dynamic analysis for axially moving viscoelastic panels abstract in this study stability and dynamic behaviour of axially moving viscoelastic panels are investigated with the help of the classical modal analysis we use the flat panel theory combined with the kelvin voigt viscoelastic constitutive model and we include the material derivative in the viscoelastic relations complex eigenvalues for the moving viscoelastic panel are studied with respect to the panel velocity and the corresponding eigenfunctions are found using central finite differences the governing equation for the transverse displacement of the panel is of fifth order in space and thus five boundary conditions are set for the problem the fifth condition is derived and set at the in flow end for clamped clamped and clamped simply supported panels the numerical results suggest that the moving viscoelastic panel undergoes divergence instability for low values of viscosity they also show that the critical panel velocity increases when viscosity is increased and that the viscoelastic panel does not experience instability with a sufficiently high viscosity coefficient for the cases with low viscosity the modes and velocities corresponding to divergence instability are found numerically we also report that the value of bending rigidity bending stiffness affects the distance between the divergence velocity and the flutter velocity the higher the bending rigidity the larger the distance

on computing the distance to stability for matrices using linear dissipative hamiltonian systems abstract in this paper we consider the problem of computing the nearest stable matrix to an unstable one we propose new algorithms to solve this problem based on a reformulation using linear dissipative hamiltonian systems we show that a matrix a is stable if and only if it can be written as a j r q where j j t r 0 and q 0 that is r is positive semidefinite and q is positive definite this reformulation results in an equivalent optimization problem with a simple convex feasible set we propose three strategies to solve the problem in variables j r q i a block coordinate descent method ii a projected gradient descent method and iii a fast gradient method inspired from smooth convex optimization these methods require o n 3 operations per iteration where n is the size of a we show the effectiveness of the fast gradient method compared to the other approaches and to several state of the art algorithms

the numerical study of the nonlinear dynamics of a light axially moving string abstract the vibration characteristics of a light axially moving band are investigated by a numerical study in the subcritical and supercritical speed ranges equations of motion for the geometrically nonlinear axially moving string are formulated using hamilton s principle and discretized by the finite element method the periodic nonlinear problem for the string is solved by the fourier galerkin newton fgn method the nonlinear dynamic behaviour of an axially moving band is examined through the dependences between the fundamental frequency axial velocity and vibration amplitude resulting from nonlinear free vibration in general the behaviour of the nonlinear axially moving string is similar to that of a nonlinear beam the most notable differences being the fact that the string does not undergo bifurcation from a straight configuration to curved equilibrium states in a supercritical transport speed regime nor does the equilibrium state remain straight

the transient amplitude of the viscoelastic travelling string an integral constitutive law abstract the problem considered is an initially stressed viscoelastic string subjected to steady state and harmonic variation of axial motion the string material is assumed to be constituted by the hereditary integral type the partial differential integral equation of motion is derived first then by applying galerkin s method the governing equation is reduced to a set of second order non linear differential integral equations which are solved by finite difference numerical integration procedures the viscoelastic string with a linear spring dashpot model is considered as the voigt element in series with a spring three parameter model the qualitative aspect of parametric excitation due to the non uniform travelling velocity of a viscoelastic string is investigated finally the results are compared with those obtained by a constitutive law of a differential type method the effects of elastic and viscoelastic parameters constant and non constant transport speeds the wave propagation speed ratio and the non linear term on the transient amplitudes are also investigated

perturbation theory for the eckart young mirsky theorem and the constrained total least squares problem abstract golub et al linear algebra appl 88 89 1987 317 327 j demmel siam j numer anal 24 1987 199 206 generalized the eckart young mirsky eym theorem which solves the problem of approximating a matrix by one of lower rank with only a specific rectangular subset of the matrix allowed to be changed based on their results this paper presents perturbation analysis for the eym theorem and the constrained total least squares problem ctls

designing structured tight frames via an alternating projection method tight frames also known as general welch bound equality sequences generalize orthonormal systems numerous applications including communications coding and sparse approximation require finite dimensional tight frames that possess additional structural properties this paper proposes an alternating projection method that is versatile enough to solve a huge class of inverse eigenvalue problems ieps which includes the frame design problem to apply this method one needs only to solve a matrix nearness problem that arises naturally from the design specifications therefore it is the fast and easy to develop versions of the algorithm that target new design problems alternating projection will often succeed even if algebraic constructions are unavailable to demonstrate that alternating projection is an effective tool for frame design the paper studies some important structural properties in detail first it addresses the most basic design problem constructing tight frames with prescribed vector norms then it discusses equiangular tight frames which are natural dictionaries for sparse approximation finally it examines tight frames whose individual vectors have low peak to average power ratio par which is a valuable property for code division multiple access cdma applications numerical experiments show that the proposed algorithm succeeds in each of these three cases the appendices investigate the convergence properties of the algorithm

a tutorial history of least squares with applications to astronomy and geodesy this article surveys the history development and applications of least squares including ordinary constrained weighted and total least squares the presentation includes proofs of the basic theory in particular unitary factorizations and singular value decompositions of matrices numerical examples with real data demonstrate how to set up and solve several types of problems of least squares the bibliography lists comprehensive sources for more specialized aspects of least squares

exponential parameter estimation in the presence of known components and noise in the determination of the natural modes of an electromagnetic scatterer the measured time series will contain desired information noise and quite often known transient components introduced by the excitation source or measuring equipment this paper describes a linearly constrained total least squares lctls prony method for extracting the exponential model parameters from observed transient data for such problems the tls criterion yields better parameter estimates than ls moreover the incorporation of known signal information via constraints leads to even greater improvements in performance mathematical connections between lctls prony and a tls variation of time series deflation tsd are used to derive constraints for higher order excitation poles also we use tsd concepts to derive numerically superior data transformations for lctls simulation studies involving idealized test data and synthetic scattering response data of a perfectly conducting sphere demonstrate the advantages of the method

structured total least squares and l2 approximation problems abstract it is shown how structured and weighted total least squares and l 2 approximation problems lead to a nonlinear generalized singular value decomposition an inverse iteration scheme to find a local minimum is proposed the emphasis of the paper is not on the convergence analysis of the algorithm rather the purpose is to illustrate its use in numerous applications in systems and control including total least squares with relative errors and or fixed elements inverse singular value problems an errors in variables variant of the kalman filter impulse response realization from noisy data h 2 model reduction h 2 system identification and calculating the largest stability radius of uncertain linear systems several numerical examples are given

on the bridge between combinatorial optimization and nonlinear optimization a family of semidefinite bounds for 0 1 quadratic problems leading to quasi newton methods this article presents a family of semidefinite programming bounds obtained by lagrangian duality for 0 1 quadratic optimization problems with linear or quadratic constraints these bounds have useful computational properties they have a good ratio of tightness to computing time they can be optimized by a quasi newton method and their final tightness level is controlled by a real parameter these properties are illustrated on three standard combinatorial optimization problems unconstrained 0 1 quadratic optimization heaviest k subgraph and graph bisection

numerical low rank approximation of matrix differential equations abstract the efficient numerical integration of large scale matrix differential equations is a topical problem in numerical analysis and of great importance in many applications standard numerical methods applied to such problems require an unduly amount of computing time and memory in general based on a dynamical low rank approximation of the solution a new splitting integrator is proposed for a quite general class of stiff matrix differential equations this class comprises differential lyapunov and differential riccati equations that arise from spatial discretizations of partial differential equations the proposed integrator handles stiffness in an efficient way and it preserves the symmetry and positive semidefiniteness of solutions of differential lyapunov equations numerical examples that illustrate the benefits of this new method are given in particular numerical results for the efficient simulation of the weather phenomenon el nino are presented

a factorization of determinant related to some random matrices we consider the expectation of the determinant det x 1for im 0 associated with some random n nmatrices and factorize it into nstieltjes transforms of probability measures moreover using this factorization we investigate the limiting behavior of the logarithm of the quantity as n

a multibody factorization method for independently moving objects the structure from motion problem has been extensively studied in the field of computer vision yet the bulk of the existing work assumes that the scene contains only a single moving object the more realistic case where an unknown number of objects move in the scene has received little attention especially for its theoretical treatment in this paper we present a new method for separating and recovering the motion and shape of multiple independently moving objects in a sequence of images the method does not require prior knowledge of the number of objects nor is dependent on any grouping of features into an object at the image level for this purpose we introduce a mathematical construct of object shapes called the shape interaction matrix which is invariant to both the object motions and the selection of coordinate systems this invariant structure is computable solely from the observed trajectories of image features without grouping them into individual objects once the matrix is computed it allows for segmenting features into objects by the process of transforming it into a canonical form as well as recovering the shape and motion of each object the theory works under a broad set of projection models scaled orthography paraperspective and affine but they must be linear so it excludes projective cameras

an iterative method for the symmetric and skew symmetric solutions of a linear matrix equation axb cyd e in this paper two efficient iterative methods are presented to solve the symmetric and skew symmetric solutions of a linear matrix equation axb cyd e respectively with real pair matrices x and y by these two iterative methods the solvability of the symmetric and skew symmetric solutions for the matrix equation can be determined automatically when the matrix equation has symmetric and skew symmetric solutions then for any initial pair matrices x 0 and y 0 symmetric and skew symmetric solutions can be obtained within finite iteration steps in the absence of roundoff errors and the minimum norm of the symmetric and skew symmetric solutions can be obtained by choosing a special kind of initial pair matrices in addition the unique optimal approximation pair solution x and y to the given matrices x and y in frobenius norm can be obtained by finding the minimum norm solution of a new matrix equation ax b cy d e where e e ax b cy d the given numerical examples demonstrate that the iterative methods are quite efficient

successive projection iterative method for solving matrix equation ax b in this paper we present a new iterative method successive projection iterative method to solve matrix equation ax b where a is a symmetric positive definite spd matrix based on this method an algorithm is proposed and proved to be convergent in addition analysis of the algorithm and numerical experiments are also given to show the efficiency of the method

iterative algorithm for solving a class of convex feasibility problem abstract in this paper we consider a class of convex feasibility problem which arises in quantum computation based on the matrix equation theory the feasible sets are characterized by exploiting the special structure of the linear constraints and its analytic expression is given by making use of the nice structure properties and the kkt condition we derive the projection formulas of a matrix onto the feasible sets the relaxed alternating projection method is designed to solve the convex feasibility problem numerical experiments show that the new method is feasible and effective

maximizing bilinear forms subject to linear constraints abstract we present methods for maximizing a bilinear form x t ay x y or its absolute value subject to linear constraints the constraints are either of the form c x d g y h or of the form c x e r d g y e r h where r d and r h represent the column spaces of matrices d and h we also show how to compute the angle between two subspaces of r m subject to these types of linear constraints and how to compute linearly constrained canonical correlations and the corresponding canonical vectors

optshrink an algorithm for improved low rank signal matrix denoising by optimal data driven singular value shrinkage the truncated singular value decomposition of the measurement matrix is the optimal solution to the representation problem of how to best approximate a noisy measurement matrix using a low rank matrix here we consider the unobservable denoising problem of how to best approximate a low rank signal matrix buried in noise by optimal re weighting of the singular vectors of the measurement matrix we exploit recent results from random matrix theory to exactly characterize the large matrix limit of the optimal weighting coefficients and show that they can be computed directly from data for a large class of noise models that includes the independent identically distributed gaussian noise case our analysis brings into sharp focus the shrinkage and thresholding form of the optimal weights the nonconvex nature of the associated shrinkage function on the singular values and explains why matrix regularization via singular value thresholding with convex penalty functions such as the nuclear norm will always be suboptimal we validate our theoretical predictions with numerical simulations develop an implementable algorithm optshrink that realizes the predicted performance gains and show how our methods can be used to improve estimation in the setting where the measured matrix has missing entries

least square solutions for inverse problems of centrosymmetric matrices let csrnxn a aij rnxn aij an 1 j i j 1 2 n in this paper we mainly discuss solving the following problems problem io given x b rnxm find a csrn n such that ax b problem i given x b rn m find a csrn n such that ax b min problem ii given a rn n find a se such that where is the frobenius norm and se is the solution set of problem i or i0 the expressions of the general solution of problems i and 10 have been derived the sufficient and necessary conditions for the solvability of problem i0 have been provided the numerical algorithm to find the optimal approximate solution and some numerical experiments have been given

an iterative method for the least squares symmetric solution of the linear matrix equation axb c abstract this paper an iterative method is presented to solve the minimum frobenius norm residual problem min axb c with unknown symmetric matrix x by this iterative method for any initial symmetric matrix x0 a solution x can be obtained within finite iteration steps in the absence of roundoff errors and the solution x with least norm can be obtained by choosing a special kind of initial symmetric matrix in addition the unique optimal approximation solution x to a given matrix x in frobenius norm can be obtained by first finding the least norm solution x of the new minimum residual problem min a x b c with unknown symmetric matrix x where c c a x x t 2 b given numerical examples are show that the iterative method is quite efficient

convergence results for projected line search methods on varieties of low rank matrices via łojasiewicz inequality the aim of this paper is to derive convergence results for projected line search methods on the real algebraic variety mathcal m le k of real m times n matrices of rank at most k such methods extend riemannian optimization methods which are successfully used on the smooth manifold mathcal m k of rank k matrices to its closure by taking steps along gradient related directions in the tangent cone and afterwards projecting back to mathcal m le k considering such a method circumvents the difficulties which arise from the nonclosedness and the unbounded curvature of mathcal m k the pointwise convergence is obtained for real analytic functions on the basis of a ojasiewicz inequality for the projection of the antigradient to the tangent cone if the derived limit point lies on the smooth part of mathcal m le k i e in mathcal m k this boils down to more or less known results but with the benefit that asymptotic convergence rate estimates for specific step sizes

symmetric confidence regions and confidence intervals for normal map formulations of stochastic variational inequalities stochastic variational inequalities svis model a large class of equilibrium problems subject to data uncertainty and are closely related to stochastic optimization problems the svi solution is usually estimated by a solution to a sample average approximation saa problem this paper considers the normal map formulation of an svi and proposes a method for building asymptotically exact confidence regions and confidence intervals for the solution of the normal map formulation based on the asymptotic distribution of saa solutions the confidence regions are single ellipsoids with high probability we also discuss the computation of simultaneous and individual confidence intervals

the change of similarity invariants under row perturbations the problem of the possible similarity classes of all the matrices obtained by small perturbations on some rows of a given complex square matrix is under consideration some necessary conditions on these similarity classes are provided the near converse problem is also studied under what conditions can a given collection of polynomials or partitions be the invariant factors or the weyr characteristic of a matrix obtained by perturbing some rows of a given matrix a complete answer is provided in some particular cases the obtained conditions can all be derived from the previously mentioned necessary conditions

approximating the nearest stable discrete time system abstract in this paper we consider the problem of stabilizing discrete time linear systems by computing a nearby stable matrix to an unstable one to do so we provide a new characterization for the set of stable matrices we show that a matrix a is stable if and only if it can be written as a s 1 u b s where s is positive definite u is orthogonal and b is a positive semidefinite contraction that is the singular values of b are less or equal to 1 this characterization results in an equivalent non convex optimization problem with a feasible set on which it is easy to project we propose a very efficient fast projected gradient method to tackle the problem in variables s u b and generate locally optimal solutions we show the effectiveness of the proposed method compared to other approaches

schur complements and statistics abstract in this paper we discuss various properties of matrices of the type s h ge 1 f which we call the schur complement of e in a e f g h the matrix e is assumed to be nonsingular when e is singular or rectangular we consider the generalized schur complements s h ge f where e is a generalized inverse of e a comprehensive account of results pertaining to the determinant the rank the inverse and generalized inverses of partitioned matrices and the inertia of a matrix is given both for schur complements and for generalized schur complements we survey the known results in a historical perspective and obtain several extensions numerous applications in numerical analysis and statistics are included the paper ends with an exhaustive bibliography of books and articles related to schur complements

steady state response of axially moving viscoelastic beams with pulsating speed comparison of two nonlinear models abstract principal parametric resonance in transverse vibration is investigated for viscoelastic beams moving with axial pulsating speed a nonlinear partial differential equation governing the transverse vibration is derived from the dynamical constitutive and geometrical relations under certain assumption the partial differential reduces to an integro partial differential equation for transverse vibration of axially accelerating viscoelastic nonlinear beams the method of multiple scales is applied to two equations to calculate the steady state response closed form solutions for the amplitude of the vibration are derived from the solvability condition of eliminating secular terms the stability of straight equilibrium and nontrivial steady state response are analyzed by use of the lyapunov linearized stability theory numerical examples are presented to highlight the effects of speed pulsation viscoelascity and nonlinearity and to compare results obtained from two equations

transverse vibrations of elastically connected double string complex system part i free vibrations this paper analyzes the forced transverse vibrations of an elastically connected double string complex continuous system the general solutions of forced vibrations of strings subjected to arbitrarily distributed continuous loads are found by using the method of expansion in a series of the mode shape functions different cases of exciting loadings are analyzed the action of stationary harmonic loads and moving concentrated forces is considered vibrations caused by the harmonic exciting forces are discussed and conditions of resonance and dynamic vibration absorption are formulated thus the string type dynamic absorber can be used to suppress the excessive vibrations of corresponding string systems the dynamic string type damper is a new concept of a continuous dynamic vibration absorber cdva it is shown that a corresponding two degree of freedom discrete system is an analogue of an elastically connected double string complex system theoretical analysis is illustrated by a numerical example

computing a nearest symmetric positive semidefinite matrix the nearest symmetric positive semidefinite matrix in the frobenius norm to an arbitrary real matrix a is shown to be b h 2 where h is the symmetric polar factor of b a at 2 in the 2 norm a nearest symmetric positive semidefinite matrix and its distance 2 a from a are given by a computationally challenging formula due to halmos we show how the bisection method can be applied to this formula to compute upper and lower bounds for 2 a differing by no more than a given amount a key ingredient is a stable and efficient test for positive definiteness based on an attempted choleski decomposition for accurate computation of 2 a we formulate the problem as one of zero finding and apply a hybrid newton bisection algorithm some numerical difficulties are discussed and illustrated by example

finding a low rank basis in a matrix subspace for a given matrix subspace how can we find a basis that consists of low rank matrices this is a generalization of the sparse vector problem it turns out that when the subspace is spanned by rank 1 matrices the matrices can be obtained by the tensor cp decomposition for the higher rank case the situation is not as straightforward in this work we present an algorithm based on a greedy process applicable to higher rank problems our algorithm first estimates the minimum rank by applying soft singular value thresholding to a nuclear norm relaxation and then computes a matrix with that rank using the method of alternating projections we provide local convergence results and compare our algorithm with several alternative approaches applications include data compression beyond the classical truncated svd computing accurate eigenvectors of a near multiple eigenvalue image separation and graph laplacian eigenproblems

technical communique boundary control of the axially moving kirchhoff string in this note the axially moving kirchhoff string is considered it is proved that this non linear string can be stabilized by the linear boundary control which is the negative feedback of the transversal velocity of the string at one end furthermore it is concluded that the axially moving kirchhoff string is stable without an active control when one of its ends is free to move transversally

a computation method for nonlinear vibration of axially accelerating viscoelastic strings a numerical algorithm is proposed for computing nonlinear vibration of axially accelerating viscoelastic strings based on independent functions the variational principle is used to discretize the governing equation into a set of differential algebraic equations numerical examples are presented

recovering low rank and sparse components of matrices from incomplete and noisy observations many problems can be characterized by the task of recovering the low rank and sparse components of a given matrix recently it was discovered that this nondeterministic polynomial time hard np hard task can be well accomplished both theoretically and numerically via heuristically solving a convex relaxation problem where the widely acknowledged nuclear norm and l 1 norm are utilized to induce low rank and sparsity this paper studies the recovery task in the general settings that only a fraction of entries of the matrix can be observed and the observation is corrupted by both impulsive and gaussian noise we show that the resulting model falls into the applicable scope of the classical augmented lagrangian method moreover the separable structure of the new model enables us to solve the involved subproblems more efficiently by splitting the augmented lagrangian function hence some splitting numerical algorithms are developed for solving the new recovery model some preliminary numerical experiments verify that these augmented lagrangian based splitting algorithms are easily implementable and surprisingly efficient for tackling the new recovery model

scaled total least squares fundamentals the standard approaches to solving overdetermined linear systems bx approx c construct minimal corrections to the vector c and or the matrix b such that the corrected system is compatible in ordinary least squares ls the correction is restricted to c while in data least squares dls it is restricted to b in scaled total least squares stls 22 corrections to both c and b are allowed and their relative sizes depend on a real positive parameter gamma stls unifies several formulations since it becomes total least squares tls when gamma 1 and in the limit corresponds to ls when gamma rightarrow 0 and dls when gamma rightarrow infty this paper analyzes a particularly useful formulation of the stls problem the analysis is based on a new assumption that guarantees existence and uniqueness of meaningful stls solutions for all parameters gamma 0 it makes the whole stls theory consistent our theory reveals the necessary and sufficient condition for preserving the smallest singular value of a matrix while appending or deleting a column this condition represents a basic matrix theory result for updating the singular value decomposition as well as the rank one modification of the hermitian eigenproblem the paper allows complex data and the equivalences in the limit of stls with dls and ls are proven for such data it is shown how any linear system bx approx c can be reduced to a minimally dimensioned core system satisfying our assumption consequently our theory and algorithms can be applied to fully general systems the basics of practical algorithms for both the stls and dls problems are indicated for either dense or large sparse systems our assumption and its consequences are compared with earlier approaches

on perturbations of some constrained subspaces perturbation bounds of subspaces such as eigen spaces singular subspaces and canonical subspaces have been extensively studied in the literature in this paper we study perturbations of some constrained subspaces of 1x2 2x1 and 2x2 block matrices in which only one of the sub matrices can be changed such problems rise from the least squares total least squares problem the constrained least squares problem and the constrained total least squares problem

qr based tls and mixed ls tls algorithms with applications to adaptive iir filtering the least squares ls total least squares tls and mixed ls tls approaches are compared as to their properties and performance on several classical filtering problems mixed ls tls is introduced as a qr decomposition based algorithm for unbiased equation error adaptive infinite impulse response iir filtering the algorithm is based on casting adaptive iir filtering into a mixed ls tls framework this formulation is shown to be equivalent to the minimization of the mean square equation error subject to a unit norm constraint on the denominator parameter vector an efficient implementation of the mixed ls tls solution is achieved through the use of back substitution and inverse iteration unbiasedness of the system parameter estimates is established for the mixed ls tls solution in the case of uncorrelated output noise and the algorithm is shown to converge to this solution ls tls and mixed ls tls performance is then compared for the problems of echo cancellation noise reduction and frequency equalization

a generalization of the eckart young mirsky matrix approximation theorem abstract the eckart young mirsky theorem solves the problem of approximating a matrix by one of lower rank however the approximation generally differs from the original in all its elements in this paper it is shown how to obtain a best approximation of lower rank in which a specified set of columns of the matrix remains fixed the paper concludes with some applications of the generalization

schmidt mirsky matrix approximation with linearly constrained singular values abstract schmidt and mirsky s theorems identify the matrix with a specified rank that lies closest to another matrix with distances measured by any matrix norm invariant under the unitary group the more general result presented here identifies the nearest matrix whose singular values satisfy any specified set of linear constraints a typical application consists in determining the oblate spheroid that most closely fits a specified ellipsoid

on the asymptotic behavior of scaled singular value and qr decompositions asymptotic expressions are derived for the singular value decompositon of a matrix some of whose columns approach zero expressions are also derived for the qr factorization of a matrix some of whose rows approach zero the expressions give insight into the method of weights for approximating the solutions of constrained least squares problems

smooth manifold structure for extreme channels a quantum channel from a system a of dimension da to a system b of dimension db is a completely positive trace preserving map from complex da da to db db matrices and the set of all such maps with kraus rank r has the structure of a smooth manifold we describe this set in two ways first as a quotient space of a subset of the rdb da dimensional stiefel manifold second as the set of all choi states of a fixed rank r these two descriptions are topologically equivalent this allows us to show that the set of all choi states corresponding to extreme channels from system a to system b of a fixed kraus rank r is a smooth submanifold of dimension 2rdadb da2 r2 of the set of all choi states of rank r as an application we derive a lower bound on the number of parameters required for a quantum circuit topology to be able to approximate all extreme channels from a to b arbitrarily well

on the statistical mechanics approach in the random matrix theory integrated density of states we consider the ensemble of random symmetricn n matrices specified by an orthogonal invariant probability distribution we treat this distribution as a gibbs measure of a mean field type model this allows us to show that the normalized eigenvalue counting function of this ensemble converges in probability to a nonrandom limit asn and that this limiting distribution is the solution of a certain self consistent equation

critical points of matrix least squares distance functions abstract a classical problem in matrix analysis and total least squares estimation is that of finding a best approximant of a given matrix by lower rank ones in this paper the critical points and the local minimum of the distance function f a x a x 2 on varieties of fixed rank symmetric skew symmetric and rectangular matrices x are determined our results extend earlier ones of eckart and young and of higham

sampling minimal subsets with large spans for robust estimation when sampling minimal subsets for robust parameter estimation it is commonly known that obtaining an all inlier minimal subset is not sufficient the points therein should also have a large spatial extent this paper investigates a theoretical basis behind this principle based on a little known result which expresses the least squares regression as a weighted linear combination of all possible minimal subset estimates it turns out that the weight of a minimal subset estimate is directly related to the span of the associated points we then derive an analogous result for total least squares which unlike ordinary least squares corrects for errors in both dependent and independent variables we establish the relevance of our result to computer vision by relating total least squares to geometric estimation techniques as practical contributions we elaborate why naive distance based sampling fails as a strategy to maximise the span of all inlier minimal subsets produced in addition we propose a novel method which unlike previous methods can consciously target all inlier minimal subsets with large spans

analysis of the generalized total least squares problemax b when some columns of a are free of error this paper studies the algebraic properties and perturbation theory of the generalized total least squares problem gtls ax b in whicha a 1 a 2 a 1 is free of error and the error contained in a 2 b is of the formec withc a given nonsingular matrix the problem was proposed by van huffel and vandewalle in 15 the solvability conditions formulas for the gtls solutions their residuals and the minimum norm correction matrices are obtained and a perturbation theory for the gtls problem is given

principal parametric resonance of axially accelerating viscoelastic strings with an integral constitutive law the steady state transverse responses and the stability of an axially accelerating viscoelastic string are investigated the governing equation is derived from the eulerian equation of motion of a continuum which leads to the mote model for transverse motion the kirchhoff model is derived from the mote model by replacing the tension with the averaged tension over the string the method of multiple scales is applied to the two models in the case of principal parametric resonance closed form expressions of the amplitudes and the existence conditions of steady state periodical responses are presented the lyapunov linearized stability theory is employed to demonstrate that the first second non trivial steady state response is always stable unstable numerical calculations show that the two models are qualitatively the same but quantitatively different numerical results are also presented to highlight the effects of the mean axial speed the axial speed fluctuation amplitude and the viscoelastic parameters

on a generalized matrix approximation problem in the spectral norm abstract in undetermined in this paper theoretical results regarding a generalized minimum rank matrix approximation problem in the spectral norm are presented an alternative solution expression for the generalized matrix approximation problem is obtained this alternative expression provides a simple characterization of the achievableminimum rank which is shown to be the same as the optimal objective value of the classical problem considered by eckart young schmidt mirsky as long as the generalized problem is feasible in addition this paper provides a result on a constrained version of the matrix approximation problem establishing that the later problem is solvable via singular value decomposition

refined error analyses of cholesky factorization this paper concentrates on the practical cholesky process for solving positive definite symmetric systems of linear equations in standard floating point arithmetic two self contained round off analyses are presented which lead to novel a posteriors error bounds of practical use and to refined a priori results of more theoretical significance unmotivated assumptions are avoided with the aim of being constructive throughout

rank constrained optimization and its applications abstract this paper investigates an iterative approach to solve the general rank constrained optimization problems rcops defined to optimize a convex objective function subject to a set of convex constraints and rank constraints on unknown rectangular matrices in addition rank minimization problems rmps are introduced and equivalently transformed into rcops by introducing a quadratic matrix equality constraint the rank function is discontinuous and nonconvex thus the general rcops are classified as np hard in most of the cases an iterative rank minimization irm method with convex formulation at each iteration is proposed to gradually approach the constrained rank the proposed irm method aims at solving rcops with rank inequalities constrained by upper or lower bounds as well as rank equality constraints proof of the convergence to a local minimizer with at least a sublinear convergence rate is provided four representative applications of rcops and rmps including system identification output feedback stabilization and structured h 2 controller design problems are presented with comparative simulation results to verify the feasibility and improved performance of the proposed irm method

semi definite matrix constraints in optimization positive semi definite matrix constraints arise in a number of optimization problems in which some or all of the elements of a matrix are variables such as the educational testing and matrix modification problems the structure of such constraints is developed including expressions for the normal cone feasible directions and their application to optimality conditions a computational framework is given within which these concepts can be exploited and which permits the quantification of second order effects the matrix of lagrange multipliers in this formulation is shown to have an important relationship to the characterization of the normal cone modifications of various iterative schemes in nonlinear programming are considered in order to develop an effective algorithm for the educational testing problem and comparative numerical experiments are described it is shown that a particular choice of the penalty parameter for an l 1 exact penalty function is appropriate for this type of problem the be

simulations of transverse vibrations of an axially moving string a modified difference approach a modified finite difference approach to simulate transverse vibrations of an axially moving string is presented the stress is treated as a new unknown in discretization of the spatial variable a set of differential algebraic equations is established based on the discreted governing equation and the constitutive relation for linear vibrations a conserved functional is employed to test the algorithm and the 1 2 3 4 term truncated modal analytical solutions are compared with the numerical solution for the free nonlinear vibration a new conserved functional is used to check the algorithm effects of the transport speed on the free and forced nonlinear vibrations are numerically investigated

low rank approximation of the symmetric positive semidefinite matrix in this paper we consider the low rank approximation of the symmetric positive semidefinite matrix which arises in machine learning quantum chemistry and inverse problem we first characterize the feasible set by x yy t email protected r n x k and then transform low rank approximation into an unconstrained optimization problem finally we use the nonlinear conjugate gradient method with exact line search to compute the optimal low rank symmetric positive semidefinite approximation of the given matrix numerical examples show that the new method is feasible and effective

a numerical method for simulating transverse vibrations of an axially moving string a modified finite difference method is presented to simulate transverse vibrations of an axially moving string by discretizing the governing equation and the stress strain relation at different frictional knots two linear sparse finite difference equations are obtained which can be computed alternatively the numerical method makes the non linear model easier to deal with and of small truncation errors it also shows stable for small initial values so it can be used in analyzing the non linear vibration of viscoelastic moving string efficiently a practical way of testing the precise of the numerical results is given by using a conservative quantity numerical examples are presented and dynamical analysis is given by using the numerical results

transient responses of an axially accelerating viscoelastic string constituted by a fractional differentiation law abstract this paper deals with the transverse vibration of an initially stressed moving viscoelastic string obeying a fractional differentiation constitutive law the governing equation is derived from newtonian second law of motion and reduced to a set of non linear differential integral equations based on galerkin s truncation a numerical approach is proposed to solve numerically the differential integral equation through developing an approximate expression of the fractional derivatives involved some numerical examples are presented to highlight the effects of viscoelastic parameters and frequencies of parametric excitations on the transient responses of the axially moving string

absolute summability of the trace relation for certain schrodinger operators a recently established general trace formula for one dimensional schrodinger operators is systematically studied in the context of short range potentials potentials which approach different spatial asymptotes sufficiently fast and appropriate impurity defect interactions in one dimensional solids we prove the absolute summability of the trace formula and establish its connections with scattering quantities such as reflection coefficients in each case

application of alternating convex projection methods for computation of positive toeplitz matrices uses alternating convex projection techniques to compute the closest positive definite toeplitz matrix that satisfies certain inequality constraints to a specified symmetric matrix some applications to signal processing and control problems are discussed

hyperspheres and hyperplanes fitted seamlessly by algebraic constrained total least squares abstract for each finite set of points in a euclidean space of any dimension the algorithm presented here determines all the algebraically best fitting circles or lines spheres or planes or hyperspheres or hyperplanes in a seamless manner from spherical through affine manifolds in particular affine submanifolds of any dimensions are not singularities of the algorithm to this end the algorithm combines projective geometry coope s and gander et al s layouts of the equations and golub et al s generalization of the schmidt mirsky matrix approximation theorem to solve the equations the resulting best fitting manifolds remain invariant under rigid transformations moreover if the best fitting manifold is affine then it coincides with golub and van loan s affine manifold of total least squares thus the algorithm can also fit hyperspheres in a manner that remains robust with data lying near a hyperplane furthermore an analysis with a theorem of wedin s shows that the fitted hyperspheres sensitivity to perturbations of the data increases as the colinearity of the data increases

rank constrained matrix best approximation problem with respect to skew hermitian matrices in this literature we study a rank constrained matrix approximation problem in the frobenius norm minr x kbxbaf2 where k is a nonnegative integer a and x are skew hermitian matrices by using the singular value decomposition and the spectrum decomposition we derive some conditions for the existence of skew hermitian solutions and establish general forms for the skew hermitian solutions to this matrix approximation problem

eigenvalues of graded matrices and the condition numbers of a multiple eigenvalue this paper concerns two closely related topics the behavior of the eigenvalues of graded matrices and the perturbation of a nondefective multiple eigenvalue we will show that the eigenvalues of a graded matrix tend to share the graded structure of the matrix and give precise conditions insuring that this tendency is realized these results are then applied to show that the secants of the canonical angles between the left and right invariant of a multiple eigenvalue tend to characterize its behavior when its matrix is slightly perturbed

hybrid methods for solving the educational testing problem abstract methods for solving the educational testing problem are considered one approach glunt 1995 is to formulate the problem as a linear convex programming problem in which the constraint is the intersection of three convex sets this method is globally convergent but the rate of convergence is slow however the method does have the capability of determining the correct rank of the solution matrix and this can be done in relatively few iterations if the correct rank of the solution matrix is known it is shown how to formulate the problem as a smooth nonlinear minimization problem for which a rapid convergence can be obtained by l1sqp method 6 this paper studies hybrid methods that attempt to combine the best features of both types of method an important feature concerns the interfacing of the component methods thus it has to be decided which method to use first and when to switch between methods difficulties such as these are addressed in the paper comparative numerical results are also reported

fitting conics of specific types to data abstract for each finite set of points in the euclidean plane and for each type of conic section elliptic hyperbolic or parabolic the algorithm presented here determines all the algebraically best fitting conics of the selected type the best ellipse the best hyperbola or the best parabola the supporting theory expands on golub hoffman and stewart s generalization of the schmidt mirsky matrix approximation theorem on bookstein s and pratt s methods to fit conics and on gander golub and strebel s method to fit ellipses because neither the set of ellipses nor the set of hyperbolae is closed the algorithm and its supporting theory must accommodate their boundary which consists of the parabolic conics the corresponding optimization problem consists in minimizing a quadratic form with two quadratic constraints which an orthogonal change of variables transforms into a least squares problem with one quadratic constraint hence analogies with geodetic coordinates identify geometric causes of numerical instability for each type of conic the resulting best fitting conic remains invariant under euclidean transformations applications include the theory and use of sundials in archaeology and astronomy