the common re nnd and re pd solutions to the matrix equations ax c and xb d abstract in this article we consider common re nnd and re pd solutions of the matrix equations ax c and xb d with respect to x where a b c and d are given matrices we give necessary and sufficient conditions for the existence of common re nnd and re pd solutions to the pair of the matrix equations and derive a representation of the common re nnd and re pd solutions to these two equations when they exist the presented examples show the advantage of the proposed approach

the reflexive and anti reflexive solutions of the matrix equation ax b abstract an n n complex matrix p is said to be a generalized reflection matrix if p h p and p 2 i an n n complex matrix a is said to be a reflexive or anti reflexive matrix with respect to the generalized reflection matrix p if a pap or a pap this paper establishes the necessary and sufficient conditions for the existence of and the expressions for the reflexive and anti reflexive with respect to a generalized reflection matrix p solutions of the matrix equation ax b in addition in corresponding solution set of the equation the explicit expression of the nearest matrix to a given matrix in the frobenius norm have been provided

a method for the integration in time of certain partial differential equations abstract a method for the numerical solution of ordinary differential equations is analyzed that is explicit and yet can conserve the quadratic quantities conserved by the equations the method can be a useful alternative to the usual leapfrog technique in that it does not suffer from the occurrence of blowup phenomena numerical examples concerning the korteweg de vries equation and the nonlinear schrodinger equation are given

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

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

alternating direction method for generalized sylvester matrix equation axb cyd e this paper presents alternating direction methods of multipliers for finding the solution the best approximate solution and the nonnegative solution of the generalized sylvester matrix equation axb cyd e where a b c d and e are given matrices of suitable sizes preliminary convergence properties of the proposed algorithms are given numerical experiments show that the proposed algorithms tend to deliver higher quality solutions with less iteration steps and less computing times than recent algorithms on the tested problems

solving general continuous single facility location problems by cutting planes abstract a unified cutting plane method is proposed to solve a large class of continuous single facility location problems these include both minisum and minimax problems with mixed norms and convex transportation costs the method provides an easy stopping rule and permits the post optimal determination of an outer approximation to near optimality regions two examples are given which show the power of the method extensive computational results in dimension two and higher are reported

the bisymmetric solutions of the matrix equation a1x1b1 a2x2b2 alxlbl c and its optimal approximation abstract a matrix a a ij r n n is said to be bisymmetric matrix if a ij a ji a n 1 j n 1 i for all 1 i j n in this paper an iterative method is constructed to find the bisymmetric solutions of matrix equation a 1 x 1 b 1 a 2 x 2 b 2 a l x l b l c where x 1 x 2 x l is real matrices group by this iterative method the solvability of the matrix equation can be judged automatically when the matrix equation is consistent for any initial bisymmetric matrix group x 1 0 x 2 0 x l 0 a bisymmetric solution group can be obtained within finite iteration steps in the absence of roundoff errors and the least norm bisymmetric solution group can be obtained by choosing a special kind of initial bisymmetric matrix group in addition the optimal approximation bisymmetric solution group to a given bisymmetric matrix group x 1 x 2 x l in frobenius norm can be obtained by finding the least norm bisymmetric solution group of new matrix equation a 1 x 1 b 1 a 2 x 2 b 2 a l x l b l c where c c a 1 x 1 b 1 a 2 x 2 b 2 a l x l b l

solutions and laws of conservation for coupled nonlinear schrodinger equations lie group analysis a set of two coupled nonlinear schr odinger equations is systematically analyzed by means of lie group technique the physical situations under consideration include nonlinear propagation in strongly birefringent and multimode optical fibers the most general lie group of point symmetries its lie algebra and a group of adjoint representations that correspond to the lie algebra are identified as a result a complete list of group invariant exact solutions is obtained and compared with earlier results the corresponding laws of conservation are derived employing noether s theorem

solving linearly constrained matrix least squares problem by lsqr abstract matrix iterative algorithm lsqr is proposed for solving the linearly constrained matrix least squares ls problem with the special properties of constraint matrix kronecker product and the coordinate mapping from the constrained space to its independent parameter space we transform the constrained matrix ls problem to the unconstrained long vector least squares problem and rewrite the corresponding vector form algorithm back to the matrix one the resulting matrix form iteration only consists of matrix matrix product and does not involve the kronecker product numerical results are reported to show the feasibility of the proposed method

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

least squares solutions of the matrix equation axb cyd e with the least norm for symmetric arrowhead matrices in this paper the least squares solution of the matrix equation axb cyd e for symmetric arrowhead matrices with the least norm is discussed by using moore penrose inverse and the kronecker product the general expression of the solution to this problem is derived a corresponding numerical algorithm and an example are also given

some properties of location problems with block and round norms the point objective problem and the weber problem are two well known formulations for locating a new facility with respect to a set of fixed facilities when locations are represented as points on a plane the point objective problem is a multiple objective formulation of minimizing the distance from a variable point to each of the fixed points similarly the weber problem is a single objective formulation of minimizing the sum of transportation costs between the variable point and the fixed points where transportation cost is a function of distance generalizing solution properties for these problems from distance measures given by the euclidean rectilinear ip and one infinity norms this paper develops solution properties under the broad classes of distance measures given by block and round norms for the point objective problem we show that i the efficient set for all round norms is the convex hull of the set of fixed points and ii the efficient set under a block norm tends to the convex hull for a sequence of block norms approaching a round norm for the weber problem we prove that i an optimal location for any block norm may be found in a finite set of intersection points belonging to the convex hull and ii this set tends to the convex hull for a sequence of block norms approaching a round norm finally we use these results to propose a synthesis of some of the main properties in continuous and network location theory

analytical best approximate hermitian and generalized skew hamiltonian solution of matrix equation axah cych f abstract in this paper we consider the matrix equation a x a h c y c h f where a c and f are given matrices with appropriate sizes and x y is an unknown hermitian and generalized skew hamiltonian matrix pair based on matrix differential calculus and projection theorem in inner product spaces we exploit the best approximate solution x y in the set s to a given matrix pair x y where s signifies the least squares hermitian and generalized skew hamiltonian solution set of the matrix equation a x a h c y c h f the analytical expression of the best approximate solution is presented by applying the canonical correlation decomposition and the generalized singular value decomposition finally a numerical algorithm and an illustrated example are given

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

sensitivity analysis in minisum facility location problems this paper considers tradeoffs between cost and perturbations of the weights in a single facility minisum location problem over a finite set of feasible points specifically the paper characterizes a tradeoff curve of variability in the weights versus the degree of optimality of a solution the general theoretical framework includes problems of location on a tree network and location using block norms special results are given for these cases

symmetric positive semidefinite and positive definite real solutions of ax xat and ax yb abstract given the equations ax xa t and ax yb with arbitrary nonzero real matrices a and b of the same size we seek all real solutions x and y which are 1 symmetric 2 symmetric and positive semidefinite and 3 symmetric and positive definite necessary and sufficient conditions for the existence of such solutions and their general forms are derived

the reflexive least squares solutions of the general coupled matrix equations with a submatrix constraint in this paper we construct an iterative method to solve the general coupled matrix email protected j 1la i jx jb i j c i i 1 2 t where x email protected r n j x n j j 1 2 l is a reflexive matrix with a specified central principal submatrix the algorithm produces suitable x 1 x 2 x l such that i 1 email protected j 1 la i jx jb i j c i min within finite iteration steps in the absence of roundoff errors we show that the algorithm is stable any case the algorithm requires little storage capacity given numerical examples show that the algorithm is efficient

the solutions to linear matrix equations ax b ya d with k involutory symmetries let rcmm and scnn be nontrivial k involutions if their minimal polynomials are both xk1 for some k2 i e rk1 r1i and sk1 s1i we say that acmn is r s symmetric if ras1 a and a is r s symmetric if ras a with 0 1 k1 and 0 let s be one of the subsets of all r s symmetric and r s symmetric matrices given xcnr ycsm bcmr and dcsn we characterize the matrices a in s that minimize axb2 yad2 frobenius norm under the assumption that r and s are unitary moreover among the set s x y b d s of the minimizers of axb2 yad2 min we find the optimal approximate matrix as x y b d that minimizes ag to a given unstructural matrix gcmn we also present the necessary and sufficient conditions such that ax b ya d is consistent in s if the conditions are satisfied we characterize the consistent solution set of all such a finally a numerical algorithm and some numerical examples are given to illustrate the proposed results

least squares solutions to the matrix equations ax b and xc d abstract the least squares solution and the least squares symmetric solution with the minimum norm of the matrix equations ax b and xc d are considered in this paper by the matrix differentiation and the spectral decomposition of matrices an explicit representation of such solution is given

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

the matrix equations ax b xc d with px sxp constraint abstract the matrix equations ax b xc d with the constraint px sxp are considered where p is a given hermitian matrix satisfying p 2 i and s 1 using an eigenvalue decomposition of p the constrained problem can be equivalently transformed to two independent unconstrained problems of matrix equations in similar forms and hence the constrained problem can be solved in terms of the eigenvectors of p in this paper a simple and eigenvector free formula of the general solutions to the constrained problem is presented using moore penrose generalized inverses of the coefficient matrices a and c a similar problem of the matrix equations with generalized constraint is discussed too we also consider the least norm solution of the constrained problems

the matrix equation axb gxd e over the quaternion field abstract using the method of complex representation this paper obtains necessary and sufficient conditions for the existence of a solution or a unique solution to the matrix equation axb gxd e over the quaternion field extends w e roth s theorem and a jameson s theorem to the quaternion field and gives the representation of the solution

the p q generalized reflexive and anti reflexive solutions of the matrix equation ax b abstract an n n complex matrix p is said to be a generalized reflection matrix if p h p and p 2 i an n n complex matrix a is said to be a p q generalized reflexive or anti reflexive matrix with respect to the generalized reflection matrix dual p q if a paq or a paq this paper establishes the necessary and sufficient conditions for the existence of and the expressions for the p q generalized reflexive and anti reflexive solutions of the matrix equation ax b in addition in corresponding solution set of the equation the explicit expression of the nearest matrix to a given matrix in the frobenius norm has been provided

using block norms for location modeling in formulating a continuous location model with facilities represented as points in rn e g typically in the plane one must characterize the distance between two points as a function of their coordinates two criteria in selecting a distance function are 1 to obtain good approximations of actual distances and 2 to obtain a mathematical model of the location problem that is easy to solve in this paper we show how a class of norms with polygonal contours called block norms can yield attractive choices as distance functions with respect to these criteria in particular we consider the following relevant properties of block norms they generalize the concepts of rectilinear or city block travel they are dense in the set of all norms they have interesting travel interpretations in the plane they can be expressed as a sum of the absolute values of linear functions they often give better approximations to actual highway distances than the most frequently used family of norms the lp norms and finally they yield linear programming formulations of certain facility location problems i e the weber problem and the rawls problem

on solutions of matrix equation axb cyd f in this paper the matrix equation with two unknown matrices x y of form axb cyd f is discussed by applying the canonical correlation decomposition ccd of matrix pairs we obtain expressions of the least squares solutions of the matrix equation and sufficient and necessary conditions for the existence and uniqueness of the solutions we also derive a general form of the solutions we also study the least squares hermitian skew hermitian solutions of equation axah cych f

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

the chebyshev solution of the linear matrix equationax yb c in this paper we investigate the properties of the chebyshev solutions of the linear matrix equationax yb c wherea b andc are given matrices of dimensionsm r s n andm n respectively wherer m ands n we separately consider two particular cases in the first case we assumem r 1 andn s 1 in the second caser s 1 andm n are arbitrary for these two cases under the assumption that the matricesa andb are full rank we formulate necessary and sufficient conditions characterizing the chebyshev solution ofax yb c and we give the formulas for the chebyshev error then we propose an algorithm which may be applied to compute the chebyshev solution ofax yb c for some particular cases some numerical examples are also given

r s conjugate solution to a pair of linear matrix equations abstract let r and s be m m and n n nontrivial real symmetric involutions an m n complex matrix a is termed r s conjugate if a ras where a denotes the conjugate of a in this paper necessary and sufficient conditions are established for the existence of the r s conjugate solution to the system of matrix equations ax c and xb d the expression is also presented for such solution to this system in addition the explicit expression of this solution to the corresponding optimal approximation problem is obtained furthermore the least squares r s conjugate solution with least norm to this system mentioned above is considered the representation of such solution is also derived finally an algorithm and numerical examples are given

ranks of least squares solutions of the matrix equation axb c for a complex matrix equation axb c we solve the following two problems 1 the maximal and minimal ranks of least square solution x to axb c and 2 the maximal and minimal ranks of two real matrices x 0 and x 1 in least square solution x x 0 ix 1 to axb c we also give a necessary and sufficient condition for matrix equations a ix ib i c i i 1 2 to have a common least square solution

real iterative algorithms for a common solution to the complex conjugate matrix equation system in this paper we propose a new algorithm for the computation of the common solution to the complex conjugate matrix equation system a i x b i c i x d i e i i 1 2 n the algorithms only need finite iteration steps requiring only real computation furthermore the algorithm can be extended to solve a more general complex matrix equation system two numerical examples are given to illustrate the effectiveness of the proposed algorithm

full column rank solutions of the matrix equation av evj necessary and sufficient conditions for the existence of at least a full column rank solution to the matrix equation av evj is obtained when the matrix pair a e is regular furthermore a simple representation of the all full column rank solutions to the matrix equation is derived if it has such solutions an example illustrates the proposed approach

a tangent algorithm for computing the generalized singular value decomposition we present two new algorithms for floating point computation of the generalized singular values of a real pair a b of full column rank matrices and for floating point solution of the generalized eigenvalue problem hx lambda mx with symmetric positive definite matrices h and m the pair a b is replaced with an equivalent pair a b and the generalized singular values are computed as the singular values of the explicitly computed matrix f a b 1 the singular values of f are computed using the jacobi method the relative accuracy of the computed singular value approximations does not depend on column scalings of a and b that is the accuracy is nearly the same for all pairs ad 1 bd 2 with d 1 d 2 arbitrary diagonal nonsingular matrices similarly the pencil h lambda m is replaced with an equivalent pencil h lambda m and the eigenvalues of h lambda m are computed as the squares of the singular values of g l h l m 1 where l h l m are the cholesky factors of h m respectively and the matrix g is explicitly computed as the solution of a linear system of equations for the computed approximation lambda delta lambda of any exact eigenvalue lambda the relative error delta lambda lambda is of order p n mbox boldmath varepsilon max min delta in cal d kappa 2 delta h delta min delta in cal d kappa 2 delta m delta where p n is a modestly growing polynomial of the dimension of the problem mbox boldmath varepsilon is the round off unit of floating point arithmetic cal d denotes the set of diagonal nonsingular matrices and kappa 2 cdot is the spectral condition number furthermore floating point computation corresponds to an exact computation with h delta h m delta m where for all i j delta h ij sqrt h ii h jj and delta m ij sqrt m ii m jj are of order of mbox boldmath varepsilon times a modest function of n

cramer s rule for a system of quaternion matrix equations with applications abstract in this paper we investigate cramer s rule for the general solution to the system of quaternion matrix equations a 1 x b 1 c 1 a 2 x b 2 c 2 and cramer s rule for the general solution to the generalized sylvester quaternion matrix equation a x b c y d e respectively as applications we derive the determinantal expressions for the hermitian solutions to some quaternion matrix equations the findings of this paper extend some known results in the literature

parametric distance functions vs nonparametric neural networks for estimating road travel distances abstract measuring and storing actual road travel distances between the points of a region is often not feasible and it is a common practice to estimate them the usual approach is to use distance estimators which are parameterized functions of the coordinates of the points we propose to use nonparametric approaches using neural networks for estimating actual distances we consider multi layer perceptrons trained with the back propagation rule and regression neural networks implementing nonparametric regression using gaussian kernels we also consider training multiple estimators and combining them using voting and stacking on a real world study using cities drawn from turkey we found out that these nonparametric approaches are more accurate than the parametric distance functions estimating actual distances has many applications in location and distribution theory

symmetric solutions of linear matrix equations by matrix decompositions abstract the symmetric solutions of linear matrix equations like ax b have been considered using the kronecker product formulation 17 and the commutation matrix k 5 11 in this paper a novel approach is followed using the singular value generalized singular value real schur and real generalized schur decompositions solvability conditions for the symmetric and nonsymmetric solutions are obtained and the numerical properties of the related solution procedures are discussed briefly

analysis of a galerkin approach applied to a system of coupled schrodinger equations abstract we study error and convergence rate of a fully discrete fourier galerkin scheme to approximate the solutions of a system consisting of two coupled cubic schrodinger equations with cross modulation which is a model for the propagation of a nonlinear pulse in a linearly birefringent kerr optical fiber we also establish a result on existence and uniqueness of a solution of the corresponding initial value problem

the solution of the matrix equations axb cxd e and ya dz yc bz e f the conditions for the existence of a unique solution of the matrix equation axb cxd e are proved to be that i the pencils a c and d b are regular and ii the spectra of the pencils have an empty intersection a numerical algorithm for solving the equation is proposed the possibility of a least squares type solution is briefly discussed the set of equations ya dz yc bz e f is proved to be equivalent to the aforementioned equation and its solution is also investigated a numerical algorithm is proposed

a new local search for continuous location problems this paper presents a new local search approach for solving continuous location problems the main idea is to exploit the relation between the continuous model and its discrete counterpart a local search is first conducted in the continuous space until a local optimum is reached it then switches to a discrete space that represents a discretisation of the continuous model to find an improved solution from there the process continues switching between the two problem formulations until no further improvement can be found in either thus we may view the procedure as a new adaption of formulation space search the local search is applied to the multi source weber problem where encouraging results are obtained this local search is also embedded within variable neighbourhood search producing excellent results

decoupled local global energy preserving schemes for the n coupled nonlinear schrodinger equations abstract we develop two local energy preserving integrators and a global energy preserving integrator for the general multisymplectic hamiltonian system when applied to the 1d and multi dimensional n coupled nonlinear schrodinger equations the given schemes have the exact preservation of the local global conservation law and are decoupled in the components n n 1 2 n i e each of the components can be solved independently the decoupled feature is significant and helpful for overcoming the computational difficulty of the n coupled n 3 nonlinear schrodinger equations especially of the multi dimensional case the composition method is employed to improve the accuracy of the schemes in time and the discrete fast fourier transform is used to reduce the computational complexity several numerical experiments are carried out to exhibit the behaviors of the wave solutions numerical results confirm the theoretical results

the reflexive and anti reflexive solutions of the matrix equation ahxb c in this paper the necessary and sufficient conditions for the solvability of matrix equation a hxb c over the reflexive or anti reflexive matrices are given and the general expression of the solution for a solvable case is obtained moreover the relative optimal approximation problem is considered the explicit expressions of the optimal approximation solution and the minimum norm solution are both provided

a deep cut ellipsoid algorithm for convex programming theory and applications this paper proposes a deep cut version of the ellipsoid algorithm for solving a general class of continuous convex programming problems in each step the algorithm does not require more computational effort to construct these deep cuts than its corresponding central cut version rules that prevent some of the numerical instabilities and theoretical drawbacks usually associated with the algorithm are also provided moreover for a large class of convex programs a simple proof of its rate of convergence is given and the relation with previously known results is discussed finally some computational results of the deep and central cut version of the algorithm applied to a min max stochastic queue location problem are reported

least squares symmetric and skew symmetric solutions of the generalized sylvester matrix equation σi 1 s a i x b i σj 1 t c j y d j e the generalized sylvester matrix equation i 1 s a i x b i j 1 t c j y d j e with unknown matrices x and y is encountered in many system and control applications in this paper a direct method is established to solve the least squares symmetric and skew symmetric solutions of the equation by using the kronecker product and the generalized inverses and the expression of the solution set s are provided moreover an optimal approximation between a given matrix pair and the affine subspace s is discussed and an explicit formula for the unique optimal approximation solution is presented finally two numerical examples are given which demonstrate that the introduced algorithm is quite efficient

error bounds for the approximative solution of restricted planar location problems abstract facility location problems in the plane play an important role in mathematical programming when looking for new locations in modeling real world problems we are often confronted with forbidden regions that are not feasible for the placement of new locations furthermore these forbidden regions may have complicated shapes even if we require them to be convex it may be more useful or even necessary to use approximations of such forbidden regions when trying to solve location problems in this paper we develop error bounds for the approximative solution of restricted planar location problems using the so called sandwich algorithm the number of approximation steps required to achieve a specified error bound is analyzed as examples of these approximation schemes we discuss round norms and polyhedral norms computational tests are also included

least squares solutions to the equations ax b xc d with some constraints abstract the least squares solutions to the equations ax b xc d with some constraints such as orthogonality symmetric orthogonality symmetric idempotence are considered by one time singular value decomposition or eigenvalue decomposition of the matrix product generated by the coefficient matrices a c and right hand side matrices b d the constrained solutions are constructed simply similar strategy is applied to the equations with corresponding p commuting constraints with given symmetric matrix p numerical examples that show the efficiency of the proposed methods are presented

finite iterative hermitian r conjugate solutions of the generalized coupled sylvester conjugate matrix equations abstract in this paper an iterative algorithm for solving a generalized coupled sylvester conjugate matrix equations over hermitian r conjugate matrices given by a 1 v b 1 c 1 w d 1 e 1 v f 1 g 1 and a 2 v b 2 c 2 w d 2 e 2 v f 2 g 2 is presented when these two matrix equations are consistent the convergence theorem shows that a solution can be obtained within finite iterative steps in the absence of round off error for any initial arbitrary hermitian r conjugate solution matrices v 1 w 1 some lemmas and theorems are stated and proved where the iterative solutions are obtained a numerical example is given to demonstrate the behavior of the proposed method and to support the theoretical results

a jacobi davidson type method for the generalized singular value problem we discuss a new method for the iterative computation of some of the generalized singular values and vectors of a large sparse matrix our starting point is the augmented matrix formulation of the gsvd the subspace expansion is performed by approximately solving a jacobi davidson type correction equation while we give several alternatives for the subspace extraction numerical experiments illustrate the performance of the method

least squares solution with the minimum norm for the matrix equation axb gxh c d based on the projection theorem in hilbert space by making use of the generalized singular value decomposition and the canonical correlation decomposition an analytical expression of the least squares solution for the matrix equation axb gxh c d with the minimum norm is derived an algorithm for finding the minimum norm solution is described and some numerical results have been given

an expression of the general common least squares solution to a pair of matrix equations with applications in this paper an explicit representation of the general common least squares solution to the pair of matrix equations a 1xb 1 c 1 and a 2xb 2 c 2 is obtained furthermore we use this result to determine the condition for the existence of a hermitian least squares solution to the matrix equation axb c and the expression of the general hermitian least squares solution is also given special attention is paid to consider the existence of hermitian 1 i inverses of a i 3 4 and the representations of the hermitian generalized inverses are presented finally a numerical example is given to illustrate our result

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

efficient algorithms for the weighted minimum circle problem the weighted minimum covering circle problem is a well known single facility location problem used in emergency facility models this paper introduces a classification scheme based on fundamental mathematical programming concepts for algorithms which solve both weighted and unweighted versions one result of this classification is proof that a recently developed method is identical to one developed in the nineteenth century also within the classification scheme efficient new algorithms are given for the weighted problem the results of some extensive computational tests identify the empirically fastest methods