a canonical form for pencils of matrices with applications to asymptotic linear programs abstract this paper studies asymptotic pencils of linear programs in which the constraint matrix right hand sides and objective function coefficients depend linearly on a parameter and we seek a basis that is optimal for all large or small enough values of the parameter the asymptotic linear programming formulation of discounted markov decision chains has this form as do many other problems for this problem we introduce a new canonical form for bases that facilitates computation of laurent expansions of the solutions of the resulting asymptotic linear programs and show how to implement the simplex method efficiently in this setting we show that the computational work to solve the asymptotic linear program is the product of the number of rows of the problem and the work required to solve an ordinary linear program of the same size this improves computational efficiency of the asymptotic simplex method of jeroslow 1973 and a related algorithm considered by lamond 1989 in essence this work generalizes that of miller and veinott 1969 and rothblum 1974 1975 for markov ordinary and markov branching decision chains to arbitrary linear programs

where is the nearest non regular pencil abstract this is a first step toward the goal of finding a way to calculate a smallest norm deregularizing perturbation of a given square matrix pencil minimal de regularizing perturbations have geometric characterizations that include a variable projection linear least squares problem and a minimax characterization reminiscent of the courant fischer theorem the characterizations lead to new computationally attractive upper and lower bounds we give a brief survey and illustrate strengths and weaknesses of several upper and lower bounds some of which are well known and some of which are new the ultimate goal remains elusive

rational and polynomial matrix factorizations via recursive pole zero cancellation we develop a recursive algorithm for obtaining factorizations of the type r r1 r2 where all three matrices are rational and r1 is nonsingular moreover the factors r1 and r2 are such that either the poles of r1 1 and r2 are in a prescribed region of the complex plane or their zeros such factorizations cover the specific cases of coprime factorization inner outer factorization gcd extraction and many more the algorithm works on the state space or generalized state space realization of r and derives in a recursive fashion the corresponding realizations of the factors

limit cycle stability reversal near a hopf bifurcation with aeroelastic applications abstract the objective of this paper is to present an analytical numerical analysis of the phenomenon of limit cycle stability reversal from unstable to stable and vice versa a singular perturbation technique the method of the normal form in the asymptotic expansion version is utilized the number of equations is then reduced to a minimal set for which the results are in good agreement with those from the original equations this minimal set is determined by the amplitude of the points a concept closely related to the small divisors in the kam theory this set is larger than that corresponding to the zero real part eigenvalues center manifold theorem the method is applied to a specific problem an aeroelastic section with cubic free play non linearities where the parameter is the flight speed numerical studies have been performed to show the dependence of the hopf bifurcation characteristics upon the structural and geometric properties of the wing section plots depicting amplitudes and frequency versus flight speed are presented

an improved algorithm for the computation of kronecker s canonical form of a singular pencil we give an 0 t algorithm for computing the kronecker structure of an arbitrary m x n pencil xe a the algorithm is shown to he numerically stable because only unitary transformations are used the improved speed over earlier unitary methods is due to the efficient use of condensed forms which are maintained throughout the recursions of the algorithm

hydrofoil vibration induced by a random flow a stochastic perturbation approach the problem of an elastic lifting hydrofoil in a randomly perturbed flow is considered it appears that in these conditions the phenomenon of hydroelastic induced vibrations is controlled by a stochastic differential operator by using the theory of stochastic perturbation a technique of solution for this class of problems is proposed leading to an effective numerical solution the fundamentals of the method are given and it is applied to the general problem of hydroelastic vibrations a numerical application to the case of an elastic control surface for a prototype high speed marine vehicle is presented comparisons between the results obtained by the stochastic perturbation method spm and those provided by standard monte carlo simulations mcs show the accuracy of the proposed method and a useful saving in computational time a method is given for comparing the computational time required by the two methods for a given statistical accuracy

eigenmodes of isospectral drums recently it was proved that there exist nonisometric planar regions that have identical laplace spectra that is one cannot hear the shape of a drum the simplest isospectral regions known are bounded by polygons with reentrant corners while the isospectrality can be proven mathematically analytical techniques are unable to produce the eigenvalues themselves furthermore standard numerical methods for computing the eigenvalues such as adaptive finite elements are highly inefficient physical experiments have been performed to measure the spectra but the accuracy and flexibility of this method are limited we describe an algorithm due to descloux and tolley comput methods appl mech engrg 39 1983 pp 37 53 that blends singular finite elements with domain decomposition and show that with a modification that doubles its accuracy this algorithm can be used to compute efficiently the eigenvalues for polygonal regions we present results accurate to 12 digits for the most famous pair of isospectral drums as well as results for another pair

ensemble energy average and energy flow relationships for nonstationary vibrating systems abstract this paper attempts to introduce a new point of view on energy analysis in structural dynamics with particular emphasis to its link with uncertainty and complexity a linear elastic system undergoing free vibrations is considered the system is subdivided into two subsystems and their respective energies together with the shared energy flow are analysed first the ensemble energy average of the two subsystems assuming uncertain the natural frequencies is investigated it is shown how the energy averages follow a simple law when observing the long term response of the system obtained by a suitable asymptotic expansion the second part of the analysis shows how the ensemble energy average of a set of random samples is representative even of the single case if the system is complex enough the two previous points combined produce a result that applies to the energy sharing between two subsystems even independently of uncertainty for complex systems a simple energy sharing law is indeed stated moreover in the case of absence of damping a nonlinear relation between the energy flow and the energy weighted difference between the two subsystems is derived on the other hand when damping is present this relationship becomes linear including two terms one is proportional to the energy weighted difference between the two subsystems the other being proportional to its time derivative therefore the approach suggests a way for deriving a general approach to energy sharing in vibration with results that in some cases are reminiscent of those met in statistical energy analysis finally computational experiments performed on systems of increasing complexity validate the theoretical results

structurally nonlinear fluttering airfoil in turbulent flow the dynamics and response of a structurally nonlinear two dimensional airfoil in turbulent flow are investigated numerically the solution is analyzed in terms of probability density function pdf response mean square power spectral density and lyapunov exponent both the longitudinal and vertical components of turbulence are considered it is shown that turbulent flow due to its longitudinal component diminishes the stability of the aeroelastic system by advancing the flutter point and decreasing the damping thus confirming results from previous work where only the longitudinal component was modeled furthermore the shift in flutter airspeed is strongly dependent on the random stiffness terms it is also observed that turbulence may advance the airspeed at which the pitch angle marginal pdf changes from uni to bimodal compared with the nonexcited deterministic case whereas the heave transition appears to be postponed this observation is attributed to the nonlinearity considered a cubic stiffness in torsion finally in terms of response mean square the system appears to he more sensitive to the presence of the longitudinal excitation component at pre rather than at postinstability airspeeds

an improved toeplitz algorithm for polynomial matrix null space computation in this paper we present an improved algorithm to compute the minimal null space basis of polynomial matrices a problem which has many applications in control and systems theory this algorithm takes advantage of the block toeplitz structure of the sylvester matrix associated with the polynomial matrix the analysis of algorithmic complexity and numerical stability shows that the algorithm is reliable and can be considered as an efficient alternative to the well known pencil state space algorithms found in the literature

ab algorithm and its modifications for the spectral problems of linear pencils of matrices this paper describes and algorithm and its modifications for solving spectral problems for linear pencils of matrices both regular as well as singular

the quasi weierstras form for regular matrix pencils regular linear matrix pencils a e k n n where k q r or c and the associated differential algebraic equation dae e u x ax are studied the wong sequences of subspaces are investigate and invoked to decompose the k n into v w where any bases of the linear spaces v and wtransform the matrix pencil into the quasi weierstras form the quasi weierstras form of the matrix pencil decouples the original dae into the underlying ode and the pure dae or in other words decouples the set of initial values into the set of consistent initial values vand pure inconsistent initial values w 0 furthermore vand ware spanned by the general ized eigenvectors at the finite and infinite eigenvalues resp the quasi weierstras form is used to show how chains of generalized eigenvectors at finite and infinite eigenvalues of a e lead to the well known weierstras form so the latter can be viewed as a generalized jordan form finally it is shown how eigenvector chains constitute a basis for the solution space of e u x ax

interconnection of kronecker canonical form and special coordinate basis of multivariable linear systems this paper establishes a straightforward interconnection between the kronecker canonical form and the special coordinate basis of linear systems such an interconnection yields an alternative approach for computing the kronecker canonical form and as a by product the smith form of the system matrix of general multivariable time invariant linear systems the overall procedure involves the transformation of a given system in the state space description into the special coordinate basis which is capable of explicitly displaying all the system structural properties such as finite and infinite zero structures as well as system invertibility structures the computation of the kronecker canonical form and smith form of the system matrix is rather simple and straightforward once the given system is put under the special coordinate basis the procedure is applicable to proper systems and singular systems

lq control of descriptor systems by cancelling structure at infinity this paper presents results concerning lq control of arbitrary descriptor systems and a numerical algorithm for achieving it the algorithm consists of two steps the first step is using the silverman algorithm orthogonal version transformation of the problem of lq control of descriptor system into a problem of lq control of a strictly proper system with singular or non singular new matrix 22 if new 22 is singular we perform the second step i e solving the lq singular control problem using again the silverman algorithm regarding the existing results on the lq singular control problem we have new results a counter example that this problem may not have a stable solution most general solvability conditions with or without stability and a new numerical algorithm based on using orthogonal matrices when stabilizability conditions are satisfied the algorithm places simultaneously stable poles in the closed loop system we show that the traditional riccati equation associated to this proble

regularity controllability observability of an nds with descriptor form subsystems and generalized lfts abstract this paper investigates regularity controllability and observability for a networked dynamic system nds with its subsystems being described in a descriptor form and system matrices of each subsystem being represented by a generalized linear fractional transformation glft of its parameters except a well posedness condition no any other constraints are put on either parameters or connections of a subsystem based on the kronecker canonical form kcf of a matrix pencil some matrix rank based necessary and sufficient conditions are established respectively for the regularity and complete controllability observability of the nds in which the associated matrix depends affinely on both subsystem parameters and subsystem connections these conditions keep the property that all the involved numerical computations are performed on each subsystem independently which is attractive in the analysis and synthesis of a large scale nds moreover some explicit and easily checkable requirements are derived for subsystem dynamics parameters with which a completely controllable observable nds can be constructed more easily

the laurent expansion of pencils that are singular at the origin abstract this paper gives two new proofs of a theorem of langenhop on the laurent expansion of a matrix pencil that is singular at the origin the second based on a decomposition of van dooren leads to a computational algorithm

pencils of complex and real symmetric and skew matrices abstract this expository paper establishes the canonical forms under congruence for pairs of complex or real symmetric or skew matrices the treatment is in the spirit of the well known book of gantmacher on matrix theory and may be regarded as a supplement to gantmacher s chapters on pencils of matrices

the wiener askey polynomial chaos for stochastic differential equations we present a new method for solving stochastic differential equations based on galerkin projections and extensions of wiener s polynomial chaos specifically we represent the stochastic processes with an optimum trial basis from the askey family of orthogonal polynomials that reduces the dimensionality of the system and leads to exponential convergence of the error several continuous and discrete processes are treated and numerical examples show substantial speed up compared to monte carlo simulations for low dimensional stochastic inputs

high order eigenvalues for the helmholtz equation in complicated non tensor domains through richardson extrapolation of second order finite differences we apply second order finite differences to calculate the lowest eigenvalues of the helmholtz equation for complicated non tensor domains in the plane using different grids which sample exactly the border of the domain we show that the results obtained applying richardson and pade richardson extrapolations to a set of finite difference eigenvalues corresponding to different grids allow us to obtain extremely precise values when possible we have assessed the precision of our extrapolations comparing them with the highly precise results obtained using the method of particular solutions our empirical findings suggest an asymptotic nature of the fd series in all the cases studied we are able to report numerical results which are more precise than those available in the literature

the generalized eigenstructure problem in linear system theory the algebraic theory of linear time invariant systems has been studied in large detail during the past few decades and numerous computational algorithms have been developed to solve problems arising in this context in this paper the numerical aspects of a certain class of such algorithms dealing with what the author calls generalized eigenstructure problems are discussed some new and or modified algorithms are presented both the nmnerical stability of the algorithms and the conditioning of the problems they solve are analyzed using numerical criteria

squaring down with zeros cancellation in generalized systems squaring down with simultaneous zeros cancellation by series compensation of a general possibly polynomial or improper linear system is investigated all static and dynamical compensators that spotlight minimal mcmillan degree are parameterized this general result is particularized to get compensators that preserve the l 2 or l inf norm of the original system either in continuous or discrete time all results are completely general numerically sound and based on general realizations allowing for poles at infinity

solving the helmholtz equation for membranes of arbitrary shape numerical results i calculate the modes of vibration of membranes of arbitrary shape using a collocation approach based on little sinc functions the matrix representation of the pde obtained using this method is explicit and does not require the calculation of integrals to illustrate the virtues of this approach i have considered a large number of examples part of them are taken from the literature and part of them new when possible i have tested the accuracy of these results by comparing them with the exact results when available or with results from the literature in particular in the case of the l shaped membrane the first example discussed in the paper i show that it is possible to extrapolate the results obtained with different grid sizes to obtain highly precise results finally i also show that the present collocation technique can be easily combined with conformal mapping to provide numerical approximations to the energies which quite rapidly converge to the exact results

matrix fraction approach for finite state aerodynamic modeling a least square procedure for the finite state approximation of the aerodynamic matrix is introduced which is more efficient and or simpler to use than those currently available this is accomplished by starting from the approximation of the aerodynamic matrix as nd 1 or d 1 n where n and d are matrices with polynomial dependence on complex reduced frequency p three different finite state realizations based on the same matrix polynomial approximation are presented the advantages of the approach and problems encountered in using the method are discussed numerical results are included

deflation in eigenvalue assignment of descriptor systems using state feedback it is shown how deflation by unitary equivalence transformations can be used to solve the eigenvalue assignment problem for descriptor systems using state feedback the system is first transformed to a form that reveals its controllability the uncontrollable part of the system if any is then discarded and the assignments are continued with the completely controllable part of the system to give insight into the approach a mathematical method for the solution is derived without considering numerical issues using the idea of deflation it is shown how this method can be modified to derive a numerically efficient algorithm the numerical stability of the algorithm is proved and a numerical example is presented

properties of numerical algorithms related to computing controllability the numerical properties of some methods for computing controllability are used in an expository way to motivate a wider understanding of numerical computations in particular the numerical rank of a matrix the numerical stability of algorithms the sensitivity of problems and the scaling of problems are discussed a numerically stable algorithm is given for computing controllability but it is pointed out that a measure of the distance of the given system from the nearest uncontrollable system would be more useful and this appears to be an open computational problem

a new look at pencils of matrix valued functions abstract matrix pencils depending on a parameter and their canonical forms under equivalence are discussed the study of matrix pencils or generalized eigenvalue problems is often motivated by applications from linear differential algebraic equations daes based on the weierstrass kronecker canonical form of the underlying matrix pencil one gets existence and uniqueness results for linear constant coefficients daes in order to study the solution behavior of linear daes with variable coefficients one has to look at new types of equivalence transformations this then leads to new canonical forms and new invariances for pencils of matrix valued functions we give a survey of recent results for square pencils and extend these results to nonsquare pencils furthermore we partially extend the results for canonical forms of hermitian pencils and give new canonicalforms there too based on these results we obtain new existence and uniqueness theorems for differential algebraic systems which generalize the classical results of weierstrass and kronecker

numerical investigation of isospectral cavities built from triangles we present computational approaches as alternatives to a recent microwave cavity experiment by s sridhar and a kudrolli phys rev lett 72 2175 1994 on isospectral cavities built from triangles a straightforward proof of isospectrality is given based on the mode matching method our results show that the experiment is accurate to 0 3 for the first 25 states the level statistics resemble those of a gaussian orthogonal ensemble when the integrable part of the spectrum is removed

a reduced order observer for descriptor systems abstract in this paper a reducer order observer is analyzed for so called generalized state space or descriptor systems based on the staircase form for generalized state space models a recursive algorithm is presented to construct a reduced order observer for a given observable descriptor system in the algorithm only the eigenfrequencies of the observer have to be specified

computation of structural invariants of generalized state space systems in this paper we develop an algorithm for computing the zeros of a generalized state space model described by the matrix 5 tuple e a b c d where e may be a singular matrix but det a he 0 the characterization of these zeros is based on the system matrix of the corresponding 5 tuple both the characterization and the computational algorithm are extensions of equivalent results for state space models described by the 4 tuples a b c d we also extend these results to the computation of infinite zeros and left and right minimal indices of the system matrix several non trivial numerical examples are included to illustrate the proposed results

a novel numerical method for exact model matching problem with stability in system design the exact model matching problem with stability consists of compensating a given system using a realizable control law of a specified structure in order to ensure the stability of the compensated system and achieve a target closed loop transfer function in this paper we develop a novel numerical method to verify the solvability of the problem for left invertible systems and further construct a desired solution our method has a complexity which is cubic in the state dimension of the system and the desired model and can be implemented in a numerically reliable way

polynomial matrices and feedback for a polynomial matrix having full row rank for all values of the parameter number of rows less than number of columns a number of extra rows is constructed such that the completed square polynomial matrix is unimodular this construction is performed in a numerically reliable way using a deadbeat control algorithm for a generalized state space system which in turn represents the given polynomial matrix

spectral and inner outer factorizations for discrete time systems the spectral and inner outer factorization problems are solved for discrete time systems with transfer matrices of arbitrary rank simple and explicit state space formulas expressed in terms of the stabilizing solution to the discrete time algebraic riccati system are derived the results are numerically effective since a reliable algorithm for computing such a stabilizing solution is available