcare which are then used to infer upper bounds for the maximal eigenvalues of the solutions of each riccati equation by utilizing the upper bounds for the maximal eigenvalues of each equation we then derive upper matrix bounds for the solutions of the ccare and dcare following the development of each bound an iterative algorithm is proposed which can be used to derive tighter upper matrix bounds finally we give numerical examples to demonstrate the effectiveness of the proposed results making comparisons with existing results

spectral collocation methods using sine functions for a rotating bose einstein condensation in optical lattices we study spectral galerkin methods sgm and spectral collocation methods scm for parameter dependent problems where the fourier sine functions are used as the basis functions when the sgm and the scm are incorporated in the context of a taylor predictor inexact newton corrector continuation algorithm for tracing solution curves of the gross pitaevskii equation gpe they can efficiently provide accurate numerical solutions for the gpe we show how the inexact newton method outperforms the classical newton method in the continuation algorithm in our numerical experiments the centered difference method cdm the sgm and scm are exploited to compute energy levels and wave functions of a rotating bose einstein condensation bec and a rotating bec in optical lattices in 2d sample numerical results are reported

the existence uniqueness and the fixed iterative algorithm of the solution for the discrete coupled algebraic riccati equation in this article applying the properties of m matrix and non negative matrix utilising eigenvalue inequalities of matrix s sum and product we firstly develop new upper and lower matrix bounds of the solution for discrete coupled algebraic riccati equation dcare secondly we discuss the solution existence uniqueness condition of the dcare using the developed upper and lower matrix bounds and a fixed point theorem thirdly a new fixed iterative algorithm of the solution for the dcare is shown finally the corresponding numerical examples are given to illustrate the effectiveness of the developed results

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