perturbation bound of singular linear systems a classical perturbation bound in numerical linear algebra is extended to more general cases of singular linear algebraic systems and weighted linear least squares problem componentwise perturbation analysis of linear least squares problem is also derived

on classification of normal operators in real spaces with indefinite scalar product abstract a real finite dimensional space with indefinite scalar product having v negative squares and v positive ones is considered the paper presents a classification of operators that are normal with respect to this product for the cases min v v 1 2 the approach used here was developed in papers by gohberg and reichstein and by the authors where a similar classification was obtained for complex spaces with v min v v 1 2 respectively

conjugate direction particle swarm optimization solving systems of nonlinear equations solving systems of nonlinear equations is a difficult problem in numerical computation for most numerical methods such as the newton s method for solving systems of nonlinear equations their convergence and performance characteristics can be highly sensitive to the initial guess of the solution supplied to the methods however it is difficult to select a good initial guess for most systems of nonlinear equations aiming to solve these problems conjugate direction particle swarm optimization cdpso was put forward which introduced conjugate direction method into particle swarm optimization pso in order to improve pso and enable pso to effectively optimize high dimensional optimization problem in one optimization problem when after some iterations pso got trapped in local minima with local optimal solution x conjugate direction method was applied with x as a initial guess to optimize the problem to help pso overcome local minima by changing high dimension function optimization problem into low dimensional function optimization problem because pso is efficient in solving the low dimension function optimization problem pso can efficiently optimize high dimensional function optimization problem by this tactic since cdpso has the advantages of method of conjugate direction cd and particle swarm optimization pso it overcomes the inaccuracy of cd and pso for solving systems of nonlinear equations the numerical results showed that the approach was successful for solving systems of nonlinear equations

recurrent neural network for computing the drazin inverse this paper presents a recurrent neural network rnn for computing the drazin inverse of a real matrix in real time this recurrent neural network rnn is composed of n independent parts subnetworks where n is the order of the input matrix these subnetworks can operate concurrently so parallel and distributed processing can be achieved in this way the computational advantages over the existing sequential algorithms can be attained in real time applications the rnn defined in this paper is convenient for an implementation in an electronic circuit the number of neurons in the neural network is the same as the number of elements in the output matrix which represents the drazin inverse the difference between the proposed rnn and the existing ones for the drazin inverse computation lies in their network architecture and dynamics the conditions that ensure the stability of the defined rnn as well as its convergence toward the drazin inverse are considered in addition illustrative examples and examples of application to the practical engineering problems are discussed to show the efficacy of the proposed neural network

the representation and approximation for the weighted moore penrose inverse in hilbert space we present a unified representation theorem of the weighted moore penrose inverse in hilbert space specific expressions and computational procedures for the weighted moore penrose inverse in hilbert space can be uniformly derived

on the augmented system approach to sparse least squares problems we study the augmented system approach for the solution of sparse linear least squares problems it is well known that this method has better numerical properties than the method based on the normal equations we use recent work by arioli et al 1988 to introduce error bounds and estimates for the components of the solution of the augmented system in particular we find that using iterative refinement we obtain a very robust algorithm and our estimates of the error are accurate and cheap to compute the final error and all our error estimates are much better than the classical or skeel s error analysis 1979 indicates moreover we prove that our error estimates are independent of the row scaling of the augmented system and we analyze the influence of the bjorck scaling 1967 on these estimates we illustrate this with runs both on large scale practical problems and contrived examples comparing the numerical behaviour of the augmented systems approach with a code using the normal equations these experiments show that while the augmented system approach with iterative refinement can sometimes be less efficient than the normal equations approach it is comparable or better when the least squares matrix has a full row and is in any case much more stable and robust

on level 2 condition number for the weighted moore penrose inverse in this paper we present characterizations for the level 2 condition number of the weighted moore penrose inverse i e cond m n a cond m n 2 a cond m n a 1 where cond m n a is the condition number of the weighted moore penrose inverse of a rectangular matrix and cond m n 2 a is the level 2 condition number of this problem this paper extends the result by cucker diao and wei f cucker h diao y wei on the level 2 condition number for moore penrose inversion 2005 unpublished report and improves the results by wei and wang y wei d wang condition numbers and perturbation of weighted moore penrose inverse and weighted linear least squares problem appl math comput 145 2003 45 58

recurrent neural network for computing outer inverse two linear recurrent neural networks for generating outer inverses with prescribed range and null space are defined each of the proposed recurrent neural networks is based on the matrix valued differential equation a generalization of dynamic equations proposed earlier for the nonsingular matrix inversion the moore penrose inversion as well as the drazin inversion under the condition of zero initial state the application of the first approach is conditioned by the properties of the spectrum of a certain matrix the second approach eliminates this drawback though at the cost of increasing the number of matrix operations the cases corresponding to the most common generalized inverses are defined the conditions that ensure stability of the proposed neural network are presented illustrative examples present the results of numerical simulations

variational problems on multiply connected thin strips ii convergence of the ginzburg landau let m be a planar embedded graph whose arcs meet transversally at the vertices let m be a strip shaped domain around m of width m except in a neighborhood of the singular points assume that the boundary of m is smooth we consider the ginzburg landau energy functional for superconductivity on m we prove that its minimizers converge in a suitable sense to the minimizers of a simpler functional on m the supercurrents in m are shown to converge to one dimensional currents in m

recurrent neural network approach based on the integral representation of the drazin inverse in this letter we present the dynamical equation and corresponding artificial recurrent neural network for computing the drazin inverse for arbitrary square real matrix without any restriction on its eigenvalues conditions that ensure the stability of the defined recurrent neural network as well as its convergence toward the drazin inverse are considered several illustrative examples present the results of computer simulations

representations and computations of 2 3 we investigate full rank representations of 2 3 and 2 4 inverses with prescribed rank as well as with prescribed range and null space in an indefinite inner product space in certain particular cases we obtain new results or generalizations related with 2 3 and 2 4 inverses in a hilbert unitary space direct and iterative methods for computing 2 3 and 2 4 inverses in an indefinite inner product space are developed numerical examples with different metric matrices are presented

a family of multi point iterative methods for solving systems of nonlinear equations we extend to n dimensional case a known multi point family of iterative methods for solving nonlinear equations this family includes as particular cases some well known and also some new methods the main advantage of these methods is they have order three or four and they do not require the evaluation of any second or higher order frechet derivatives a local convergence analysis and numerical examples are provided

some topics on weighted moore penrose inverse weighted least squares and weighted regularized tikhonov problems it is well known that the normwise relative condition numbers measure the sensitivity of matrix inversion and the solution of linear systems the classical normwise relative condition number formulas are given by higham linear algebra appl 214 1995 193 here we consider the condition number formulas for the weighted moore penrose inverse of a rectangular matrix and give explicit expressions for the weighted condition numbers of the singular linear systems ax b moreover we consider the nearness to rank deficiency finally we give some decomposition tools and their application in the regularized weighted tikhonov problems

a hardy inequality in twisted waveguides we show that twisting of an infinite straight three dimensional tube with non circular cross section gives rise to a hardy type inequality for the associated dirichlet laplacian as an application we prove certain stability of the spectrum of the dirichlet laplacian in locally and mildly bent tubes namely it is known that any local bending no matter how small generates eigenvalues below the essential spectrum of the laplacian in the tubes with arbitrary cross sections rotated along a reference curve in an appropriate way in the present paper we show that for any other rotation some critical strength of the bending is needed in order to induce a non empty discrete spectrum

the weighted moore penrose inverse of modified matrices the weighted moore penrose inverse of modified matrix a cb is derived under some range conditions where cb is a full rank factorization the generalized singular value decomposition of a matrix is used to derive systematically the weighted moore penrose inverse for a modified matrix where cb need not be a full rank factorization the results extend the earlier works by various authors

new perturbation results for equality constrained least squares problems abstract we present some perturbation results for least squares problems with equality constraints relative errors are obtained on perturbed solutions and lagrange multipliers of the problem based on the equivalence of the problem to a consistent system of linear equations

recurrent neural networks for computing weighted moore penrose inverse1this project is supported by the national nature science foundation of china doctoral point foundation of china and youth science foundation of universities in shanghai of china 1 three recurrent neural networks are presented for computing the weighted moore penrose inverse of rank deficient matrices the first recurrent neural network has the dynamical equation similar to the one proposed earlier for matrix inversion and is capable of weighted moore penrose inverse under the condition of zero initial states the second recurrent neural network consists of an array of neurons corresponding to a weighted moore penrose inverse matrix with decaying self connections and constant connections in each row or column the third recurrent neural network consists of two layers of neuron arrays corresponding respectively to a weighted moore penrose inverse and a lagrangian matrix with constant connections

expression for the perturbation of the weighted moore penrose inverse abstract we consider the perturbation formula for the weighted moore penrose inverse of a rectangular matrix and give an explicit expression for the weighted moore penrose inverse of a perturbed matrix under the weakest rank condition this explicit expression extends the earlier work of several authors

a new filled function method for an unconstrained nonlinear equation in this paper we transform an unconstrained system of nonlinear equations into a special optimization problem a new filled function is constructed by employing the special properties of the transformed optimization problem theoretical and numerical properties of the proposed filled function are investigated and a solution of the algorithm is proposed under some conditions we can find a solution or an approximate solution to the system of nonlinear equations in finite iterations the implementation of the algorithm on six test problems is reported with satisfactory numerical results

similarity invariants for pairs of upper triangular toeplitz matrices abstract the problem considered is that of simultaneous reduction to simple forms of pairs of upper triangular toeplitz matrices in this context simple matrices are those that are equal to or differ only slightly from a power of the upper triangular nilpotent jordan block the problem is related to that of complementary triangularization of pairs of matrices and therefore has a background in systems theory the main results are concerned with existence and uniqueness of the reduced form also a similarity invariant for pairs of upper triangular toeplitz matrices is obtained the invariant consists of a pair involving a complex number and an integer the results can be generalized to a class of matrices strictly larger than that formed by the upper triangular toeplitz matrices

homotopies exploiting newton polytopes for solving sparse polynomial systems this paper is concerned with the problem of finding all isolated solutions of a polynomial system the bkk bound defined as the mixed volume of the newton polytopes of the polynomials in the system is a sharp upper bound for the number of isolated solutions in mathbb c 0 n mathbb c 0 mathbb c backslash 0 of a polynomial system with a sparse monomial structure first an algorithm is described for computing the bkk bound following the lines of bernshtei n s proof the algorithmic construction of the cheater s homotopy or the coefficient homotopy is obtained the mixed homotopy methods can be combined with the random product start systems based on a generalized bezout number applications illustrate the effectiveness of the new approach

perturbation bounds for constrained and weighted least squares problems we derive perturbation bounds for the constrained and weighted linear least squares ls problems both the full rank and rank deficient cases are considered the analysis generalizes some results of earlier works

particle swarm algorithm for solving systems of nonlinear equations solving systems of nonlinear equations is one of the most difficult problems in all of numerical computation and in a diverse range of engineering applications newton s method for solving systems of nonlinear equations can be highly sensitive to the initial guess of the solution in this study a new particle swarm optimization algorithm is proposed to solve systems of nonlinear equations some standard systems are presented to demonstrate the efficiency of this method

condition number for the drazin inverse and the drazin inverse solution of singular linear system with their condition numbers in this paper we investigate the condition number of drazin inverse and drazin inverse solution of singular linear system ax b where a is a nxn rank deficient matrix and b a real vector of size n x a real vector let a and b be two positive real numbers when we consider the weighted frobenius norm aa bb p q f on the data we get the formula of condition number of the drazin inverse solution of singular linear system for the normwise condition number the sensitivity of the relative condition number itself is studied the componentwise perturbation is also investigated

imperialist competitive algorithm for solving systems of nonlinear equations solving systems of nonlinear equations is a relatively complicated problem in which arise a diverse range of sciences there are a number of different approaches that have been proposed in this paper we employ the imperialist competitive algorithm ica for solving systems of nonlinear equations some well known problems are presented to demonstrate the efficiency of this new robust optimization method in comparison to other known methods

a characterization and representation of the generalized inverse a 2 t s and its applications abstract this paper presents an explicit expression for the generalized inverse a 2 t s based on this we established the characterization the representation theorem and the limiting process for a 2 t s as an application we estimate the error bound of the iterative method for approximating a 2 t s

an accelerated iterative method for computing weighted moore penrose inverse the goal of this paper is to present an accelerated iterative method for computing weighted moore penrose inverse analysis of convergence is included to show that the proposed scheme has sixth order convergence using a proper initial matrix a sequence of iterates will be produced which is convergent to the weighted moore penrose inverse numerical experiments are reported to show the efficiency of the new method

classes of normal matrices in indefinite inner products abstract we study classes of matrices defined by various normality properties with respect to an indefinite complex inner product the relationships between many such properties all of them equivalent to the normality in case of a definite inner product are described in particular a canonical form is developed for the class of matrices that are polynomials of a self adjoint matrix

normal matrices an update abstract a list of seventy conditions on an n n complex matrix a equivalent to its being normal published nearly ten years ago by grone johnson sa and wolkowicz has proved to be very useful hoping that in an extended form it will be even more helpful we compile here another list of about twenty conditions they either have been overlooked by the authors of the original list or have appeared during the last decade

the condition numbers for weighted moore penrose inverse and weighted linear least squares problem condition numbers play an important role in numerical analysis classical normise condition numbers are used to measure the size of both input perturbations and output errors in this paper we study the weighted normwise relative condition numbers for the weighted moore penrose inverse and the weighted linear least squares wls problems in the case of the full column rank matrix the bounds or formulas for the weighted condition numbers are presented the obtained results can be viewed as extensions of the earlier works studied by others

perturbation of least squares problem in hilbert spaces let h 1 h 2 be two hilbert spaces over the same field and let t h 1 h 2 be a bounded linear operator with closed range we present some results of the perturbation analysis for the least squares solutions to the operator equation tx y

unitary equivalence in an indefinite scalar product an analogue of singular value decomposition if h1 is an n n and h2 an m m invertible hermitian matrix and x and y are arbitrary complex m n matrices we call the last matrices equivalent if x u2yu1 for some h1 unitary matrix u1 and some h2 unitary matrix u2 it is well known that if h1 and h2 are positive definite then without loss of generality we can assume that they are identities and x and y are equivalent if and only if the diagonalizable matrices x x and y y have the same spectrum in the present paper we show that in general the jordan form of x x where x is the h1 h2 adjoint of x x h 11x h2 defines a finite number of nonequivalent classes of matrices and that each such class is defined by its integer matrix explicit formulas for all classes having the same jordan form of x x are presented the necessary and sufficient conditions for an operator to be presentable as uz with h unitary u and h self adjoint h consistent z h polar decomposition of the operator are found

chaotic quantum behaved particle swarm optimization algorithm for solving nonlinear system of equations abstract this study proposes a novel chaotic quantum behaved particle swarm optimization algorithm for solving nonlinear system of equations different chaotic maps are introduced to enhance the effectiveness and robustness of the algorithm several benchmark studies are carried out logistic map gives the best results and is utilized in solving nonlinear equation sets nine well known problems are solved with our algorithm and results are compared with quantum behaved particle swarm optimization intelligent tuned harmony search gravitational search algorithm and literature studies comparison results reveal that the proposed algorithm can cope with the highly non linear problems and outperforms many algorithms which exist in the literature

classification of normal operators in spaces with indefinite scalar product of rank 2 abstract a finite dimensional complex space with indefinite scalar product having v 2 negative squares and v 2 positive ones is considered the paper presents a classification of operators that are normal with respect to this product it relates to the study by gohberg and reichstein in which the similar classification was obtained for the case v min v v 1

pcr algorithm for parallel computing minimum norm t least squares s solution of inconsistent linear equations this paper presents a new highly parallel algorithm for computing the minimum norm t least squares s solution of inconsistent linear equations ax b email protected r m x n r email protected r a by this algorithm the solution x a s t b is obtained in t 1 m 1 log 2m n 6 log 2 n r 1 log 2m log 2n r 1 log 2n steps with p mn processors when m 2 n 1 and with p 2n n 1 processors otherwise

positive real matrices in indefinite inner product spaces and invariant maximal semidefinite subspaces in this paper positive real matrices in indefinite inner product spaces are studied this class of matrices is intimately connected to dissipative matrices in the complex case first the complex case and then the real case are treated an explicit construction is given for invariant semidefinite subspaces

asymptotics of dirichlet eigenvalues and eigenfunctions of the laplacian on thin domains in rd abstract we consider the laplace operator with dirichlet boundary conditions on a domain in r d and study the effect that performing a scaling in one direction has on the eigenvalues and corresponding eigenfunctions as a function of the scaling parameter around zero this generalizes our previous results in two dimensions and as in that case allows us to obtain an approximation for dirichlet eigenvalues for a large class of domains under very mild assumptions as an application we derive a three term asymptotic expansion for the first eigenvalue of d dimensional ellipsoids

thin waveguides with robin boundary conditions we consider the laplace operator in a thin three dimensional tube with a robin type condition on its boundary and study asymptotically the spectrum of such operator as the diameter of the tube s cross section becomes infinitesimal in contrast with the dirichlet condition case 2 we evidence different behaviors depending on a symmetry criterium for the fundamental mode in the cross section if that symmetry condition fails then we prove the localization of lower energy levels in the vicinity of the minimum point of a suitable function on the tube s axis depending on the curvature and the rotation angle in the symmetric case the behavior of lower energy modes is shown to be ruled by a one dimensional sturm liouville problem involving an effective potential given in explicit form

condition numbers and perturbation of the weighted moore penrose inverse and weighted linear least squares problem it is well known that the normwise relative condition numbers measure the sensitivity of matrix inversion and the solution of nonsingular linear systems here we consider the condition number formulas for the weighted moore penrose inverse of a rectangular matrix and give explicit expressions for the weighted condition numbers of the singular linear systems ax b these explicit expressions extend the earlier work of several authors as we know the computed weighted least squares solution x of min email protected email protected m in the presence of the round off error satisfies the perturbed equation min email protected a e y b f m finally an upper bound of email protected n is derived for the case where the weighted matrix and vector norms and the assumption rank a e rank a are used

condition numbers and their condition numbers abstract various normwise relative condition numbers that measure the sensitivity of matrix inversion and the solution of linear systems are characterized new results are derived for the cases where two common noninduced matrix norms are used and where different vector norms are used for the domain and range of the matrix condition numbers that respect the structure of symmetric problems are also analyzed the sensitivity of the condition number itself is then investigated and we obtain sharp examples of demmel s general result that for certain problems in numerical analysis the condition number of the condition number is the condition number finally upper bounds are derived for the sensitivity of componentwise condition numbers

stability of self adjoint square roots and polar decompositions in indefinite scalar product spaces continuity properties of factors in polar decompositions of matrices with respect to indefinite scalar products are studied the matrix having the polar decomposition and the indefinite scalar product are allowed to vary closely related properties of a self adjoint with respect to an indefinite scalar product perturbed matrix to have a self adjoint square root or to have a representation of the form xx are also

moore penrose invertible normal and hermitian elements in rings in this paper we present several new characterizations of normal and hermitian elements in rings with involution in purely algebraic terms and considerably simplify proofs of already existing characterizations

vector least squares solutions for coupled singular matrix equations the weighted least squares solutions of coupled singular matrix equations are too difficult to obtain by applying matrices decomposition in this paper a family of algorithms are applied to solve these problems based on the kronecker structures subsequently we construct a computationally efficient solutions of coupled restricted singular matrix equations furthermore the need to compute the weighted drazin and weighted moore penrose inverses and the use of tian s work and lev ari s results are due to appearance in the solutions of these problems the several special cases of these problems are also considered which includes the well known coupled sylvester matrix equations finally we recover the iterative methods to the weighted case in order to obtain the minimum d norm g vector least squares solutions for the coupled sylvester matrix equations and the results lead to the least squares solutions and invertible solutions as a special case

extension and generalization properties of the weighted minkowski inverse in a minkowski space for an arbitrary matrix the weighted minkowski inverse a m n m n m related to the positive definite matrices m m m and n m n of an arbitrary matrix a m m n including singular and rectangular is one of the important generalized inverses for solving matrix equations in minkowski space in this paper the results are introduced in the following three ways first we establish some new and attractive properties of the weighted minkowski inverse a m n in a minkowski space second new representations and conditions for the continuity of the weighted minkowski inverse a m n in a minkowski space are discussed finally some illustrated counterexamples are also studied to show that some well known properties of the weighted moore penrose inverse a m n in a hilbert space h are not valid in the minkowski inverse a m n in a minkowski space

weighted tikhonov filter matrices for ill posed problems we consider the concept of weighted tikhonov filter matrices in connection to discrete ill posed and rank deficient linear problems some properties of the weighted tikhonov filter matrices are given together with their filtering and regularization effects we also present perturbation identities for the weighted tikhonov regularized linear least squares problem using weighted filter matrices generalizing well known weighted perturbation identities for the weighted linear least squares problem and weighted pseudoinverses

successive matrix squaring algorithm for parallel computing the weighted generalized inverse amn we derive a successive matrix squaring sms algorithm to approximate the weighted generalized inverse which can be expressed in the form of successive squaring of a composite matrix t given an m by n matrix a with m n we show that the weighted generalized inverse of a can be computed in parallel time ranging from o log n to o log 2n provided that there are enough processors to support matrix multiplication in time o log n

on indecomposable normal matrices in spaces with indefinite scalar product abstract finite dimensional linear spaces both complex and real with indefinite scalar product are considered upper and lower bounds are given for the size of an indecomposable matrix that is normal with respect to this scalar product in terms of specific functions of v min v v where v v is the number of negative positive squares of the form x x all the bounds except for one are proved to be strict

iterative methods for computing the generalized inverses a 2 t s of a matrix a abstract in this paper the iterative methods for computing the generalized inverse a t s 2 the unique matrix x such that xax x r x t and n x s are established and the iterative methods for computing many important generalized inverses of matrices such as a a mn a d a a d w a l 1 a l and ap l are constructed

a simple proof of the product theorem for ep matrices abstract we give a simple proof of a recent theorem of hartwig and katz on the product of ep matrices

the perturbation of consistent least squares problems abstract suppose that the linear system ax b is consistent and a a a b b b are perturbed from a b respectively where a and are m n matrices b and b are m dimensional vectors let a f e a 2 b 2 e b 2 and a 2 a 2 be the condition number of a it is shown that if e 1 then the least squares solution x of a x b is close to a solution x of ax b even when a is not of full rank and rank rank a

polar decompositions in finite dimensional indefinite scalar product spaces general theory abstract several classes of polar decompositions of real and complex matrices with respect to a given indefinite scalar product are studied matrices that admit such polar decompositions are described in various ways in particular a full description of all polar decompositions of a given matrix up to the natural similarity between polar decompositions is given

the representation and approximation for the weighted moore penrose inverse we present a unified representation theorem for the weighted moore penrose inverse specific expressions and computational procedures for the weighted moore penrose inverse can be uniformly derived

singular value decomposition in minkowski space abstract the well known singular value decomposition theorem states that every matrix m can be written in the form m qdr where d is diagonal and q and r are orthogonal we explore the analogous decomposition where q and r are required to be orthogonal with respect to the minkowski rather than the euclidean metric

on the spectrum and weakly effective operator for dirichlet laplacian in thin deformed tubes abstract we study the laplacian in deformed thin bounded or unbounded tubes in r 3 i e tubular regions along a curve r s whose cross sections are multiplied by an appropriate deformation function h s 0 one of the main requirements on h s is that it has a single point of global maximum we find the asymptotic behaviors of the eigenvalues and weakly effective operators as the diameters of the tubes tend to zero it is shown that such behaviors are not influenced by some geometric features of the tube such as curvature torsion and twisting and so a huge amount of different deformed tubes are asymptotically described by the same weakly effective operator

normal generalized selfadjoint operators in krein spaces abstract a full description is given of n selfadjoint normal operators in krein spaces of finite defect as well as in general krein spaces both real and complex krein spaces are considered