on chaotic simulated annealing chen and aihara 1995 proposed a chaotic simulated annealing approach to solving optimization problems by adding a negative self coupling to a network model proposed earlier by aihara et al and gradually removing this negative self coupling they used the transient chaos for searching and self organizing thereby achieving great improvement over other neural network approaches to optimization problems with or without simulated annealing in this paper we suggest a new approach to chaotic simulated annealing with guaranteed convergence and minimization of the energy function by gradually reducing the time step in the euler approximation of the differential equations that describe the continuous hopfield neural network this approach eliminates the need to carefully select other system parameters we also generalize the convergence theorems of chen and aihara to arbitrarily increasing neuronal input output functions and to less restrictive and yet more compact forms

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

particle swarm optimization with stochastic selection of perturbation based chaotic updating system in this paper we consider the particle swarm optimization pso in particular we focus on an improved pso called the cpso vqo which uses a perturbation based chaotic system and a threshold based method of selecting from the standard and chaotic updating systems for each particle on the basis of the difference vector between its pbest and the gbest although it was reported that the cpso vqo performs well it is not easy to select an amplitude of the perturbation and a threshold appropriately for an effective search this is because the bifurcation structure of the chaotic system depends on the difference vector and the difference vector varies widely between different stages of the search and between different problems therefore we improve the cpso vqo by proposing a modified chaotic system whose bifurcation structure is irrelevant to the difference vector and show theoretically desirable properties of the modified system we also propose a new stochastic method that selects the updating system according to the ratio between the components of the difference vector for each particle and restarting and acceleration techniques to develop the standard updating system used in the proposed pso model the proposed methods can maintain an appropriate balance between the identification and diversification aspects of the search moreover we perform numerical experiments to evaluate the performance of the proposed psos pso tpc pso spc pso sdpc ipso spc and ipso sdpc in particular we demonstrate that the ipso sdpc finds high quality solutions and is robust against variations in its parameter values

a generalized finite difference gfd ale scheme for incompressible flows around moving solid bodies on hybrid meshfree cartesian grids a scheme using the mesh free generalized finite differencing gfd on flows past moving bodies is proposed the aim is to devise a method to simulate flow past an immersed moving body that avoids the intensive remeshing of the computational domain and minimizes data interpolation associated with the established computational fluid methodologies as such procedures are time consuming and are a significant source of error in flow simulation in the present scheme the moving body is embedded and enveloped by a cloud of mesh free nodes which convects with the motion of the body against a background of cartesian nodes the generalized finite difference gfd method with weighted least squares wls approximation is used to discretize the two dimensional viscous incompressible navier stokes equations at the mesh free nodes while standard finite difference approximations are applied elsewhere the convecting motion of the mesh free nodes is treated by the arbitrary lagrangian eulerian ale formulation of the flow equations which are solved by a second order crank nicolson based projection method the proposed numerical scheme was tested on a number of problems including the decaying vortex flow external flows past moving bodies and body driven flows in enclosures

massively parallel finite element computation of incompressible flows involving fluid body interactions abstract we describe our massively parallel finite element computations of unsteady incompressible flows involving fluid body interactions these computations are based on the deforming spatial domain stabilized space time dsd sst finite element formulation unsteady flows past a stationary naca 0012 airfoil are computed for reynolds numbers 1000 5000 and 100 000 significantly different flow patterns are observed for these three cases the method is then applied to computation of the dynamics of an airfoil falling in a viscous fluid under the influence of gravity it is observed that the location of the center of gravity of the airfoil plays an important role in determining its pitch stability computations are reported also for simulation of the dynamics of a two dimensional projectile that has a certain initial velocity specially designed mesh moving schemes are employed to eliminate the need for remeshing all these computations were carried out on the thinking machines cm 200 and cm 5 supercomputers with major speed ups compared to traditional supercomputers the implicit equation systems arising from the finite element discretizations of these large scale problems are solved iteratively by using the gmres update technique with diagonal preconditioners the finite element formulations and their parallel implementations assume unstructured meshes

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

chaotic simulated annealing by a neural network with a variable delay design and application in this paper we have three goals the first is to delineate the advantages of a variably delayed system the second is to find a more intuitive lyapunov function for a delayed neural network and the third is to design a delayed neural network for a quadratic cost function for delayed neural networks most researchers construct a lyapunov function based on the linear matrix inequality lmi approach however that approach is not intuitive we provide a alternative candidate lyapunov function for a delayed neural network on the other hand if we are first given a quadratic cost function we can construct a delayed neural network by suitably dividing the second order term into two parts a self feedback connection weight and a delayed connection weight to demonstrate the advantage of a variably delayed neural network we propose a transiently chaotic neural network with variable delay and show numerically that the model should possess a better searching ability than chen aihara s model wang s model and zhao s model we discuss both the chaotic and the convergent phases during the chaotic phase we simply present bifurcation diagrams for a single neuron with a constant delay and with a variable delay we show that the variably delayed model possesses the stochastic property and chaotic wandering during the convergent phase we not only provide a novel lyapunov function for neural networks with a delay the lyapunov function is independent of the lmi approach but also establish a correlation between the lyapunov function for a delayed neural network and an objective function for the traveling salesman problem

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

fast chaotic optimization algorithm based on spatiotemporal maps for global optimization recently many researches have tackled chaos optimization algorithms coas as an attractive method of global optimization considering the statistical property such as the probability density function pdf of the chaotic sequences the search ability of coa can improve the global searching capability by escaping the local solutions than classical stochastic optimization algorithms this paper proposes a novel method for global optimization using spatiotemporal map to improve the performance of the coa the experimental results of typical nonlinear multimodal benchmark functions optimization show that spatiotemporal coa map scoa improves the convergence and high efficiency compared to five hybrid optimization algorithms which are the monte carlo bfgs algorithm mc bfgs logistic map based chaos bfgs algorithm lm bfgs skew tent map based chaos bfgs algorithm stm bfgs coa based on the logistic map lm coa and coa based on the skew tent map stm coa

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

bisymmetric and centrosymmetric solutions to systems of real quaternion matrix equations in this paper we consider bisymmetric and centrosymmetric solutions to certain matrix equations over the real quaternion algebra necessary and sufficient conditions are obtained for the matrix equation ax c and the following systemsa 1x c 1 a 1x c 1 xb 3 c 3 a 2x c 2 to have bisymmetric solutions and the systema 1x c 1 a 3xb 3 c 3 to have centrosymmetric solutions the expressions of such solutions of the matrix and the systems mentioned above are also given moreover a criterion for a quaternion matrix to be bisymmetric is established and some auxiliary results on other sets over are also mentioned

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 monolithic approach for interaction of incompressible viscous fluid and an elastic body based on fluid pressure poisson equation this paper describes a new monolithic approach based on the fluid pressure poisson equation ppe to solve an interaction problem of incompressible viscous fluid and an elastic body the ppe is derived so as to be consistent with the coupled equation system for the fluid structure interaction fsi based on this approach we develop two kinds of efficient monolithic methods in both methods the fluid pressure is derived implicitly so as to satisfy the incompressibility constraint and all other unknown variables are derived fully explicitly or partially explicitly the coefficient matrix of the ppe for the fsi becomes symmetric and positive definite and its condition is insensitive to inhomogeneity of material properties the arbitrary lagrangian eulerian ale method is employed for the fluid part in order to take into account the deformable fluid structure interface to demonstrate fundamental performances of the proposed approach the developed two monolithic methods are applied to evaluate the added mass and the added damping of a circular cylinder as well as to simulate the vibration of a rectangular cylinder induced by vortex shedding copyright 2005 john wiley sons ltd

simulation and experimental validation of the motion of immersed rigid bodies in viscous flows abstract in this work we present the numerical analysis of the motion of a rigid body immersed in a viscous fluid the fluid flow induced by the gravity driven motion of a rigid body is particularly analysed using cylinders and spheres the navier stokes equations coupled with the interface motion are solved in the framework of a finite element formulation the interface of the rigid body is represented using a moving lagrangian interface technique mlit the proposed formulation is applied to describe the two dimensional motion of a cylinder between parallel walls as well as the axisymmetric representation of a sphere assessing in both cases the effects of the walls distance on the velocity developed by such bodies the numerical predictions are verified using different models for the analysis of the problem and are validated by comparison with theoretical empirical correlations moreover the computed body velocities are compared with experimental data reported in the literature as well as measurements obtained from experiments carried out in the framework of the present work

chaos and asymptotical stability in discrete time neural networks abstract this paper aims to theoretically prove by applying marotto s theorem that both transiently chaotic neural networks tcnn and discrete time recurrent neural networks drnn have chaotic structure a significant property tcnn and drnn is that they have only one bounded fixed point when absolute values of the self feedback connection weights in tcnn and the difference time in drnn are sufficiently large we show that this unique fixed point can actually evolve into a snap back repeller which generates chaotic structure if several conditions are satisfied on the other hand by using the lyapunov functions we also derive sufficient conditions on asymptotical stability for symmetrical versions of both tcnn and drnn under which tcnn and drnn asymptotically converge to a fixed point furthermore related bifurcations are also considered in this paper since both tcnn and drnn are not special but simple and general the obtained theoretical results hold for a wide class of discrete time neural networks to demonstrate the theoretical results of this paper better several numerical simulations are provided as illustrating examples

an efficient method for special least squares solution of the complex matrix equation axb cxd e f abstract in this paper we propose an efficient method for special least squares solution of the complex matrix equation a x b c x d e f by using the real representation matrices of complex matrices the particular structure of the real representation matrices the moore penrose generalized inverse and the kronecker product we obtain the explicit expression of the minimal norm least squares hermitian solution of the complex matrix equation a x b c x d e f which was studied by a product of matrices and vectors in wang et al 2016 our resulting formulas only involve real matrices and the corresponding algorithm only performs real arithmetic therefore our proposed method is more effective and portable finally we give three numerical examples to illustrate the effectiveness of our proposed method

a chaotic particle swarm optimization exploiting a virtual quartic objective function based on the personal and global best solutions the particle swarm optimization method pso is one of population based optimization techniques for global optimization where a number of candidate solutions called particles simultaneously move toward the tentative solutions found by particles so far which are called the personal and global bests respectively since in the pso the exploration ability is important to find a desirable solution various kinds of methods have been investigated to improve it in this paper we propose a pso with a new chaotic system derived from the steepest descent method for a virtual quartic objective function with perturbations having its global minima at the personal and global bests where elements of each particle s position are updated by the proposed chaotic system or the standard update formula thus the proposed pso can search for solutions around the personal and global bests intensively without being trapped at any local minimum due to the chaoticness moreover we show approximately the sufficient condition of parameter values of the proposed system under which the system is chaotic through computational experiments we verify the performance of the proposed pso by applying it to some global optimization problems

a discontinuous galerkin conservative level set scheme for interface capturing in multiphase flows the accurate conservative level set acls method of desjardins et al o desjardins v moureau h pitsch an accurate conservative level set ghost fluid method for simulating turbulent atomization j comput phys 227 18 2008 8395 8416 is extended by using a discontinuous galerkin dg discretization dg allows for the scheme to have an arbitrarily high order of accuracy with the smallest possible computational stencil resulting in an accurate method with good parallel scaling this work includes a dg implementation of the level set transport equation which moves the level set with the flow field velocity and a dg implementation of the reinitialization equation which is used to maintain the shape of the level set profile to promote good mass conservation a near second order converging interface curvature is obtained by following a height function methodology common amongst volume of fluid schemes in the context of the conservative level set various numerical experiments are conducted to test the properties of the method and show excellent results even on coarse meshes the tests include zalesak s disk two dimensional deformation of a circle time evolution of a standing wave and a study of the kelvin helmholtz instability finally this novel methodology is employed to simulate the break up of a turbulent liquid jet

global optimization by an improved differential evolutionary algorithm a hybrid differential evolutionary de algorithm for global optimization is proposed in the new algorithm the stochastic properties of chaotic systems are used to spread the individuals in search spaces as much as possible the pattern search method is employed to speed up the local exploiting and the de operators are used to jump to a better point the global convergence is proved three typical chaotic systems are investigated in detail numerical experiments on benchmark examples including 13 high dimensional functions demonstrate that the new method achieved an improved success rate and final solution with less computational effort

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

streamline upwind petrov galerkin formulations for convection dominated flows with particular emphasis on the incompressible navier stokes equations abstract a new finite element formulation for convection dominated flows is developed the basis of the formulation is the streamline upwind concept which provides an accurate multidimensional generalization of optimal one dimensional upwind schemes when implemented as a consistent petrov galerkin weighted residual method it is shown that the new formulation is not subject to the artificial diffusion criticisms associated with many classical upwind methods the accuracy of the streamline upwind petrov galerkin formulation for the linear advection diffusion equation is demonstrated on several numerical examples the formulation is extended to the incompressible navier stokes equations an efficient implicit pressure explicit velocity transient algorithm is developed which accomodates several treatments of the incompressibility constraint and allows for multiple iterations within a time step the effectiveness of the algorithm is demonstrated on the problem of vortex shedding from a circular cylinder at a reynolds number of 100

arbitrary lagrangian eulerian formulation for fluid rigid body interaction a new formulation for two dimensional fluid rigid body interaction problems is developed in particular vortex induced oscillations of a rigid body in viscous incompressible flow are studied the incompressible navier stokes equations are used to describe the motion of the fluid while it is assumed that the rigid body is mounted on a system consisting of a spring and a dashpot an arbitrary lagrangian eulerian formulation ale is used in order to account for large boundary motion a general formulation for the coupled problem is obtained by uncoupling the translation motion of the body from its rotational motion and developing a specific algorithm to efficiently handle the nonlinear dependence of the rotations this general formulation can be easily applied to multi body problems two numerical examples involving either translations and rotations are presented as an illustration of the proposed methodologies for fluid rigid body interaction

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

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

positive solutions to the equations ax c and xb d for hilbert space operators abstract the paper studies the equation a x c for bounded linear operators between hilbert spaces gives conditions for the existence of hermitian solutions and positive solutions and obtains the formula for the general form of these solutions then the common hermitian and positive solutions to the equations a x c and x b d are studied and new representations of the general solutions are given many results for matrices are recovered as special cases and the results of phadke and thakare s v phadke n k thakare generalized inverses and operator equations linear algebra appl 23 1979 191 199 are corrected

symmetric and skew antisymmetric solutions to systems of real quaternion matrix equations in this paper we consider symmetric and skew antisymmetric solutions to certain matrix equations over the real quaternion algebra h first a criterion for a quaternion matrix to be symmetric and skew antisymmetric is given then necessary and sufficient conditions are obtained for the matrix equation ax c and the following system a 1x c 1xb 3 c 3 to have symmetric and skew antisymmetric solutions the expressions of such solutions of the matrix equation and the system mentioned above are also given

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

discrete time convergence theory and updating rules for neural networks with energy functions we present convergence theorems for neural networks with arbitrary energy functions and discrete time dynamics for both discrete and continuous neuronal input output functions we discuss systematically how the neuronal updating rule should be extracted once an energy function is constructed for a given application in order to guarantee the descent and minimization of the energy function as the network updates we explain why the existing theory may lead to inaccurate results and oscillatory behaviors in the convergence process we also point out the reason for and the side effects of using hysteresis neurons to suppress these oscillatory behaviors

a self adaptive chaotic differential evolution algorithm using gamma distribution for unconstrained global optimization evolutionary algorithms eas have yielded promising results for solving nonlinear non differentiable and multi modal optimization problems due to its population based nature eas can avoid being trapped in a local optimum and consequently have the ability to find global optimal solutions as a novel evolutionary technique differential evolution de has received increasing attention and wide applications in a variety of fields de algorithm uses an efficient way of self adapting mutation using small populations for function optimization over continuous spaces the potentialities of de are its simple structure easy use convergence property quality of solution and robustness in this paper an effective self adaptive de algorithm based on gaussian probability distribution gamma distribution and chaotic sequence degc for solving continuous global optimization problems is proposed the proposed degc algorithm is tested on several benchmark functions from the usual literature numerical results comparisons with a classical de approach and a self adaptive de approach demonstrate the effectiveness and efficiency of the proposed degc algorithm

aerodynamic and aeroelastic analysis of bundled cables by numerical simulation abstract a numerical model for the aerodynamic and aeroelastic analysis of bundled cables commonly used in energy transmission lines is presented in this work the bundles were idealized by a sectional model representing the section at the mid span between two supporting towers a slightly compressible viscous fluid was considered and the two dimensional flow was analyzed using a two step explicit method with an arbitrary eulerian lagrangean description a taylor series expansion was used in time and the classical galerkin technique with the finite element method were used for space discretization turbulence was modeled using large eddy simulation with the classical smagorinsky s sub grid scale model the set of cables forms a single body with elastic constrains working mechanically coupled being each cable linked to the others by spacers the fluid structure interaction was taken into account considering equilibrium and compatibility conditions at the fluid solid interfaces and the resulting dynamical equilibrium equation was solved using the newmark s method

the matrix equations ax c xb d for the pair of matrix equations ax c xb d this paper gives common solutions of minimum possible rank and also other feasible specified ranks

a computational approach for predicting the hydroelasticity of flexible structures based on the pressure poisson equation this work is concerned with the modeling of the interaction of fluid flow with flexible solid structures the flow under consideration is governed by the navier stokes equations for incompressible viscous fluids and modeled with low order velocity pressure finite elements the motion of the fluid domain is accounted for by the arbitrary lagrangian eulerian formulation the structure is represented by means of an appropriate standard finite element formulation the spring smooth analogy is used to mesh control the time integrating algorithm is based on the predictor multi corrector algorithm an important aspect of the present work is the introduction of a new monolithic approach based on the fluid pressure poisson equation ppe to solve the hydroelasticity problem of an incompressible viscous fluid with an elastic body that is vibrating due to flow excitation the ppe is derived to be consistent with the coupled system equation for the fluid structure interaction fsi based on this approach an efficient monolithic method is adopted to simulate hydroelasticity between the flexible structure and the flow the fluid pressure is implicitly derived to satisfy the incompressibility constraint and the other unknown variables are explicitly derived the coefficient matrix of the ppe for the fsi becomes symmetric and positive definite to demonstrate the performance of the proposed approach two working examples a beam immersed in incompressible fluid and a guide vane of a francis turbine passage were used the results show the validity of the proposed approach copyright 2007 john wiley sons ltd

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

a weak coupling algorithm for seabed wave interaction analysis this paper proposes a weak coupling algorithm for the seabed wave interaction analysis this algorithm is based on the error integration over a time interval and space along fluid seabed interfaces with this weak coupling algorithm seabed wave interaction analysis can be easily carried out with a fluid solver in fluid domain and a meshless solver in seabed domain independently the fluid solver is based on a two step projection method for navier stokes equation the free surface of seawaters during wave propagation is traced through the volume of fluid vof the meshless solver is based on the biot s consolidation theory of saturated poroelasticity which considers the interaction of soil deformation and pore water pressure in seabed data exchange at the interface between fluid and porous medium domains is carried out through a point interpolation method interface condition is approximately satisfied in a variational sense through an iteration scheme between two solvers this algorithm does allow not only non matching mesh along the interface but also non matching time step in different subdomains numerical examples show that this weak coupling algorithm is able to capture the deformation of interface and improve the computation accuracy and efficiency 2004 elsevier b v all rights reserved

a novel chaotic neural network with the ability to characterize local features and its application to provide an ability to characterize local features for the chaotic neural network cnn gauss wavelet is used for the self feedback of the cnn with the dilation parameter acting as the bifurcation parameter the exponentially decaying dilation parameter and the chaotically varying translation parameter not only govern the wavelet self feedback transform but also enable the cnn to generate complex dynamics behavior preventing the network from being trapped in the local minima analysis of the energy function of the cnn indicates that the local characterization ability of the proposed cnn is effectively provided by the wavelet self feedback in the manner of inverse wavelet transform and that the proposed cnn can achieve asymptotical stability the experimental results on traveling salesman problem tsp suggest that the proposed cnn has a higher average success rate for obtaining globally optimal or near optimal solutions

the skew symmetric orthogonal solutions of the matrix equation ax b abstract an n n real matrix x is said to be a skew symmetric orthogonal matrix if xt x and xtx i using the special form of the c s decomposition of an orthogonal matrix with skew symmetric k k leading principal submatrix this paper establishes the necessary and sufficient conditions for the existence of and the expressions for the skew symmetric orthogonal 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 furthermore the procrustes problem of skew symmetric orthogonal matrices is considered and the formula solutions are provided finally an algorithm is proposed for solving the first and third problems numerical experiments show that it is feasible

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

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

cooperative updating in the hopfield model we propose a new method for updating units in the hopfield model with this method two or more units change at the same time so as to become the lowest energy state among all possible states since this updating algorithm is based on the detailed balance equation convergence to the boltzmann distribution is guaranteed if our algorithm is applied to finding the minimum energy in constraint satisfaction and combinatorial optimization problems then there is a faster convergence than those with the usual algorithm in the neural network this is shown by experiments with the travelling salesman problem the four color problem the n queen problem and the graph bi partitioning problem in constraint satisfaction problems for which earlier neural networks are effective in some cases our updating scheme works fine even though we still encounter the problem of ending up in local minima our updating scheme has a great advantage compared with the usual updating scheme used in combinatorial optimization problems also we discuss parallel computing using our updating algorithm