construction of square root factor for solution of the lyapunov matrix equation abstract the lyapunov matrix equation is considered in this paper where the solution is a nonnegative definite matrix i e a matrix admitting decomposition in square root factors an algorithm for findings the square root factor without preliminary finding the solution itself is given

fast frequency sweep of fem models via the balanced truncation proper orthogonal decomposition a fast frequency sweep method for wideband antennas and infinite arrays based on a singular value decomposition svd krylov model reduction method for frequency domain tangential vector finite elements tvfems is presented reduced models are constructed using balanced congruence transformations constructed from the dominant invariant subspace of the system s hankel matrix traditionally forming such matrix requires the intensive computation of gramians the proposed method only forms their low rank cholesky factors via a novel adaptive proper orthogonal decomposition pod sampling strategy leading to significant savings unlike some other model reduction methods balanced truncation pod bt pod is directly applicable to lossy and dispersive electromagnetic models numerical studies on large scale wideband antennas and infinite arrays show that the method is stable error controllable and without memory overheads capable of up to two orders of magnitude speed ups

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

well conditioned asymptotic waveform evaluation for finite elements the frequency domain finite element method fem results in matrix equations that have polynomial dependence on the frequency of excitation for a wide band fast frequency sweep technique based on a moment matching model order reduction more process researchers generally take one of two approaches the first is to linearize the polynomial dependence which will either limit the bandwidth of accuracy or require the introduction of extra degrees of freedom and then use a well conditioned krylov subspace technique the second approach is to work directly with the polynomial matrix equation and use one of the available but ill conditioned asymptotic waveform evaluation awe methods for large scale fem simulations introducing extra degrees of freedom and therefore increasing the length of the more vectors and the amount of memory required is not desirable therefore the first approach is not alluring on the other hand an ill conditioned awe process is unattractive this paper presents a novel more technique for polynomial matrix equations that circumvents these problematic issues first this novel process does not require any additional unknowns second this process is well conditioned along with the presentation of the novel algorithm which is called well conditioned awe wcawe numerical examples modeled using the fem are given to illustrate its accuracy

a multilevel technique for the approximate solution of operator lyapunov and algebraic riccati equations we consider multigrid or more appropriately multilevel techniques for the numerical solution of operator lyapunov and algebraic riccati equations the riccati equation which is quadratic plays an essential role in the solution of linear quadratic optimal control problems the linear lyapunov equation is important in the stability theory for linear systems and its solution is the primary step in the newton kleinman nk algorithm for the solution of algebraic riccati equations both equations are operator equations when the underlying linear system is infinite dimensional in this case finite dimensional discretization is required however as the level of discretization increases the convergence rate of the standard iterative techniques for solving high order matrix lyapunov and riccati equations decreases to deal with this multileveling is introduced into the iterative nk method for solving the algebraic riccati equation and smith s method for solving matrix lyapunov equations theoretical result

balanced model reduction via the proper orthogonal decomposition a new method for performing a balanced reduction of a high order linear system is presented the technique combines the proper orthogonal decomposition and concepts from balanced realization theory the method of snapshotsisused to obtainlow rank reduced rangeapproximationsto thesystemcontrollability and observability grammiansineitherthetimeorfrequencydomain theapproximationsarethenusedtoobtainabalancedreducedorder model the method is particularly effective when a small number of outputs is of interest it is demonstrated for a linearized high order system that models unsteady motion of a two dimensional airfoil computation of the exact grammians would be impractical for such a large system for this problem very accurate reducedorder models are obtained that capture the required dynamics with just three states the new models exhibit far superiorperformancethanthosederived using a conventionalproperorthogonal decomposition although further development is necessary the concept also extends to nonlinear systems

a bayesian approach to managing learning curve uncertainty this paper introduces a bayesian decision theoretic model of optimal production in the presence of learning curve uncertainty the well known learning curve model is extended to allow for random variation in the learning process with uncertainty regarding some parameter of the variation a production run generates excess value above its current revenue for a bayesian manager in two ways it pushes the firm further along the learning curve increasing the likelihood of lower costs for future runs and it provides information through the observed costs that reduces the uncertainty regarding the rate at which costs are decreasing we provide conditions under which one of the classical deterministic learning curve results namely that optimal production exceeds the myopic level carries over to the extended framework we demonstrate that another classical deterministic learning curve result namely that optimal production increases with cumulative production does not hold in the bayesian setting

krylov subspace methods for solving large unsymmetric linear systems some algorithms based upon a projection process onto the krylov subspace k sub m span r sub 0 ar sub 0 a sup m 1 r sub 0 are developed generalizing the method of conjugate gradients to unsymmetric systems these methods are extensions of arnoldi s algorithm for solving eigenvalue problems the convergence is analyzed in terms of the distance of the solution to the subspace k sub m and some error bounds are established showing in particular a similarity with the conjugate gradient method for symmetric matrices when the eigenvalues are real several numerical experiments are described and discussed

a generalized adi iterative method the adi iterative method for the solution of sylvester s equationax xb c proceeds by strictly alternating between the solution of the two equations begin gathered left a delta k 1 i right x 2k 1 x 2k left b delta k 1 i right c hfill x 2k 2 left b tau k 1 i right left a tau k 1 i right x 2k 1 c hfill end gathered fork 0 1 2 herex o is a given initial approximate solution and the k and k are real or complex parameters chosen so that the computed approximate solutionsx k converge rapidly to the solutionx of the sylvester equation ask increases this paper discusses the possibility of solving one of the equations in the adi iterative method more often than the other one i e relaxing the strict alternation requirement in order to achieve a higher rate of convergence our analysis based on potential theory shows that this generalization of the adi iterative method can give faster convergence than when strict alternation is required

optimal alternating direction implicit parameters for nonsymmetric systems of linear equations the determination of optimal parameters for the well known alternating direction implicit adi method leads to a minimization problem for rational functions of given degree l on compact sets e and f that contain the eigenvalues of the matrices involved if e and f are real intervals the solution of this problem was given by zolotarev more than 100 years ago and by wachspress in the 1960s however in the complex case e g if the linear system arises from the discretization of non self adjoint elliptic differential equations little is known about this problem in this paper two approaches to the zolotarev problem in the complex plane are considered first generalized leja points are considered as an example for rational functions which are asymptotically minimal in addition for rectangular eigenvalue domains e and f the exact minimal solutions for l 2 are constructed and it is shown by numerical examples that the results obtained using the optimal parameters of degree 2 are comparable to tho

principal component analysis in linear systems controllability observability and model reduction kalman s minimal realization theory involves geometric objects controllable unobservable subspaces which are subject to structural instability specifically arbitrarily small perturbations in a model may cause a change in the dimensions of the associated subspaces this situation is manifested in computational difficulties which arise in attempts to apply textbook algorithms for computing a minimal realization structural instability associated with geometric theories is not unique to control it arises in the theory of linear equations as well in this setting the computational problems have been studied for decades and excellent tools have been developed for coping with the situation one of the main goals of this paper is to call attention to principal component analysis hotelling 1933 and an algorithm golub and reinsch 1970 for computing the singular value decompositon of a matrix together they form a powerful tool for coping with structural instability in dynamic systems as developed in this paper principal component analysis is a technique for analyzing signals singular value decomposition provides the computational machinery for this reason kalman s minimal realization theory is recast in terms of responses to injected signals application of the signal analysis to controllability and observability leads to a coordinate system in which the internally balanced model has special properties for asymptotically stable systems this yields working approximations of x c x bar o the controllable and unobservable subspaces it is proposed that a natural first step in model reduction is to apply the mechanics of minimal realization using these working subspaces

numerical solution of the lyapunov equation by approximate power iteration abstract we present the approximate power iteration api algorithm for the computation of the dominant invariant subspace of the solution x of large order lyapunov equations ax xa t q 0 without first computing the matrix x itself the api algorithm is an iterative procedure that uses krylov subspace bases in computing estimates of matrix vector products x v in a power iteration sequence application of the api algorithm requires that a a t x admits a good low rank solution then api provides an orthogonal basis of a subspace that closely approximates the dominant x invariant subspace of corresponding dimension analytical convergence results are also presented

feje r walsh points for rational functions and their use in the adi iterative method abstract a new approach to the rational zolotarev problem in the complex plane is presented this minimization problem for rational functions arises from the determination of optimal parameters for the adi iterative method applied to nonsymmetric systems of linear equations based on a generalization due to walsh of the fejer points which are well known in connection to polynomial interpolation in the complex plane we are able to construct rational functions which are almost optimal for the rational zolotarev problem in an asymptotic sense for the construction of these fejer walsh points we make use of the schwarz christoffel mapping for doubly connected polygonal regions due to akhiezer and its recent implementation by dappen a numerical example illustrates the usefulness of the fejer walsh points as adi parameters

krylov subspace methods for solving large lyapunov equations this paper considers several methods for calculating low rank approximate solutions to large scale lyapunov equations of the form ap pa bb 0 the interest in this problem stems from model reduction where the task is to approximate high dimensional models by ones of lower order the two recently developed krylov subspace methods exploited in this paper are the arnoldi method saad math comput 37 1981 pp 105 126 and the generalised minimum residual method gmres saad and schultz siam j sci statist comput 7 1986 pp 856 869 exact expressions for the approximation errors incurred are derived in both cases the numerical solution of the low dimensional linear matrix equation arising from the gmres method is discussed and an algorithm for its solution is proposed low rank solutions of discrete time lyapunov equations and continuous time algebraic riccati equations are also considered throughout this paper the authors tackle problems in which b has more than one column with the

the inverse problem for the hill operator a direct approach we consider the hill operator t yd2 dx2 v x on l2 r where v 2 l2 0 1 is an even 1 periodic real potential and r 1 0 v x dx 0 it is wellknown that the spectrum of t is absolutely continuous and consists of intervals s1 s2 such that sn any1 an any1 an an n 1 these intervals are separated by gaps g1 g2 where gn an an if a gap degenerates that is gn then the corresponding segments sn sn 1 merge let u x e v x e v be the solutions of the equation yf 00 vf ef e 2 c 1 1

inverse problem and the trace formula for the hill operator ii consider the hill operator t d 2 dx 2 q x on l 2 mathbb r where q in l 2 0 1 is a 1 periodic real potential and int 0 1q x dx 0 the spectrum of t is absolutely continuous and consists of intervals separated by gaps gamma n a n a n n geq 1 let mu n n geq 1 be the dirichlet eigenvalue of the equation y qy ly on the interval 0 1 introduce the vector g n g cn g sn in mathbb r 2 with components g cn a n a n 2 mu n and g sn gamma n 2 4 g cn 2 s n where the sign s n or s n for all n geq 1 using nonlinear functional analysis in hilbert spaces we show that the mapping g q to g q g n 1 iy in ell 2 oplus ell 2 is a real analytic isomorphism in the second part a trace formula for q in l 2 0 1 is proved

periodic jacobi operator with finitely supported perturbations the inverse resonance problem abstract we consider a periodic jacobi operator h with finitely supported perturbations on z we solve the inverse resonance problem we prove that the mapping from finitely supported perturbations to the scattering data the inverse of the transmission coefficient and the jost function on the right half axis is one to one and onto we consider the problem of reconstruction of the scattering data from all eigenvalues resonances and the set of zeros of r 1 where r is the reflection coefficient

a fuzzy set approach for evaluating and enhancing the mid term competitiveness of a semiconductor factory fuzzy set theory is applied in this study to establish a systematic procedure for evaluating the mid term competitiveness of a semiconductor manufacturing factory subsequently to enhance the overall competitiveness of the semiconductor manufacturing factory a capacity re allocation mechanism is designed to improve the competitiveness of less competitive products without raising the expense of the other products and diminishing their competitiveness for this purpose a fuzzy nonlinear programming fnp model is constructed and is then converted into an equivalent np problem that can be solved with the existing optimization problem solving package to prove the usefulness of the proposed methodology a real case containing the data collected from a dram manufacturing factory was investigated after reallocating its capacity the competitiveness of the whole factory was indeed enhanced besides the capacity plan developed by the capacity re allocation mechanism was shown to be both efficient and practical

inverse problem for the discrete 1d schrodinger operator with small periodic potentials consider the discrete 1d schrodinger operator on with an odd 2k periodic potential q for small potentials we show that the mapping q heights of vertical slits on the quasi momentum domain similar to the marchenko ostrovski maping for the hill operator is a local isomorphism and the isospectral set consists of 2 k distinct potentials finally the asymptotics of the spectrum are determined as q 0

controlling a stochastic process with unknown parameters the problem of controlling a stochastic process with unknown parameters over an infinite horizon with discounting is considered agents express beliefs about unknown parameters in terms of distributions under general conditions the sequence of beliefs converges to a limit distribution the limit distribution may or may not be concentrated at the true parameter value in some cases complete learning is optimal in others the optimal strategy does not imply complete learning the paper concludes with examination of some special cases and a discussion of a procedure for generating examples in which incomplete learning is optimal copyright 1988 by the econometric society

multiproduct production planning in the presence of work force learning this paper explores the multiproduct production planning problem in the presence of work force learning mppl in this setting the work force productivity reflects a learning curve effect which can either increase or decrease learn or forget as a function of previous production volume this research advances the understanding of mppl along a number of dimensions we formulate mppl as a nonlinear mixed integer programming problem and establish problem complexity demonstrating that it is strongly np hard we also discuss some important economic properties of production planning in the presence of learning we then define a branch and bound algorithm for mppl as well as a tabu search heuristic tsh in addition we consider a previously defined heuristic for mppl and demonstrate that its performance can be arbitrarily bad computational experiments on a large set of test problems were performed to assess the performance of the algorithm and the tsh procedure and also to study the behavior of mppl problems the results of the experiments indicate that despite the underlying problem complexity the algorithm is able to solve reasonably large problems thus providing a basis for evaluating the heuristic solution quality the tsh consistently obtains high quality solutions to the test problems suggesting that the tabu search methodology offers an effective approach to complex production planning problems mppl problem difficulty is seen to vary with the number of periods in the planning horizon the relative degree of labor intensity and the relative demand behavior of products

on electromagnetics and information theory some connections are described between electromagnetic theory and information theory identifying some unavoidable limitations imposed by the laws of electromagnetism to communication systems starting from this result the role of the degrees of freedom of the field in radiating systems is investigated different classes of antennas use the available degrees of freedom in different ways in particular a multiple input multiple output antenna is a radiating system conveying statistically independent information on more than one degree of freedom of the field applications of the theory to antenna synthesis and antenna characterization in complex environments are shown

krylov subspace methods for the sylvester equation abstract we describe galerkin and minimal residual algorithms for the solution of sylvester s equation ax xb c the algorithms use krylov subspaces forwhich orthogonal bases are generated by the arnoldi process for certain choices of krylov subspaces the computation of the solution splits into the solution of many independent subproblems this makes the algorithms suitable for implementation on parallel computers

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

a schur method for balanced truncation model reduction it is shown that a not necessarily balanced state space realization of the moore reduced model can be computed directly without balancing via projections defined in terms of arbitrary bases for the left and right eigenspaces associated with the large eigenvalues of the product pq of the reachability and controllability grammians two specific methods for computing these bases are proposed one based on the ordered schur decomposition of pq and the other based on the cholesky factors of p and q the algorithms perform reliably even for nonminimal models

multipoint galerkin asymptotic waveform evaluation for model order reduction of frequency domain fem electromagnetic radiation problems in this paper a model order reduction technique is presented this technique known as galerkin asymptotic waveform evaluation gawe or multipoint galerkin asymptotic waveform evaluation mgawe if multiple expansion points are considered simultaneously can be used to reduce matrices describing electromagnetic em phenomena generated through the finite element method fem to a smaller space while still accurately approximating the characteristics of the original responses the resulting solution procedure of using gawe or mgawe to solve fem equations allows for wideband frequency simulations with a reduction in total computation time numerical simulations using these methods are shown along with traditional methods such as using an lu decomposition at each frequency point of interest and asymptotic waveform evaluation awe comparisons in accuracy as well as computation time are also given