a hamiltonian jacobi algorithm the nonsymmetric jacobi iteration is adapted to the special structure of hamiltonian matrices this hamiltonian jacobi algorithm uses symplectic unitary similarity transformations to solve algebraic riccati equations through the hamiltonian schur form it preserves hamiltonian structure without using a condensed form although it converges too slowly for use on conventional serial computers it may be attractive for some highly parallel architectures

sufficient conditions for pseudo lipschitz property in convex semi infinite vector optimization problems abstract this paper is devoted to the study of the pseudo lipschitz property of the efficient pareto solution map for the perturbed convex semi infinite vector optimization problem csvo we establish sufficient conditions for the pseudo lipschitz property of the efficient solution map of csvo under continuous perturbations of the right hand side of the constraints and functional perturbations of the objective function examples are given to illustrate the obtained results

a new method for computing the closed loop eigenvalues of a discrete time algebraic riccati equation abstract we present a fast method for computing the closed loop eigenvalues of a discrete time algebraic riccati equation the method is close to the symplectic method for finding all the eigenvalues of a hamiltonian matrix and is based on a g orthogonal transformation which preserves structure and has desirable numerical properties the algorithm requires about one fourth the number of floating point operations and one half the storage of the qz algorithm

numerical solution of projected algebraic riccati equations we consider the numerical solution of projected algebraic riccati equations using newton s method such equations arise for instance in model reduction of descriptor systems based on positive real and bounded real balanced truncation we also discuss the computation of low rank cholesky factors of the solutions of projected riccati equations numerical examples are given that demonstrate the properties of the proposed algorithms

some numerical considerations in h control abstract certain h control problems are considered and potential numerical difficulties are discussed an alternative test for checking that the solution to an algebraic riccati equation is positive semi definite which is robust to rounding error even when the matrix is singular is presented finally the h loop shaping method is summarised and its numerical properties are shown to be satisfactory

numerical algorithms for undamped gyroscopic systems abstract the solutions of a gyroscopic vibrating system oscillating about an equilibrium position with no external applied forces and no damping forces are completely determined by the quadratic eigenvalue problem 2 i m i g k x i 0 for i 1 2 n where m g and k are real n n matrices and m is symmetric positive definite denoted by m 0 g is skew symmetric and either k 0 or k 0 gyroscopic system in motion about a stable equilibrium position with k 0 are well understood two lanczos type algorithms the pseudo skew symmetric lanczos algorithm and the j lanczos algorithm are studied for computing some extreme eigenpairs for solving gyroscopic systems in motion about an unstable equilibrium position with k 0 shift and invert strategies error bounds implementation issues and numerical results for both algorithms are presented in details

fourth order time stepping for stiff pdes a modification of the exponential time differencing fourth order runge kutta method for solving stiff nonlinear pdes is presented that solves the problem of numerical instability in the scheme as proposed by cox and matthews and generalizes the method to nondiagonal operators a comparison is made of the performance of this modified exponential time differencing etd scheme against the competing methods of implicit explicit differencing integrating factors time splitting and fornberg and driscoll s sliders for the kdv kuramoto sivashinsky burgers and allen cahn equations in one space dimension implementation of the method is illustrated by short matlab programs for two of the equations it is found that for these applications with fixed time steps the modified etd scheme is the best

some remarks on the complex j symmetric eigenproblem abstract the eigenproblem for complex j symmetric matrices h c a c d a t a c c t d d t c n n is considered a proof of the existence of a transformation to the complex j symmetric schur form proposed in 21 is given the complex symplectic unitary qr decomposition and the complex symplectic sr decomposition are discussed it is shown that a qr like method based on the complex symplectic unitary qr decomposition is not feasible here a complex symplectic sr algorithm is presented which can be implemented such that one step of the sr algorithm can be carried out in o n arithmetic operations based on this a complex symplectic lanczos method can be derived moreover it is discussed how the 2 n 2 n complex j symmetric matrix h c can be embedded in a 4 n 4 n real hamiltonian matrix

spectral factorization via hermitian pencils abstract the spectral factorization problem is solved for state space systems via results on the canonical forms and inertia of hermitian matrix pencils these algebraic results then give a deflation method for spectral factorization

curved fronts in the belousov zhabotinskii reaction diffusion systems in r 2 in this paper we consider a diffusion system with the belousov zhabotinskii bz for short chemical reaction following brazhnik and tyson 4 and perez munuzuri et al 45 who predicted v shaped fronts theoretically and discovered v shaped fronts by experiments respectively we give a rigorous mathematical proof of their results we establish the existence of v shaped traveling fronts in r 2 by constructing a proper supersolution and a subsolution furthermore we establish the stability of the v shaped front in r 2 c 2018 elsevier inc all rights reserved

a fourth order numerical scheme for solving the modified burgers equation a finite difference scheme based on fourth order rational approximants to the matrix exponential term in a two time level recurrence relation is proposed for the numerical solution of the modified burgers equation the resulting nonlinear system which is analyzed for stability is solved using an already known modified predictor corrector scheme the results arising from the experiments are compared with the corresponding ones known from the available literature

an exponential time differencing method of lines for burgers fisher and coupled burgers equations abstract an exponential time differencing scheme using the method of lines is developed in this article for the numerical solution of the generalized burgers fisher and coupled burgers equations in each of the aforementioned cases the resulting nonlinear systems are solved explicitly using a second order modified predictor corrector method the efficiency of the method introduced is tested by comparing our experimental results with others selected from the available literature

a front dynamics approach to curvature dependent flow a front dynamics approach is developed to study the evolution of planar curves whose normal speed depends on curvature the formulation is similar to whitham s shock dynamics theory for the propagation of shock wave in gases but assumes a different propagation rule equations that describe the motion of the front are obtained and these are evolution equations for the normal direction and local arc length of the front the solution of these equations leads to the front positions using an appropriate integration along rays a similarity solution of the equations is found for the evolution of an initial corner free boundary problems for the motion of a junction connecting front segments are discussed a numerical method is presented to calculate the evolution of any number of front segments the segments can be closed or open connected to wall boundaries or not or connected to other segments at 3 segment junctions several sample problems are considered to illustrate the method an extension of the metho

lower semi continuity of the pareto solution map in quasiconvex semi infinite vector optimization abstract this paper is concerned with the lower semi continuity of the efficient pareto solution map for the perturbed quasiconvex semi infinite vector optimization problem qcsvo we establish sufficient conditions for the lower semi continuous property of the efficient solution map of qcsvo under functional perturbations of both the objective function and the constraints examples are designed to analyze the obtained results

a treatment of breakdowns and near breakdowns in a reduction of a matrix to upper j hessenberg form and related topics abstract the reduction of a matrix to an upper j hessenberg form is a crucial step in the s r algorithm which is a q r like algorithm structure preserving for computing eigenvalues and vectors for a class of structured matrices this reduction may be handled via the algorithm jhess or via the recent algorithm jhmsh and its variants the main drawback of jhess or jhmsh is that it may suffer from a fatal breakdown causing a brutal stop of the computations and hence the s r algorithm does not run jhess may also encounter near breakdowns source of serious numerical instability in this paper we focus on these aspects we first bring light on the necessary and sufficient condition for the existence of the s r decomposition which is intimately linked to j hessenberg reduction then we will derive a strategy for curing fatal breakdowns and also for treating near breakdowns hence the j hessenberg form may be obtained numerical experiments are given demonstrating the efficiency of our strategies to cure and treat breakdowns or near breakdowns

adomian decomposition method for burger s huxley and burger s fisher equations the approximate solutions for the burger s huxley and burger s fisher equations are obtained by using the adomian decomposition method solving frontier problems of physics the decomposition method kluwer boston 1994 the algorithm is illustrated by studying an initial value problem the obtained results are presented and only few terms of the expansion are required to obtain the approximate solution which is found to be accurate and efficient

a quasi steady state solver for the stiff ordinary differential equations of reaction kinetics abstract a quasi steady state method is presented that integrates stiff differential equations arising from reaction kinetics this predictor corrector method is a stable for linear equations and second order accurate the method is used for all species regardless of the time scales of the individual equations and it works well for problems typical of hydrocarbon combustion start up costs are low making the method ideal for use in process split reacting flow simulations which require the solution of an initial value problem in every computational cell for every global time step the algorithm is described and error analysis and linear stability analysis are included the algorithm is also applied to several test problems and the results are compared to those of the stiff integrator chemeq the method which we call qss is more stable more accurate and less costly than chemeq

the symplectic eigenvalue problem the butterfly form the sr algorithm and the lanczos method abstract we discuss some aspects of the recently proposed symplectic butterfly form which is a condensed form for symplectic matrices any 2n 2n symplectic matrix can be reduced to this condensed form which contains 8n 4 nonzero entries and is determined by 4n 1 parameters the symplectic eigenvalue problem can be solved using the sr algorithm based on this condensed form the sr algorithm preserves this form and can be modified to work only with the 4n 1 parameters instead of the 4n2 matrix elements the reduction of symplectic matrices to symplectic butterfly form has a close analogy to the reduction of arbitrary matrices to hessenberg form a lanczos like algorithm for reducing a symplectic matrix to butterfly form is also presented

a symplectic qr like algorithm for the solution of the real algebraic riccati equation a method is presented to solve the real algebraic riccati equation xnx xa a t x k 0 where k k t and n n t the solution for the corresponding eigenvalue problem mx lambda x where m is a hamiltonian matrix is computed by an algorithm similar to the qr algorithm special symplectic matrices are used for the transformation of m such that the hamiltonian form is preserved during the computations

on hamiltonian and symplectic lanczos processes abstract large sparse hamiltonian eigenvalue problems arise in a variety of contexts these problems can be attacked directly or they can first be transformed to problems having some related structure such as symplectic or skew hamiltonian in the interest of efficiency stability and accuracy such problems should be solved by methods that preserve the structure whether it be hamiltonian skew hamiltonian or symplectic the present work outlines krylov subspace methods for computing partial eigensystems of skew hamiltonian hamiltonian and symplectic matrices and records some of the relationships between them the ordinary unsymmetric lanczos process is a structure preserving method for skew hamiltonian matrices the hamiltonian and symplectic lanczos processes developed here are condensed versions of the processes that have been published previously the condensed hamiltonian lanczos process applied to h is equivalent to the unsymmetric lanczos process applied to the skew hamiltonian h 2 but costs half as much to execute the condensed symplectic lanczos process applied to s is equivalent to the unsymmetric lanczos process applied to the skew hamiltonian matrix s s 1 but also costs half as much to execute implicit restarts of the hamiltonian and symplectic lanczos processes can be effected by the sr algorithm because of the known relationship between the sr and hr algorithms the much simpler hr algorithm can be used to restart the condensed symplectic and hamiltonian lanczos processes each of these restart procedures is equivalent to restarting the unsymmetric lanczos process using the hr algorithm the hr algorithm is most effectively implemented as an implicit bulge chasing hz algorithm on a symmetric tridiagonal diagonal pencil t d

the tanh method for generalized forms of nonlinear heat conduction and burgers fisher equations generalized forms of the kdv the nonlinear heat conduction and burgers fisher equations are investigated the analysis rests mainly on the standard tanh method the focus is on the case where the parameter m is noninteger a variety of exact travelling wave solutions of distinct physical structures are formally derived

on traveling wave solutions of fisher s equation in two spatial dimensions it is shown for the quadratic fisher equation in two spatial dimensions that along with a plane wave there exist several other traveling waves with nontrivial front geometry some of the solutions are found in explicit form others are constructed approximately the dispersion relationship and velocity curvature dependence generated by these solutions are studied

variational analysis in semi infinite and infinite programming ii necessary optimality conditions this paper concerns applications of advanced techniques of variational analysis and generalized differentiation to problems of semi infinite and infinite programming with feasible solution sets defined by parameterized systems of infinitely many linear inequalities of the type intensively studied in the preceding development canovas et al siam j optim 20 2009 pp 1504 1526 from the viewpoint of robust lipschitzian stability the main results establish necessary optimality conditions for broad classes of semi infinite and infinite programs where objectives are generally described by nonsmooth and nonconvex functions on banach spaces and where infinite constraint inequality systems are indexed by arbitrary sets the results obtained are new in both smooth and nonsmooth settings of semi infinite and infinite programming we illustrate our model and results by considering a practically meaningful model of water resource optimization via systems of reservoirs

on the sensitivity of the sr decomposition abstract first order componentwise and normwise perturbation bounds for the sr decomposition are presented the new normwise bounds are at least as good as previously known results in particular for the r factor the normwise bound can be significantly tighter than the previous result

the effects of obstacle size on periodic travelling waves in oscillatory reaction diffusion equations many natural populations undergo multi year cycles and field studies have shown that these can be organized into periodic travelling waves ptws mathematical studies have shown that large scale landscape obstacles represent a natural mechanism for wave generation here we investigate how the amplitude and wavelength of the selected waves depend on the obstacle size we firstly consider a large circular obstacle in an infinite domain for a reaction diffusion system of l u type we use perturbation theory to derive a leading order approximation to the wave generated by the obstacle this shows the dependence of the wave properties on both parameter values and obstacle size we find that the limiting values of the amplitude and wavelength are approached algebraically with distance from the obstacle edge rather than exponentially in the case of a flat boundary we use our results to predict the properties of waves generated by a large circular obstacle for an oscillatory predator prey system via a reduction of the predator prey model to normal form close to hopf bifurcation our predictions compare well with numerical simulations we also discuss the implications of these results for wave stability and briefly investigate the effects of obstacles with elliptical geometries

a symplectic acceleration method for the solution of the algebraic riccati equation on a parallel computer abstract we give a cubic acceleration method for improving the current symplectic jacobi like algorithm for computing the hamiltonian schur decomposition of a hamiltonian matrix and finding the positive semidefinite solution of the riccati equation the acceleration method can speed up the rate of convergence at the end of the symplectic jacobi like process when the norm of the current strictly j lower triangle has become sufficiently small it has high parallelism and takes o n computational time when implemented on a mesh connected n n array processor system a quantitative analysis of convergence and numerical comparisons of one jacobi sweep versus one correction step are presented

a schur decomposition for hamiltonian matrices abstract a schur type decomposition for hamiltonian matrices is given that relies on unitary symplectic similarity transformations these transformations preserve the hamiltonian structure and are numerically stable making them ideal for analysis and computation using this decomposition and a special singular value decomposition for unitary symplectic matrices a canonical reduction of the algebraic riccati equation is obtained which sheds light on the sensitivity of the nonnegative definite solution after presenting some real decompositions for real hamiltonian matrices we look into the possibility of an orthogonal symplectic version of the qr algorithm suitable for hamiltonian matrices a finite step initial reduction to a hessenberg type canonical form is presented however no extension of the francis implicit shift technique was found and reasons for the difficulty are given

properties of a quadratic matrix equation and the solution of the continuous time algebraic riccati equation abstract we discuss some properties of a quadratic matrix equation with some restrictions then use these results on the algebraic riccati equation to get a new algorithm the algorithm sufficiently takes account of the structure of the associated matrix hence it is very effective

exponential time differencing for stiff systems we develop a class of numerical methods for stiff systems based on the method of exponential time differencing we describe schemes with second and higher order accuracy introduce new runge kutta versions of these schemes and extend the method to show how it may be applied to systems whose linear part is nondiagonal we test the method against other common schemes including integrating factor and linearly implicit methods and show how it is more accurate in a number of applications we apply the method to both dissipative and dispersive partial differential equations after illustrating its behavior using forced ordinary differential equations with stiff linear parts

optimal symplectic householder transformations for sr decomposition the sr factorization is a key step for some important structure preserving eigenproblems in this work we introduce symplectic householder transformations and show their main features furthermore we construct a new algorithm for computing the sr factorization based on these transformations the new algorithm is the analogous of the classical qr factorization via householder transformations unlike the latter it involves free parameters we show how to choose these parameters in an optimal way and derive an algorithm implementing this some illustrating numerical experiments comparisons and computational aspects are given

orthosymmetric block reflectors we develop a general theory of reflectors and block reflectors in a class of non euclidean scalar product spaces generated by orthosymmetric scalar product matrices j these j reflectors are generalizations of ordinary householder transformations and we show that they can always be expressed in a householder like representation reflection and mapping properties of j reflectors are completely described block j reflectors can be used for block annihilation in qr like factorizations where q is j unitary and r is block upper triangular

the parameterized sr algorithm for symplectic butterfly matrices the sr algorithm is a structure preserving algorithm for computing the spectrum of symplectic matrices any symplectic matrix can be reduced to symplectic butterfly form a symplectic matrix b in butterfly form is uniquely determined by 4n 1 parameters using these 4n 1 parameters we show how one step of the symplectic sr algorithm for b can be carried out in o n arithmetic operations compared to o n 3 arithmetic operations when working on the actual symplectic matrix moreover the symplectic structure which will be destroyed in the numerical process due to roundoff errors when working with a symplectic butterfly matrix will be forced by working just with the parameters

convex analysis and global optimization of joint actuator location and control problems it is shown that the optimal value of the continuous time linear quadratic problem regarded as a function of the system model and index parameters exhibits properties convexity concavity and monoticity specially suitable for optimization purposes based on this fact a procedure for the global solution determination of eventually nonconvex problems involving the above mentioned function is proposed such problems embody some known designs such as filtering under noise uncertainty or precision constraints and optimal actuator sensor location the last problem is deeply analyzed and two practical applications namely satellite attitude control and large flexible system control are included

matrix factorizations for symplectic qr like methods abstract symplectic qr like methods use symplectic or unitary symplectic similarity transformations instead of the usual unitary similarity transformations in the qr process a fundamental problem for the development of such methods is the choice of a suitable type of decomposition a sr corresponding to the qr decomposition where s is symplectic or unitary symplectic decompositions of this type are studied with regard to their application in a qr like process

a class of hamiltonian symplectic methods for solving the algebraic riccati equation abstract the algebraic riccati equation can be solved by finding a certain invariant subspace of a related hamiltonian matrix g ammar and v mehrmann devised a method for calculating this invariant subspace that exploits the hamiltonian structure of the matrix by using unitary symplectic similarity transformations this paper discusses a class of methods that find the invariant subspace by using symplectic similarity transformations that are not necessarily unitary the method of ammar and mehrmann is a special case

sr and sz algorithms for the symplectic butterfly eigenproblem abstract sr and sz algorithms for the symplectic generalized eigenproblem that are based on the reduction of a symplectic matrix to symplectic butterfly form are discussed a 2 n 2 n symplectic butterfly matrix has 8 n 4 generically nonzero entries which are determined by 4 n 1 parameters while the sr algorithm operates directly on the matrix entries the sz algorithm works with the 4 n 1 parameters the algorithms are made more compact and efficient by using laurent polynomials instead of standard polynomials to drive the iterations

a numerical simulation and explicit solutions of the generalized burgers fisher equation we consider solitary wave solutions of the generalized burgers fisher bf equation u t pu ru x u x x qu 1 u r 0 in this paper by considering the decomposition scheme we first obtain the exact solutions of the generalized bf equation for the initial condition without using any classical transformations and then its numerical solutions are constructed without using any discretization technique the numerical solutions are compared with the known analytical solutions its remarkable accuracy is finally demonstrated in the study of some values r 2 of the generalized bf equation

a jacobi like method for solving algebraic riccati equations on parallel computers an algorithm to solve continuous time algebraic riccati equations through the hamiltonian schur form is developed it is an adaption for hamiltonian matrices of an asymmetric jacobi method of eberlein 1987 it uses unitary symplectic similarity transformations and preserves the hamiltonian structure of the matrix each iteration step needs only local information about the current matrix thus admitting efficient parallel implementations on certain parallel architectures convergence performance of the algorithm is compared with the hamiltonian jacobi algorithm of byers 1990 the numerical experiments suggest that the method presented here converges considerably faster for non hermitian hamiltonian matrices than byers hamiltonian jacobi algorithm besides that numerical experiments suggest that for the method presented here the number of iterations needed for convergence can be predicted by a simple function of the matrix size

the moser veselov equation abstract we study the orthogonal solutions of the matrix equation xj jx t m where j is symmetric positive definite and m is skew symmetric this equation arises in the discrete version of the dynamics of a rigid body investigated by moser and veselov commun math phys 139 1991 217 we show connections between orthogonal solutions of this equation and solutions of a certain algebraic riccati equation this will bring out the symplectic geometry of the moser veselov equation and also reduces most computational issues about solutions to finding invariant subspaces of a certain hamiltonian matrix necessary and sufficient conditions for the existence of orthogonal solutions and methods to compute them are presented our method is contrasted with the moser veselov approach commun math phys 139 1991 217 we also exhibit explicit solutions of a particular case of the moser veselov equation which appears associated with the continuous version of the dynamics of a rigid body

some numerical considerations and newton s method revisited for solving algebraic riccati equations analyzes some of the numerical aspects of solving the algebraic riccati equation are this analysis applies to both the symmetric and unsymmetric cases the author reconsiders the numerically relevant problems of balancing the are and the conditioning properties of the are and shows how these can be exploited by a solution algorithm he proposes an estimator for the condition number of the sylvester equation ax xb c based on iterative refinement also he interprets newton s method as a sequence of similarity transformations on the underlying system matrix this closes the gap between so called global and iterative methods for solving the are and also suggests an altogether revised implementation of newton s method one of the advantages of this revised implementation is that in the case where newton s method converges to a solution different from the desired solution enough information emerges to allow a switch to the desired solution the author examines the roundoff properties of the new algorithm and provides implementation considerations and numerical examples to highlight pros and cons

continuity properties of solutions of vector optimization this paper characterizes upper semicontinuity and lower semicontinuity of weakly efficient solution mapping sw and efficient solution mapping s on the space cm x of continuous m dimension vector real valued objective functions on a nonempty compact set x as applications some more interesting results about the properties of weakly efficient solutions and efficient solutions are obtained such as the upper semicontinuity of s at f in cm x implies the closedness and connectedness of the set s f of efficient solutions any essential weakly efficient solution of f can be approximated by a sequence of efficient solutions of f i e ew f s f and in the sense of baire categories most of vector optimization problems in cm x for efficient solutions are essential and so on

two connections between the sr and hr eigenvalue algorithms abstract the sr and hr algorithms are members of the family of gr algorithms for calculating eigenvalues and invariant subspaces of matrices this paper makes two connections between the sr and hr algorithms 1 an iteration of the sr algorithm on a 2 n 2 n symplectic butterfly matrix using shifts i 1 i i 1 k is equivalent to an iteration of the hr algorithm on an n n tridiagonal sign symmetric matrix using shifts i 1 i i 1 k 2 an iteration of the sr algorithm on a 2 n 2 n j tridagonal hamiltonian matrix using shifts i i i 1 k is equivalent to an iteration of the hr algorithm on an n n tridiagonal sign symmetric matrix using shifts 2 i i 1 k

an implicitly restarted symplectic lanczos method for the hamiltonian eigenvalue problem abstract an implicitly restarted symplectic lanczos method for the hamiltonian eigenvalue problem is presented the lanczos vectors are constructed to form a symplectic basis the inherent numerical difficulties of the symplectic lanczos method are addressed by inexpensive implicit restarts the method is used to compute eigenvalues eigenvectors and invariant subspaces of large and sparse hamiltonian matrices and low rank approximations to the solution of continuous time algebraic riccati equations with large and sparse coefficient matrices

general matrix pencil techniques for the solution of algebraic riccati equations a unified approach we present a unified theory of matrix pencil techniques for solving both continuous and discrete time algebraic riccati equations ares under fairly general conditions on the coefficient matrices the theory applies to a large class of ares and riccati like equations arising from the singular h sup spl infin and h sup 2 control problems singular linear quadratic control the 4 block nehari problem or from singular j spectral factorizations the underlying concept is the so called proper deflating subspace of a possibly singular matrix pencil in terms of which necessary and sufficient conditions for the solvability of riccati equations are given it is shown that these conditions can be checked and the solutions computed by a numerically sound algorithm

matrix factorizations and their perturbations there are several ways in which a matrix can be factorized as a product of two special matrices these factorizations are often used in numerical analysis and many perturbation bounds useful in such analysis have been proved by various authors in this paper a simple method which leads to these results and several new ones is discussed in the last section related results on the lipschitz continuity of the matrix absolute value are surveyed

qr like algorithms for eigenvalue problems in the year 2000 the dominant method for solving matrix eigenvalue problems is still the qr algorithm this paper discusses the family of gr algorithms with emphasis on the qr algorithm included are historical remarks an outline of what gr algorithms are and why they work and descriptions of the latest highly parallelizable versions of the qr algorithm now that we know how to parallelize it the qr algorithm seems likely to retain its dominance for many years to come

a hamiltonian krylov schur type method based on the symplectic lanczos process we discuss a krylov schur like restarting technique applied within the symplectic lanczos algorithm for the hamiltonian eigenvalue problem this allows to easily implement a purging and locking strategy in order to improve the convergence properties of the symplectic lanczos algorithm the krylov schur like restarting is based on the sr algorithm some ingredients of the latter need to be adapted to the structure of the symplectic lanczos recursion we demonstrate the efficiency of the new method for several hamiltonian eigenproblems

product eigenvalue problems many eigenvalue problems are most naturally viewed as product eigenvalue problems the eigenvalues of a matrix a are wanted but a is not given explicitly instead it is presented as a product of several factors a a k a k 1 cdots a 1 usually more accurate results are obtained by working with the factors rather than forming a explicitly for example if we want eigenvalues vectors of b t b it is better to work directly with b and not compute the product the intent of this paper is to demonstrate that the product eigenvalue problem is a powerful unifying concept diverse examples of eigenvalue problems are discussed and formulated as product eigenvalue problems for all but a couple of these examples it is shown that the standard algorithms for solving them are instances of a generic gr algorithm applied to a related cyclic matrix

self similar flows abstract large classes of self similar isospectral flows can be viewed as continuous analogues of certain matrix eigenvalue algorithms in particular there exist families of flows associated with the qr lr and cholesky eigenvalue algorithms this paper uses lie theory to develop a general theory of self similar flows which includes the qr lr and cholesky flows as special cases also included are new families of flows associated with the sr and hr eigenvalue algorithms the basic theory produces analogues of unshifted single step eigenvalue algorithms but it is also shown how the theory can be extended to include flows which are continuous analogues of shifted and multiple step eigenvalue algorithms

on the decoupling of dichotomic linear hamiltonians considerations on integrating symmetric differential riccati equations abstract the main purpose of this paper is to analyze some questions related to the decoupling of a class of linear hamiltonian systems such as those of optimal control applications we call these systems dichotomic hamiltonian systems we are thus led rather naturally to consider the numerical integration of symmmetric differential riccati equations sdres we first briefly consider general time dependent transformation for linear hamiltonians and then the sdre our goal is to understand which integration schemes are well suited for this equation it is natural to suspect that so called symplectic schemes should be good candidates however possibly contrary to intuition we will see that the sdre is not generally a hamiltonian system by looking at which properties an integration scheme for the sdre should have we realize that among many of the known schemes the best suited seems to be the simple backward euler similar results are obtained for the integration of the symmetric lyapunov equation we also discuss the interpolation problem for the computed solution and the solution procedure with newton s method for the nonlinear discrete systems we show that newton s method is a fully symplectic algorithm for our problem that is it can be interpreted as a chain of symplectic transformations on a constant hamiltonian matrix finally we briefly discuss implementation aspects other choices and extensions of the results herein

metric regularity in convex semi infinite optimization under canonical perturbations this paper is concerned with the lipschitzian behavior of the optimal set of convex semi infinite optimization problems under continuous perturbations of the right hand side of the constraints and linear perturbations of the objective function in this framework we provide a sufficient condition for the metric regularity of the inverse of the optimal set mapping this condition consists of the slater constraint qualification together with a certain additional requirement in the karush kuhn tucker conditions for linear problems this sufficient condition turns out to be also necessary for the metric regularity and it is equivalent to some well known stability concepts

an sdr algorithm for the solution of the generalized algebraic riccati equation an algorithm is presented for solving certain classes of the algebraic riccati equation by using the mdr transformation and extending the symplectic qr like or sr method of a bunse gerstner linear alg applic vol 83 p 49 78 1986 and v mehrmann ph d diss univ bielefeld w germany to the generalized eigenvalue problem the riccati equations arise from linear systems of the form edx dt ax bu where e is nonsingular but for numerical reasons e sup 1 a and e sup 1 b cannot be computed reliably

a symplectic method for approximating all the eigenvalues of a hamiltonian matrix a fast method for computing all the eigenvalues of a hamiltonian matrix m is given the method relies on orthogonal symplectic similarity transformations which preserve structure and have desirable numerical properties the algorithm is about four times faster than the standard q r algorithm the computed eigenvalues are shown to be the exact eigenvalues of a matrix m e where vert e vert depends on the square root of the machine precision the accuracy of a computed eigenvalue depends on its condition and its magnitude larger eigenvalues typically being more accurate

conditional full stability of positivity preserving finite difference scheme for diffusion advection reaction models this work has been partially supported by the ministerio de economia y competitividad spanish grant mtm2017 89664 p

stability theory for linear inequality systems ii upper semicontinuity of the solution set mapping this paper deals with the upper semicontinuity of the solution set mapping for linear inequality systems complementing a previous work on lower semicontinuity and related stability concepts the main novelty of our approach is that we are not assuming any standard hypothesis about the set indexing the inequalities in the system this set possibly infinite has no topological structure and therefore the functional dependence between the linear inequalities and their associated indices has no qualification at all the space of consistent systems over a fixed index set is endowed with the uniform topology derived from the pseudometric of chebyshev which turns out to be a natural way to measure the size of the perturbations in this context we provide some necessary and some sufficient conditions for the upper semicontinuity of the feasible set map at a given system whose solution set is not necessarily bounded

a numerical method for computing the hamiltonian schur form we derive a new numerical method for computing the hamiltonian schur form of a hamiltonian matrix mathcal m in mathbb r 2n times 2n that has no purely imaginary eigenvalues we demonstrate the properties of the new method by showing its performance for the benchmark collection of continuous time algebraic riccati equations despite the fact that no complete error analysis for the method is yet available the numerical results indicate that if no eigenvalues of mathcal m are close to the imaginary axis then the method computes the exact hamiltonian schur form of a nearby hamiltonian matrix and thus is numerically strongly backward stable the new method is of complexity mathbf o n 3 and hence it solves a long standing open problem in numerical analysis

symplectic householder transformations for a qr like decomposition a geometric and algebraic approaches the aim of this paper is to show how geometric and algebraic approaches lead us to a new symplectic elementary transformations the 2 d symplectic householder transformations their features are studied in details their interesting properties allow us to construct a new algorithm for computing a sr factorization this algorithm is based only on these 2 d symplectic householder transformations its new features are highlighted the study shows that in the symplectic case the new algorithm is the corresponding one to the classical qr factorization algorithm via the householder transformations some numerical experiments are given

a symmetric structure preserving γqr algorithm for linear response eigenvalue problems abstract in this paper we present an efficient qr algorithm for solving the linear response eigenvalue problem h x x where h is symmetric with respect to 0 diag i n i n based on newly introduced orthogonal transformations the qr algorithm preserves the symmetric structure of h throughout the whole process and thus guarantees the computed eigenvalues to appear pairwise as they should with the help of a newly established implicit orthogonality theorem we incorporate the implicit multi shift technique to accelerate the convergence of the qr algorithm numerical experiments are given to show the effectiveness of the algorithm

a numerical treatment of the coupled viscous burgers equation in the presence of very large reynolds number abstract a numerical investigation of the coupled viscous burgers equation for very large reynolds numbers do not exist in the literature coupled viscous burgers equations are solved numerically in the presence of very large reynolds numbers for this case numerical approach to the coupled viscous burgers equation via contributions of two effective methods is used the first component of the mixed method is finite difference method and the second one is differential quadrature method for this process the third order modified cubic b spline functions are used as base function to display the effectiveness of the present mixed method four different test problems have been investigated for various values of the coefficients more particularly for very large reynolds numbers in other words for the very small value of kinematic viscosity parameters solutions are obtained error norms are calculated and compared with analytical results and also with numerical results of the related literature present results display that the present mixed method obtains high accurate solutions and in compatibility with both of the analytical and numerical results

the shift inverted j lanczos algorithm for the numerical solutions of large sparse algebraic riccati equations abstract the goal of solving an algebraic riccati equation is to find the stable invariant subspace corresponding to all the eigenvalues lying in the open left half plane the purpose of this paper is to propose a structure preserving lanczos type algorithm incorporated with shift and invert techniques named shift inverted j lanczos algorithm for computing the stable invariant subspace for large sparse hamiltonian matrices the algorithm is based on the j tridiagonalization procedure of a hamiltonian matrix using symplectic similarity transformations we give a detailed analysis on the convergence behavior of the j lanczos algorithm and present error bound analysis and paige type theorem numerical results for the proposed algorithm applied to a practical example arising from the position and velocity control for a string of high speed vehicles are reported

a new class of time discretization schemes for the solution of nonlinear pdes we consider issues of stability of time discretization schemes withexacttreatment of thelinear part elp schemes for solving nonlinear pdes a distinctive feature of elp schemes is the exact evaluation of the contribution of the linear term that is if the nonlinear term of the equation is zero then the scheme reduces to the evaluation of the exponential function of the operator representing the linear term computing and applying the exponential or other functions of operators with variable coefficients in the usual manner requires evaluating dense matrices and is highly inefficient it turns out that computing the exponential of strictly elliptic operators in the wavelet system of coordinates yields sparse matrices for a finite but arbitrary accuracy this observation makes our approach practical in a number of applications in particular we consider applications of elp schemes to advection diffusion equations we study the stability of these schemes and show that both explicit and implicit elp schemes have distinctly different stability properties if compared with known implicit explicit schemes for example we describe explicit schemes with stability regions similar to those of typical implicit schemes used for solving advection diffusion equations

the pfaff lattice on symplectic matrices the pfaff lattice is an integrable system arising from the sr group factorization in an analogous way to how the toda lattice arises from the qr group factorization in our earlier paper kodama and pierce 2007 int math res not arxiv 0705 0510 we studied the pfaff lattice hierarchy for the case where the lax matrix is defined to be a lower hessenberg matrix in this paper we deal with the case of a symplectic lower hessenberg lax matrix this forces the lax matrix to take a 2 2 block tridiagonal shape we then show that the odd members of the pfaff lattice hierarchy are trivial while the even members are equivalent to the indefinite toda lattice hierarchy defined in kodama and ye 1996 physica d 91 321 39 this is analogous to the case of the toda lattice hierarchy in relation to the kac van moerbeke system in the case with the initial matrix having only real or imaginary eigenvalues the fixed points of the even flows are given by 2 2 block diagonal matrices with zero diagonals we also consider a family of skew orthogonal polynomials with a symplectic recursion relation related to the pfaff lattice and find that they are succinctly expressed in terms of orthogonal polynomials appearing in the indefinite toda lattice

stability of semi infinite vector optimization problems without compact constraints this paper is devoted to the continuity of solution maps for perturbation semi infinite vector optimization problems without compact constraint sets the sufficient conditions for lower semicontinuity and upper semicontinuity of solution maps under functional perturbations of both objective functions and constraint sets are established some examples are given to analyze the assumptions in the main result

defect correction method for the solution of algebraic riccati equations the solution of discrete and continuous algebraic riccati equations is considered it is shown that if an approximate solution is obtained then the defect for this solution again solves an algebraic riccati equation of the same form and that the system properties of detectability and stabilizability are inherited by this defect equation on the basis of these results a general defect correction method is proposed and numerical examples are given for the use of this method in combination with a bunse gerstner and v mehrmann s 1986 sr method

the locally extrapolated exponential time differencing lod scheme for multidimensional reaction diffusion systems this paper introduces the local extrapolation of first order locally one dimensional exponential time differencing scheme for numerical solution of multidimensional nonlinear reaction diffusion systems this novel scheme has the benefit of solving multidimensional problems in locally one dimensional fashion by implementing sequences of tridiagonal matrix solvers instead of solving a banded system the storage size needed for solving systems in higher dimensions with this scheme is similar to that needed for one spatial dimension systems the stability monotonicity and convergence of the locally extrapolated exponential time differencing scheme have been examined stability analysis shows that the scheme is strongly stable l stable and is particularly beneficial to nonlinear partial differential equations with irregular initial data or discontinuity involving initial and boundary conditions due to its ability to damp spurious oscillations caused by high frequency components in the solution the performance of the novel scheme has been investigated by testing it on a two dimensional schnakenberg model two and three dimensional brusselator models and a three dimensional enzyme kinetics of michaelis menten type reaction diffusion model numerical experiments demonstrate the efficiency accuracy and reliability of the scheme locally extrapolated lod exponential time differencing scheme is developed multidimensional nonlinear reaction diffusion systems are considered monotonicity and stability are analyzed the order of convergence is examined

an upper j hessenberg reduction of a matrix through symplectic householder transformations abstract in this paper we introduce a reduction of a matrix to a condensed form the upper j hessenberg form via elementary symplectic householder transformations which are rank one modification of the identity features of the reduction are highlighted and a general algorithm is derived then we study different possibilities to specify the general algorithm in order to built better versions we are led to two variants numerically more stables that we compare to jhess algorithm also some numerical experiments for comparing the different algorithms are given

convergence of algorithms of decomposition type for the eigenvalue problem we develop the theory of convergence of a generic gr algorithm for the matrix eigenvalue problem that includes the qr lr sr and other algorithms as special cases our formulation allows for shifts of origin and multiple gr steps the convergence theory is based on the idea that the gr algorithm performs nested subspace iteration with a change of coordinate system at each step thus the convergence of the gr algorithm depends on the convergence of certain sequences of subspaces it also depends on the quality of the coordinate transformation matrices as measured by their condition numbers we show that with a certain obvious shifting strategy the gr algorithm typically has a quadratic asymptotic convergence rate for matrices possessing certain special types of structure cubic convergence can be achieved