the congruence class of the solutions of some matrix equations the congruence class of a least square solution for the following matrix equations ax b a xa d axb dand axxb ef is presented also we derive necessary and sufficient conditions for the existence of a least square solution and present a general form of such solutions using the singular value decomposition svd and generalized singular value decomposition gsvd

on the use of quarter point boundary elements for stress intensity factor computations this communication studies a procedure for stress intensity factor computations using traction singular quarter point boundary elements opening mode stress intensity factors are computed from the tractions nodal values at the crack tip a comparison is made between the factors calculated using this procedure and those obtained by previously recommended methods which made use of the nodal values of the displacements the proposed procedure was seen to be less discretization sensitive than any other of the considered methods accurate results were obtained even in the case of coarse meshes

towards a generalized singular value decomposition we suggest a form for and give a constructive derivation of the generalized singular value decomposition of any two matrices having the same number of columns we outline its desirable characteristics and compare it to an earlier suggestion by van loan siam j numer anal 13 1976 pp 76 83 the present form largely follows from the work of van loan but is slightly more general and computationally more amenable than that in the paper cited we also prove a useful extension of a theorem of stewart siam rev 19 1977 pp 634 662 on unitary decompositions of submatrices of a unitary matrix

singular value and generalized singular value decompositions and the solution of linear matrix equations abstract the solution of the linear matrix equations i axb cyd e and ii axb fxg e h are considered new necessary and sufficient conditions for the consistency of the equations are derived some using the generalized singular value decomposition special cases iii ax yd e and iv axb e are treated using the singular value decomposition numerical algorithms for the solutions are also suggested

computing thecs decomposition of a partitioned orthonormal matrix this paper describes an algorithm for simultaneously diagonalizing by orthogonal transformations the blocks of a partitioned matrix having orthonormal columns

a pair of simultaneous linear matrix equations a1xb1 c1 a2xb2 c2 and a matrix programming problem necessary and sufficient conditions are obtained for a pair of matrix equations a1xb1 c1 a2xb2 c2 on a general field f to have a common solution along with the expression for a general common solution when certain conditions hold common solutions of minimum rank are described a matrix programming problem is solved en route

finite iterative algorithms for a common solution to a group of complex matrix equations abstract an iterative algorithm is constructed to give a common solution to a group of complex matrix equations by using the proposed algorithm the existence of a common solution can be determined automatically when a common solution exists for this group of matrix equations it is proven by using a real inner product in complex matrix spaces as a tool that a solution can be obtained within finite iteration steps for any initial values in the absence of round off errors the algorithm is also generalized to solve a more general case a numerical example is given to illustrate the effectiveness of the proposed method

three dimensional curved crack in an elastic body abstract the boundary integral equations which give the relation between the crack opening displacement and traction on the surface of a crack embedded in an infinite isotropic elastic body are formulated the integral equations are transformed into spherical and cylindrical coordinates in the cases of cracks curved in the shape of spherical and cylindrical surfaces respectively so that these boundary integral equations may be converted into a system of algebraic equations by the boundary element method the dependence of stress intensity factors on the curvature of crack has been numerically calculated for the spherical crack with circular contour under a constant load

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

a representation of the general common solution to the matrix equations a1xb1 c1 and a2xb2 c2 with applications we give new necessary and sufficient conditions for the existence of a common solution to the pair of linear matrix equations a1xb1 c1 and a2xb2 c2 and derive a new representation of the general common solution to these two equations we apply this result to determine new necessary and sufficient conditions for the existence of an hermitian solution and a representation of the general hermitian solution to the matrix equation axb c

partitioning boundary integral equations and exact green s functions discretization of boundary integral equations leads to large full systems of algebraic equations in practice partitioning is a method for solving such systems by breaking them down into smaller systems it may be viewed merely as a technique from linear algebra however it is profitable to view it as arising directly from partitions of the boundary these partitions could be natural such as two separate boundaries but they need not be we investigate partitioning in the context of multiple scattering of acoustic waves by two sound hard obstacles the ideas extend to other physical situations specifically we make a connection between partitioning and the use of the exact green s function for a single obstacle in isolation this suggests computing the action of this green s function once and for all storing it perhaps on a compact disc and then using it to solve other problems in which the second obstacle is altered one example of this approach is computing the stress distribution around a cavity of a standard but complicated shape inside a structure whose shape is varied the theoretical foundation for these ideas is given as well as a connection with the use of generalized born series for multiple scattering problems important distinctions between the partitioning green s function idea in this paper and seemingly similar ideas such as substructuring multi zoning and domain decomposition are made

the symmetric solution of the matrix equations ax ya c axat bybt c and atxa btxb c d abstract the necessary and sufficient conditions for the existence of and the expressions for the symmetric solutions of matrix equations i ax ya c ii axa t byb t c and iii a t xa b t xb c d are derived in addition the minimum 2 norm least squares symmetric solution of equation i the minimum 2 norm symmetric solution of equation ii and the least squares solution of equation iii are obtained

the matrix nearness problem for symmetric matrices associated with the matrix equation atxa btxb c d abstract a direct method based on the projection theorem in inner products spaces the generalized singular value decomposition and the canonical correlation decomposition is presented for finding the optimal approximate solution x in the set s e to a given matrix x where s e denotes the least squares symmetric solution set of the matrix equation atxa btxb c d the analytical expression of the optimal approximate solution x is obtained and an algorithm for finding this solution is also suggested

three dimensional crack analysis using singular boundary elements a multi domain method of solving three dimensional elastic crack problems in an infinite elastic body using the boundary element method is proposed the displacement and traction behaviours near a crack front are incorporated in special crack elements the elimination of singularities arising from the term combined with kelvin s kernel for displacement in the integrals is discussed in detail stress intensity factors of modes i ii and iii are obtained directly from crack front nodal values without any extrapolation as in some other methods no differentiation of conventional boundary integral equations with kelvin s tensor kernels is necessary in the current approach this method is applicable to cracks of arbitrary shape infinite bodies are modelled precisely as such not approximated as large finite bodies numerical solutions of stress intensity factors are given for several problems involving a penny shaped crack

linear matrix equations from an inverse problem of vibration theory abstract the symmetric positive semidefinite and positive definite real solutions of matrix equations a t xa d and a t xa xa y ad d 0 are considered necessary and sufficient conditions for the existence of such solutions and their general forms are derived using the singular value decomposition the theory is motivated and illustrated with a problem of vibration theory

submatrix constrained least squares inverse problem for symmetric matrices from the design of vibrating structures abstract given a full column rank matrix x r n m a matrix b r m m and a symmetric matrix a 0 r p p in structural dynamic model updating yuan and dai 2007 considered the matrix equation x t a x b with a leading principal submatrix a 0 constraint for a given matrix a r n n they updated the mass and stiffness matrix in the frobenius norm sense such that the corrected matrices satisfy the generalized eigenvalue equation and orthogonality conditions but due to measurement errors the measured mass and stiffness matrices will not always satisfy these requirements since they still contain some useful information we would like to retrieve their least squares approximations to correct these matrices then we obtain the least squares symmetric solutions of the equation x t a x b with a trailing principal submatrix a 0 constraint by using the matrix differential calculus and canonical correlation decomposition furthermore by applying the generalized singular value decomposition and projection theorem we get the best frobenius norm approximate symmetric solution of this equation according to a given matrix a r n n with a 0 as its trailing principal submatrix finally a numerical algorithm for computing the best approximate solution is established some illustrated numerical examples are also presented

a boundary integral equation method for radiation and scattering of elastic waves in three dimensions etude du rayonnement et de la diffusion d ondes elastiques par des obstacles de forme arbitraire

on the computation of two dimensional stress intensity factors using the boundary element method general two dimensional linear elastic fracture problems are investigated using the boundary element method the r displacement and 1 r traction behaviour near a crack tip are incorporated in special crack elements stress intensity factors of both modes i and ii are obtained directly from crack tip nodal values for a variety of crack problems including straight and curved cracks in finite and infinite bodies a multidomain approach is adopted to treat cracks in an infinite body the body is subdivided into two regions an infinite part with a finite hole and a finite inclusion numerical results compared with exact solution whenever possible are accurate even with a coarse discretization

improvement of a large analytical model using test data a method has been developed which uses measured normal modes and natural frequencies to improve an analytical mass and stiffness matrix model of a structure the method directly identifies without iteration a set of minimum changes in the analytical matrices which force the eigensolutions to agree with the test measurements an application is presented in which the analytical model had 508 degrees of freedom and 19 modes were measured at 101 locations on the structure the resulting changes in the model are judged to be small compared to expectations of error in the analysis thus the improved model is accepted as a reasonable model of the structure with improved dynamic response characteristics in addition it is shown that the procedure may be a useful tool in identifying apparent measured modes which are not true normal modes of the structure nomenclature analytical matrix matrix of changes identity matrix full improved stiffness and mass matrices n x n full analytical k m matrices n x n partitions of ka ma corresponding to test coordinates partitions of ka ma corresponding to coupling elements partitions of ka ma corresponding to unmea sured coordinates number of measured modes number of degrees of freedom in the model measures of changes eqs 15 17 matrix norm sum of the squares of all the elements rectangular modal matrix normalized n x m th mode th column of measured and unmeasured partitions of diagonal matrix of measured natural frequencies m xm natural frequency of th mode 12 17 sum of the squares of all elements of matrix

perturbation analysis of the canonical correlations of matrix pairs abstract we present a perturbation analysis of the canonical correlations of matrix pairs absolute perturbation bounds for normwise as well as componentwise perturbations are derived we demonstrate by way of examples that small relative perturbation of a matrix pair does not necessarily imply small relative perturbation in its canonical correlations and we identify a class of matrix pairs for which good relative perturbation bounds can be derived under certain conditions

lsqr iterative common symmetric solutions to matrix equations axb e and cxd f abstract a new matrix based iterative method is presented to compute common symmetric solution or common symmetric least squares solution of the pair of matrix equations axb e and cxd f by this iterative method for any initial matrix x0 a solution x can be obtained within finite iteration steps if exact arithmetic was used and the solution x with the minimum frobenius norm can be obtained by choosing a special kind of initial matrix in addition the unique nearest common symmetric solution or common symmetric least squares solution x to given matrix x in frobenius norm can be obtained by first finding the minimum frobenius norm common symmetric solution or common symmetric least squares solution of the new pair of matrix equations the given numerical examples show that the matrix based iterative method proposed in this paper has faster convergence than the iterative methods proposed in 1 2 to solve the same problems

on the existence of a common solution x to the matrix equations aixbj cij i j γ in this paper conditions are derived for the existence of a common solution x to the matrix equations aixbj cij i j where the matrices ai bj cij and x have suitable dimensions and the i j s are index pairs in some set the purpose of this paper is to present for certain specific sets of index pairs verifiable necessary and sufficient solvability conditions that are stated directly in terms of the matrices and that do not use kronecker products