backward perturbation analysis of certain characteristic subspaces this paper gives optimal backward perturbation bounds and the accuracy of approximate solutions for subspaces associated with certain eigenvalue problems such as the eigenvalue problemax x the generalized eigenvalue problem ax bx and the singular value decomposition of a matrixa this paper also gives residual bounds for certain eigenvalues generalized eigenvalues and singular values

a note on the backward errors for hermite eigenvalue problems we discuss the backward error analysis of the hermite eigenproblems three kinds of problems are included generalized eigenproblem matrix polynomial eigenproblem and multiparameter eigenproblem in the case of the approximate eigenvalue being real these problems have already been solved by higham and higham siam j matrix anal 20 1998 493 tisseur lin alg appl 309 2002 339 and hachstenbach and plestenjak lin alg appl 375 2003 63 this note as a complement of their results considers the complex case

non linear vibrations of viscoelastic moving belts part i free vibration analysis abstract the non linear free vibration of viscoelastic moving belts is studied based on the linear viscoelastic differential constitutive law the generalized equation of motion is derived for a moving belt with geometric non linearities the method of multiple scales is applied directly to the governing equation which is in the form of continuous autonomous gyroscopic systems with weak non linearity this direct treatment does not involve a prior assumption regarding the spatial solutions the non linear natural frequencies and free response amplitude for autonomous systems are predicted by the perturbation method the results obtained with the quasi static assumption and those without this assumption are compared the effects of elastic and viscoelastic parameters axial moving speed and the geometric non linearity on natural frequencies and amplitudes of the free response are investigated from numerical examples

inverse eigenproblem for centrosymmetric and centroskew matrices and their approximation in this paper we first give the solvability condition for the following inverse eigenproblem iep given a set of vectors xi i 1m in cn and a set of complex numbers i i 1m find a centrosymmetric or centroskew matrix c in rn n such that xi i 1m and i i 1m are the eigenvectors and eigenvalues of c respectively we then consider the best approximation problem for the ieps that are solvable more precisely given an arbitrary matrix b in rn n we find the matrix c which is the solution to the iep and is closest to b in the frobenius norm we show that the best approximation is unique and derive an expression for it

on the generalized bi skew symmetric solutions of a linear matrix equation and its procrust problems in this paper the solvability conditions and the explicit expressions of the generalized bisymmetric and bi skew symmetric solutions of the matrix equation ax b are respectively established by applying two methods then the maximal and minimal ranks of the solutions are derived if the solvability conditions are not satisfied the generalized bisymmetric and bi skew symmetric least squares solutions of the matrix equation are considered and the generalized bisymmetric and bi skew symmetric least squares solutions with the minimum norm are also obtained in addition two algorithms are provided to compute the generalized bi skew symmetric least squares solution and some examples are given to illustrate that the algorithms are feasible

supercritical speed stability of the trivial equilibrium of an axially moving string on an elastic foundation the stability of an axially moving string supported by a discrete or distributed elastic foundation is examined analytically particular attention is directed at the distribution of the critical speeds and identifying the divergence instability of the trivial equilibrium the elastically supported string shows unique stability behavior that is considerably different from unsupported axially moving string in particular any elastic foundation discrete or distributed leads to multiple critical speeds and a single region of divergence instability above the first critical speed whereas the unsupported string has one critical speed and is stable at all supercritical speeds additionally the elastically supported string critical speeds are bounded above and the maximum critical speed is the upper bound of the divergent speed region the analysis draws on the self adjoint eigenvalue problem for the critical speeds and a perturbation analysis about the critical speeds neither numerical solution nor spatial discretization which can produce substantially incorrect results are required the system falls in the class of dispersive gyroscopic continua and its behavior provides a useful comparison for general gyroscopic systems the analytical findings also serve as a benchmark for approximate methods applied to gyroscopic continua

the inverse eigenproblem with a submatrix constraint and the associated approximation problem for r s symmetric matrices abstract let r r n n and s r n n be nontrivial involutions i e r r 1 i and s s 1 i a matrix a r n n is called r s symmetric if r a s a this paper presents a r s symmetric matrix solution to the inverse eigenproblem with a leading principal submatrix constraint the solvability condition of the constrained inverse eigenproblem is also derived the existence the uniqueness and the expression of the r s symmetric matrix solution to the best approximation problem of the constrained inverse eigenproblem are achieved respectively an algorithm is presented to compute the r s symmetric matrix solution to the best approximation problem two numerical examples are given to illustrate the effectiveness of our results

minimization problems for r s symmetric and r s skew symmetric matrices let r cm m and s cn n be nontrivial involutions thus r r 1 i and s s 1 i we say that a cm n is r s symmetric r s skew symmetric if ras a ras a let s be the class of m n r s symmetric matrices or the class of m n r s skew symmetric matrices let z cn q and w cm q we study the following problems i give necessary and sufficient conditions for the existence of an a s such that az w and find all such matrices if the conditions are met ii find z w mina s az w and characterize the class s z w a s az w z w iii if b cm n is arbitrary find z w b mina s z w a b and find a s z w such that a b z w b we obtain explicit formulas for z w b z w and all the matrices in question

on the kahan parlett jiang theorem a globally optimal backward perturbation error for two sided invariant subspaces abstract a crucial question in large non hermitian eigenproblems is whether or not any residual bounds can be very useful in practical situations for this problem the famous kahan parlett jiang theorem kahan et al 1982 15 seems to be the best compromise between generality and sharpness in essence it presents an a posteriori error bound on approximate solutions to eigenproblems which provides a powerful tool to evaluate quality of the computed two sided invariant subspaces of large scale non hermitian matrices however the perturbation error determined by this theorem is only locally optimal rather than globally optimal in this work we revisit this problem and derive a globally optimal backward perturbation error for given two sided approximate invariant subspaces our theorem enhances the kahan parlett jiang theorem and numerical experiments demonstrate the effectiveness of our theoretical results

semidefinite inverse eigenvalue problems with prescribed entries and partial eigendata in this paper we study the semidefinite inverse eigenvalue problem of reconstructing a real n by n matrix c such that it is nearest to the original pre estimated real n by n matrix c o in the frobenius norm and satisfies the measured partial eigendata where the required matrix c should preserve the symmetry positive semidefiniteness and the prescribed entries of the pre estimated matrix c o we propose the alternating direction method of multipliers for solving the semidefinite inverse eigenvalue problem where three related iterative algorithms are presented we also extend our method to the case of lower bounds numerical experiments are reported to illustrate the efficiency of the proposed method for solving semidefinite inverse eigenvalue problems

symmetric matrix pencils abstract a significant number of matrix eigenvalue problems quadratic or linear are best reformulated as pencils a m in which both a and m are real and symmetric some examples are given and then the canonical forms are re examined to explain the role of the sign characteristic attached to real eigenvalues in addition we examine the limitations on the use of the rayleigh quotient functional x ax x mx in describing the eigenvalues this sheds new light on the class of definite pencils and the stability of their eigenvalues under perturbations the reduction of indefinite pencils to useful sparse forms is mentioned

three dimensional green s function for time harmonic dynamics in a viscoelastic layer abstract this paper presents green s function for time harmonic elastodynamic problems for a single layer domain three dimensional region bounded by two parallel planes with traction free boundary conditions the semi analytic solution is built in three steps a potential displacement representation b angular fourier series c radial hankel transform reflection matrices are presented for the plate domain kernels are integrally split into a singular closed form term the static half space solution plus an incremental solution in order to compute the inverse hankel transform for displacements and stress components a modified complex integration path is required theoretical considerations allow an adequate delimitation of such a complex path a specific treatment is proposed for low excitation frequencies where asymmetric lamb waves play a major role a series of numerical benchmarks are presented to validate the implementation of the functions displacements and tractions

structured mapping problems for linearly structured matrices abstract given an appropriate class of structured matrices s we characterize matrices x and b for which there exists a matrix a s such that a x b and determine all matrices in s mapping x to b we also determine all matrices in s mapping x to b and having the smallest norm we use these results to investigate structured backward errors of approximate eigenpairs and approximate invariant subspaces and structured pseudospectra of structured matrices

a note on stewart s theorem for definite matrix pairs abstract let a and b be n n hermitian matrices the matrix pair a b is called definite pair and the corresponding eigenvalue problem ax bx is definite if c a b inf x 1 h a ib x 0 in this note we develop a uniform upper bound for differences of corresponding eigenvalues of two definite pairs and so improve a result which is obtained by g w stewart 2 moreover we prove that this upper bound is a projective metric in the set of n n definite pairs

a note on the approximation of two sided rayleigh quotient to eigenvalues abstract we consider the two sided rayleigh quotient introduced by ostrowski as convergence acceleration tool for eigenvalue solvers an interesting property of the two sided rayleigh quotient is established its approximation to eigenvalues depends on the eigenvalue condition number this fact is valid not only for matrix but also for matrix polynomial multiparameter system and periodic matrix pairs

eigenvalues of rayleigh quotient matrices support thata is ann n hermitian matrix andx 1 y 1 aren 1 matrices satisfyingx 1 h x 1 y 1 h y 1 i 1 l n such that the column space x 1 is an eigenspace ofa and y 1 is an approximation of x 1 let a 1 x 1 h ax 1 h 1 y 1 h ay 1 r ay 1 y 1 h 1 and theta arccos x 1 h y 1 y 1 h x 1 1 mathord left vphantom 1 2 right kern nulldelimiterspace 2 geqslant 0 the matricesa 1 andh 1 are called the rayleigh quotient matrices ofa with respect tox 1 andy 1 respectively this paper derives new upper bounds of the difference between h 1 and a 1 the set of the eigenvaluesh 1 anda 1 in terms of the quantities sin and or r in a simple way here denotes a unitarily invariant norm

on the alternating direction method of multipliers for nonnegative inverse eigenvalue problems with partial eigendata we consider the nonnegative inverse eigenvalue problem with partial eigendata which aims to find a nonnegative matrix such that it is nearest to a pre estimated nonnegative matrix and satisfies the prescribed eigendata in this paper we propose several iterative schemes based on the alternating direction method of multipliers for solving the nonnegative inverse problem we also extend our schemes to the symmetric case and the cases of prescribed lower bounds and of prescribed entries numerical tests including a practical engineering application in vibrations show the efficiency of the proposed iterative schemes

the hermitian p k 1 anti reflexive solutions of a linear matrix equation in this paper we first establish the necessary and sufficient conditions for the existence and the explicit expressions of the hermitian p k 1 anti reflexive solutions of the matrix equation a x b and meanwhile the best approximation solution is considered then if the solvability conditions are not satisfied the least squares hermitian p k 1 anti reflexive solutions and the least squares hermitian p k 1 anti reflexive solutions with the minimum norm of the above matrix equation are respectively derived in addition two algorithms are shown to compute the least squares hermitian p k 1 anti reflexive solutions and the corresponding numerical examples are also given to illustrate the feasibility of the algorithms

residual bounds of approximate solutions of the algebraic riccati equation let tilde x geq 0 approximate the unique hermitian positive semi definite solution x to the algebraic riccati equation are where f g geq 0 a f is stabilizable and a g is detectable let be the residual of the are with respect to tilde x and define the linear operator vec l by by applying a new forward perturbation bound to the optimal backward perturbation corresponding to the approximate solution tilde x we obtained the following result if a f tilde x is stable and if 4 vec l 1 vec l 1 hat r f 1 for any unitarily invariant norm then

continuity properties of the 1 inverse and perturbation bounds for the drazin inverse there are known results about the continuity of the important pseudoinverses we extend some well known results to the 1 inverse as for the drazin inverse which is not a 1 inverse we present new perturbation bounds

on the variation of the spectrum of a normal matrix abstract let a and a be two n n normal matrices with spectra and j then by the hoffman wielandt theorem there is a permutation of 1 2 n such that n j 1 j j 2 a a f where f denotes the frobenius norm however if a is normal but a nonnormal it may be asked how to relate the eigenvalues of a to those of a an answer is given in this paper there is a permutation of 1 2 n such that n j 1 j j 2 n a a f and the factor n is best possible as a corollary we have j j n a a 2 for the spectral norm 2 thus the known upper bound 2n 1 a a 2 is reduced by a factor of about two

travelling load response of an axially moving string abstract the classical moving load problem in which a mass traverses an elastic structure is extended to systems in which the guideway also has prescribed axial motion a monocable ropeway such as a tramway or ski lift is an example of such a traveling structure traveling load system and it is modeled herein as an axially moving string that transports an attached discrete mass between two supports the green s function methods for the two constituent systems the particle and the uniform traveling string are combined to derive a volterra integral equation with delay that governs the interaction force in the coupled system the calculated response of the traveling string particle system to arbitrary initial conditions and external excitation is compared to that predicted by the derived asymptotic solution for vanishing discrete mass repeated scattering by the particle of transverse waves in the string dominates the motion transient vibration as the particle is transported between the supports is maximized at a transport speed that can be substantially lower than the critical speed of divergence instability for the uniform traveling string agreement between responses measured in the laboratory and those predicted by the traveling structure traveling load theory supports the analysis

inexact and incremental bilinear lanczos components algorithms for high dimensionality reduction and image reconstruction recently matrix based methods have gained wide attentions in pattern recognition and machine learning communities the generalized low rank approximations of matrices glram and the bilinear lanczos components blc algorithm are two popular algorithms that treat data as the native two dimensional matrix patterns however these two algorithms often require heavy computation time and memory space in practice especially for large scale problems in this paper we propose inexact and incremental bilinear lanczos components algorithms for high dimensionality reduction and image reconstruction we first introduce the thick restarting strategy to the blc algorithm and present a thick restarted lanczos components trblc algorithm in this algorithm we use the ritz vectors as approximations to dominant eigenvectors instead of the lanczos vectors in our implementation the iterative matrices are neither formed nor stored explicitly thanks to the characteristic of the lanczos procedure then we explore the relationship between the reconstruction error and the accuracy of the ritz vectors so that the computational complexities of eigenpairs can be reduced significantly as a result we propose an inexact thick restarted lanczos components inex trblc algorithm moreover we investigate the problem of incremental generalized low rank approximations of matrices and propose an incremental and inexact trblc incr trblc algorithm numerical experiments illustrate the superiority of the new algorithms over the glram algorithm and its variations as well as the blc algorithm for some real world image reconstruction and face recognition problems highlightswe proposed an inexact algorithm for image reconstructionthis algorithm is efficient both in storage and running time we analyze its convergence and backward errors an incremental version of the algorithm is given to handle dynamic increased data set

relative eigenvalues of a definite matrix pair abstract let a and b be hermitian matrices and let c a b min x 2 1 xh a ib x the matrix pair a b is called a definite pair and the corresponding eigenvalue problem ax bx is definite if c a b 0 the relationship between the eigenvalues of a b and those of a definite pair of the form xh1ax1 xh1bx1 is studied in this paper the eigenvalues of xh1ax1 xh1bx1 are said to be the relative eigenvalues of a b with respect to the subspace r x1 from the main result of this paper one can deduce a corresponding conclusion on hermitian matrices

perturbation theorems for the generalized eigenvalue problem abstract given a matrix pair z a b the perturbation of its eigenvalues is studied considering two pairs z w as points of the grassman manifold g n 2 n and its eigenvalues as points in g 1 2 the projective complex plane the distance of the spectra measured in the chordal metric in g 1 2 is bounded by some distance of the matrix pairs in g n 2 n analogs of the bauer fike theorem henrici s theorem and the hoffman weilandt theorem are obtained from which the classical results can be derived

structured backward errors for kkt systems karush kuhn tucker kkt systems are linear systems with coefficient matrices of the form graphics where h is symmetric a normwise structured backward error for kkt systems is defined and a com

free in plane vibration of an axially moving membrane the free in plane vibration of an axially moving membrane is studied considering the effects of both the translating speed and aspect ratio of the membrane two sets of boundary conditions which are free and fixed constraints in the lateral direction at boundaries with mass transport are discussed in this study from the extended hamilton principle the coupled equations of the longitudinal and lateral motions are derived these equations are then discretized by using the galerkin method from the discretized equations the natural frequencies and mode shapes are obtained for the variations of the translating speed and aspect ratio the results show that the translating speed aspect ratio and boundary conditions have significant effects on the in plane vibrations of the moving membrane

on estimating the separation of two regular matrix pairs let a b and c d be two n times n n n and m times m m m regular matrix pairs respectively the main contribution of this paper is to derive lower and upper bounds of the separation between a b and c d in terms of their generalized schur decompositions and weierstrass canonical forms the results are illustrated by numerical examples

on a class of alternating coefficient matrices quadratic eigenvalue problem in this paper we consider a class of alternating coefficient matrices quadratic eigenvalue problem aqep appearing in a wide range of applications at first the aqep is reformulated as a generalized eigenvalue problem gep by applying tisseur s linearization technique then we point out a property about skew symmetric tridiagonal matrix eigenvalue problem consequently we give that the corresponding gep can be solved by applying the cholesky ql algorithm or lanczos algorithm without complex arithmetic the structured backward error for qep would be discussed because of the special structure of the polynomial eigenvalue problems we develop structured condition number the upper bound and the lower bound are given some computational experiments and some concluding remarks are provided

structured backward errors for generalized saddle point systems abstract in this paper we investigate structured backward errors for three kinds of generalized saddle point systems where the matrix is not symmetric and its 2 2 block is not zero and has perturbations computable formula of backward errors are derived the expressions are useful for testing the stability of practical algorithms

eigenvector free solutions to ax b with px xp and xh sx constraints abstract the matrix equation ax b with px xp and xh sx constraints is considered where p is a given hermitian involutory matrix and s 1 by an eigenvalue decomposition of p we equivalently transform the constrained problem to two well known constrained problems and represent the solutions in terms of the eigenvectors of p using moore penrose generalized inverses of the products generated by matrices a b and p the involved eigenvectors can be released and eigenvector free formulas of the general solutions are presented similar strategy is applied to the equations ax b xc d with the same constraints

dynamic responses of the in plane and out of plane vibrations for an axially moving membrane the dynamic responses of both the in plane and out of plane vibrations are investigated for an axially moving membrane using the extended hamilton principle equations of motion are derived for the moving membrane with no slip boundary conditions at rollers the equations of motion are discretized by using the galerkin method and then the generalized a time integration method is applied to compute the dynamic responses of the membrane based on the computed results the responses are compared between the in plane and out of plane vibrations in addition the effects of translating velocity acceleration critical speeds and velocity profiles on the dynamic behaviours for displacements and stresses are investigated and discussed r 2006 elsevier ltd all rights reserved

stability analysis of an axially accelerating string the dynamic response of an axially accelerating string is investigated the time dependent velocity is assumed to vary harmonically about a constant mean velocity approximate analytical solutions are sought using two different approaches in the first approach the equations are discretized first and then the method of multiple scales is applied to the resulting equations in the second approach the method of multiple scales is applied directly to the partial differential system principal parametric resonances and combination resonances are investigated in detail stability boundaries are determined analytically it is found that instabilities occur when the frequency of velocity fluctuations is close to two times the natural frequency of the constant velocity system or when the frequency is close to the sum of any two natural frequencies when the velocity variation frequency is close to zero or to the difference of two natural frequencies however no instabilities are detected up to the first order of perturbation numerical results are presented for a band saw and a threadline problem

fem analysis of transverse vibrations of an axially moving membrane immersed in ideal fluid transverse vibrations of an axially moving travelling in its own plane membrane submerged in ideal fluid are analysed using the fem the discrete equations differ from the equations of the common coupled problem in that there appear two additional inertia terms due to the convective acceleration the effect of the density of the element mesh the effect of the truncation distance and the effect of various lumping techniques on the accuracy of the results are studied the fem results are compared with available analytical and experimental results the agreement seems to be rather good