iterative algorithms for solving a class of complex conjugate and transpose matrix equations this paper is concerned with iterative solutions to a class of complex matrix equations which include some previously investigated matrix equations as special cases by applying the hierarchical identification principle an iterative algorithm is constructed to solve this class of matrix equations a sufficient condition is presented to guarantee that the proposed algorithm is convergent for an arbitrary initial matrix with a real representation of a complex matrix as tools by using some properties of the real representation a convergence condition that is easier to compute is also given in terms of original coefficient matrices a numerical example is employed to illustrate the effectiveness of the proposed methods

spectra of sparse random matrices we compute the spectral density for ensembles of sparse symmetric random matrices using replica our formulation of the replica symmetric ansatz shares the symmetries of that suggested in a seminal paper by rodgers and bray symmetry with respect to permutation of replica and rotation symmetry in the space of replica but uses a different representation in terms of superpositions of gaussians it gives rise to a pair of integral equations which can be solved by a stochastic population dynamics algorithm remarkably our representation allows us to identify pure point contributions to the spectral density related to the existence of normalizable eigenstates our approach is not restricted to matrices defined on graphs with poissonian degree distribution matrices defined on regular random graphs or on scale free graphs are easily handled we also look at matrices with row constraints such as discrete graph laplacians our approach naturally allows us to unfold the total density of states into contributions coming from vertices of different local coordinations and an example of such an unfolding is presented our results are well corroborated by numerical diagonalization studies of large finite random matrices

iterative solutions to the extended sylvester conjugate matrix equations this paper is concerned with iterative solutions to a class of complex matrix equations by applying the hierarchical identification principle an iterative algorithm is constructed to solve this class of complex matrix equations the range of the convergence factor is given to guarantee that the proposed algorithm is convergent for arbitrary initial matrix by applying a real representation of a complex matrix as a tool by using some properties of the real representation a sufficient convergence condition that is easier to compute is also given by original coefficient matrices two numerical examples are given to illustrate the effectiveness of the proposed methods

on matrix equations x axf c and x axf c with the help of the concept of kronecker map an explicit solution for the matrix equation x axf c is established this solution is neatly expressed by a symmetric operator matrix a controllability matrix and an observability matrix in addition the matrix equation email protected f c is also studied an explicit solution for this matrix equation is also proposed by means of the real representation of a complex matrix this solution is neatly expressed by a symmetric operator matrix two controllability matrices and two observability matrices

stochastic stability for markovian jump linear systems associated with a finite number of jump times abstract this paper deals with a stochastic stability concept for discrete time markovian jump linear systems the random jump parameter is associated to changes between the system operation modes due to failures or repairs which can be well described by an underlying finite state markov chain in the model studied a fixed number of failures or repairs is allowed after which the system is brought to a halt for maintenance or for replacement the usual concepts of stochastic stability are related to pure infinite horizon problems and are not appropriate in this scenario a new stability concept is introduced named stochastic stability that is tailored to the present setting necessary and sufficient conditions to ensure the stochastic stability are provided and the almost sure stability concept associated with this class of processes is also addressed the paper also develops equivalences among second order concepts that parallels the results for infinite horizon problems

gradient based iterative solutions for general linear matrix equations in this paper we present a gradient based iterative algorithm for solving general linear matrix equations by extending the jacobi iteration and by applying the hierarchical identification principle convergence analysis indicates that the iterative solutions always converge fast to the exact solutions for any initial values and small condition numbers of the associated matrices two numerical examples are provided to show that the proposed algorithm is effective

iterative solutions of the generalized sylvester matrix equations by using the hierarchical identification principle abstract in this paper by extending the well known jacobi and gauss seidel iterations for ax b we study iterative solutions of matrix equations axb f and generalized sylvester matrix equations axb cxd f including the sylvester equation ax xb f as a special case and present a gradient based and a least squares based iterative algorithms for the solution it is proved that the iterative solution always converges to the exact solution for any initial values the basic idea is to regard the unknown matrix x to be solved as the parameters of a system to be identified and to obtain the iterative solutions by applying the hierarchical identification principle finally we test the algorithms and show their effectiveness using a numerical example

unifying model for random matrix theory in arbitrary space dimensions a sparse random block matrix model suggested by the hessian matrix used in the study of elastic vibrational modes of amorphous solids is presented and analyzed by evaluating some moments benchmarked against numerics differences in the eigenvalue spectrum of this model in different limits of space dimension d and for arbitrary values of the lattice coordination number z are shown and discussed as a function of these two parameters and their ratio z d the most studied models in random matrix theory erdos renyi graphs effective medium and replicas can be reproduced in the various limits of block dimensionality d remarkably the marchenko pastur spectral density which is recovered by replica calculations for the laplacian matrix is reproduced exactly in the limit of infinite size of the blocks or d ensuremath rightarrow ensuremath infty which clarifies the physical meaning of space dimension in these models we feel that the approximate results for d 3 provided by our method may have many potential applications in the future from the vibrational spectrum of glasses and elastic networks to wave localization disordered conductors random resistor networks and random walks

the general coupled matrix equations over generalized bisymmetric matrices abstract in the present paper by extending the idea of conjugate gradient cg method we construct an iterative method to solve the general coupled matrix equations j 1 p a ij x j b ij m i i 1 2 p including the generalized coupled lyapunov and sylvester matrix equations as special cases over generalized bisymmetric matrix group x 1 x 2 x p by using the iterative method the solvability of the general coupled matrix equations over generalized bisymmetric matrix group can be determined in the absence of roundoff errors when the general coupled matrix equations are consistent over generalized bisymmetric matrices a generalized bisymmetric solution group can be obtained within finite iteration steps in the absence of roundoff errors the least frobenius norm generalized bisymmetric solution group of the general coupled matrix equations can be derived when an appropriate initial iterative matrix group is chosen in addition the optimal approximation generalized bisymmetric solution group to a given matrix group x 1 x 2 x p in frobenius norm can be obtained by finding the least frobenius norm generalized bisymmetric solution group of new general coupled matrix equations the numerical results indicate that the iterative method works quite well in practice

localization in random bipartite graphs numerical and empirical study we investigate adjacency matrices of bipartite graphs with a power law degree distribution motivation for this study is twofold first vibrational states in granular matter and jammed sphere packings second graphs encoding social interaction especially electronic commerce we establish the position of the mobility edge and show that it strongly depends on the power in the degree distribution and on the ratio of the sizes of the two parts of the bipartite graph at the jamming threshold where the two parts have the same size localization vanishes we found that the multifractal spectrum is nontrivial in the delocalized phase but still near the mobility edge we also study an empirical bipartite graph namely the amazon reviewer item network we found that in this specific graph the mobility edge disappears and we draw a conclusion from this fact regarding earlier empirical studies of the amazon network

spectra of modular random graphs we compute spectra of symmetric random matrices defined on graphs exhibiting a modular structure modules are initially introduced as fully connected sub units of a graph by contrast inter module connectivity is taken to be incomplete two different types of inter module connectivities are considered one where the number of intermodule connections per node diverges and one where this number remains finite in the infinite module size limit in the first case the results can be understood as a perturbation of a superposition of semicircular spectral densities one would obtain for uncoupled modules in the second case matters can be more involved and depend in detail on inter module connectivities for suitable parameters we even find near triangular shaped spectral densities similar to those observed in certain scale free networks in a system consisting of just two coupled modules analytic results are presented for the infinite module size limit they are well corroborated by numerical simulations

new symmetry preserving method for optimal correction of damping and stiffness matrices using measured modes measured and analytical data are unlikely to be equal due to measured noise model inadequacies structural damage etc it is necessary to update the physical parameters of analytical models for proper simulation and design studies starting from simulated measured modal data such as natural frequencies and their corresponding mode shapes a new computationally efficient and symmetry preserving method and associated theories are presented in this paper to update the physical parameters of damping and stiffness matrices simultaneously for analytical modeling a conjecture which is proposed in y x yuan h dai a generalized inverse eigenvalue problem in structural dynamic model updating j comput appl math 226 2009 42 49 is solved two numerical examples are presented to show the efficiency and reliability of the proposed method it is more important that some numerical stability analysis on the model updating problem is given combining with numerical experiments

optimization problems and replica symmetry breaking in finite connectivity spin glasses a formalism capable of handling the first step of hierarchical replica symmetry breaking rsb in finite connectivity models is introduced the emerging order parameter is claimed to be a probability distribution over the space of field distributions or equivalently magnetization distributions inside the cluster of states the approach is shown to coincide with previous works in the replica symmetric case and in the two limiting cases m 0 and 1 where m is parisi s break point as an application to the study of optimization problems the gs properties of the random 3 satisfiability problem are investigated and we present a first rsb solution improving replica symmetric results

finitely coordinated models for low temperature phases of amorphous systems we introduce models of heterogeneous systems with finite connectivity defined on random graphs to capture finite coordination effects on the low temperature behaviour of finite dimensional systems our models use a description in terms of small deviations of particle coordinates from a set of reference positions particularly appropriate for the description of low temperature phenomena a born von karman type expansion with random coefficients is used to model effects of frozen heterogeneities the key quantity appearing in the theoretical description is a full distribution of effective single site potentials which needs to be determined self consistently if microscopic interactions are harmonic the effective single site potentials turn out to be harmonic as well and the distribution of these single site potentials is equivalent to a distribution of localization lengths used earlier in the description of chemical gels for structural glasses characterized by frustration and anharmonicities in the microscopic interactions the distribution of single site potentials involves anharmonicities of all orders and both single well and double well potentials are observed the latter with a broad spectrum of barrier heights the appearance of glassy phases at low temperatures is marked by the appearance of asymmetries in the distribution of single site potentials as previously observed for fully connected systems double well potentials with a broad spectrum of barrier heights and asymmetries would give rise to the well known universal glassy low temperature anomalies when quantum effects are taken into account

reduced rank gradient based algorithms for generalized coupled sylvester matrix equations and its applications in this paper by constructing an objective function and using the gradient search full rank and reduced rank gradient based algorithms are suggested for solving generalized coupled sylvester matrix equations it is proved that the reduced rank iterative algorithm is convergent for proper initial iterative values by analyzing the spectral radius of the related matrices the convergence properties are studied and the optimal convergence factor of the reduced rank algorithm is determined the relationship between the reduced rank algorithm and the full rank algorithm is discussed consequently the computation load can be reduced greatly for solving a class of matrix equation a numerical example is provided to illustrate the effectiveness of the proposed algorithms and testify the conclusions suggested in this paper

replica theory for learning curves for gaussian processes on random graphs we use a statistical physics approach to derive accurate predictions for the challenging problem of predicting the performance of gaussian process regression performance is quantified by the learning curve defined as the average error versus number of training examples we assume the gaussian process prior is defined by a random walk kernel inputs are vertices on a random graph and the outputs are noisy function values we show that replica techniques can be used to obtain exact performance predictions in the limit of large graphs after first rewriting the average error in terms of a graphical model conventionally the gaussian process kernel is only globally normalized so that the prior variance can differ between vertices as a more principled alternative we also consider local normalization where the prior variance is uniform the normalization constants for the prior then have to be defined as thermal averages in an unnormalized model and this requires the introduction of a second auxiliary set of replicas our results for both types of kernel normalization apply generically to all random graph ensembles constrained by a fixed but arbitrary degree distribution we compare with numerically simulated learning curves and find excellent agreement a significant improvement over existing approximations

new proof of the gradient based iterative algorithm for the sylvester conjugate matrix equation abstract by applying the hierarchical identification principle the gradient based iterative algorithm is suggested for solving the sylvester conjugate matrix equation with the real representation of a complex matrix a new convergence proof is given the necessary and sufficient conditions for the convergence factor is determined to guarantee the convergence of the algorithm for any initial iterative matrix also a conjecture by wu et al 2010 is solved a numerical example is offered to illustrate the effectiveness of the suggested algorithm and verify some conclusions proposed in this paper

the mgpbicg method for solving the generalized coupled sylvester conjugate matrix equations in this paper we extend the generalized product type bi conjugate gradient gpbicg method for solving the generalized sylvester conjugate matrix equations a 1 x b 1 c 1 y d 1 s 1 a 2 x b 2 c 2 y d 2 s 2 by the real representation of the complex matrix and the properties of kronecker product and vectorization operator some numerical experiments demonstrate that the introduced iteration approach is efficient

a fundamental notion of equivalence for linear multivariable systems a fundamental form of equivalence of polynomial matrix descriptions of linear multivariable systems is defined based on the existence of a bijective map between the finite and infinite solution sets of the differential equations describing the two systems the connection with the system matrix relationship of full system equivalence is established

dynamical replica analysis of processes on finitely connected random graphs i vertex covering we study the stochastic dynamics of ising spin models with random bonds interacting on finitely connected poissonian random graphs we use the dynamical replica method to derive closed dynamical equations for the joint spin field probability distribution and solve these within the replica symmetry ansatz although the theory is developed in a general setting with a view to future applications in various other fields in this paper we apply it mainly to the dynamics of the glauber algorithm extended with cooling schedules when running on the so called vertex cover optimization problem our theoretical predictions are tested against both monte carlo simulations and known results from equilibrium studies in contrast to previous dynamical analyses based on deriving closed equations for only a small number of scalar order parameters the agreement between theory and experiment in the present study is nearly perfect

top eigenpair statistics for weighted sparse graphs we develop a formalism to compute the statistics of the top eigenpair of weighted sparse graphs with finite mean connectivity and bounded maximal degree framing the problem in terms of optimisation of a quadratic form on the sphere and introducing a fictitious temperature we employ the cavity and replica methods to find the solution in terms of self consistent equations for auxiliary probability density functions which can be solved by population dynamics this derivation allows us to identify and unpack the individual contributions to the top eigenvector s components coming from nodes of degree k the analytical results are in perfect agreement with numerical diagonalisation of large weighted adjacency matrices and are further cross checked on the cases of random regular graphs and sparse markov transition matrices for unbiased random walks

spin models on random graphs with controlled topologies beyond degree constraints we study ising spin models on finitely connected random interaction graphs which are drawn from an ensemble in which not only the degree distribution p k can be chosen arbitrarily but which allows for further fine tuning of the topology via preferential attachment of edges on the basis of an arbitrary function q k k of the degrees of the vertices involved we solve these models using finite connectivity equilibrium replica theory within the replica symmetric ansatz in our ensemble of graphs phase diagrams of the spin system are found to depend no longer only on the chosen degree distribution but also on the choice made for q k k the increased ability to control interaction topology in solvable models beyond prescribing only the degree distribution of the interaction graph enables a more accurate modeling of real world interacting particle systems by spin systems on suitably defined random graphs

gradient based and least squares based iterative algorithms for matrix equations axb cxtd f abstract this paper develops a gradient based and a least squares based iterative algorithms for solving matrix equation axb cx t d f the basic idea is to decompose the matrix equation system under consideration into two subsystems by applying the hierarchical identification principle and to derive the iterative algorithms by extending the iterative methods for solving ax b and axb f the analysis shows that when the matrix equation has a unique solution under the sense of least squares the iterative solution converges to the exact solution for any initial values a numerical example verifies the proposed theorems

replica field theory for deterministic models i binary sequences with low autocorrelation we study systems without quenched disorder with a complex landscape and we use replica symmetry theory to describe them we discuss the golay bernasconi derrida approximation of the low autocorrelation model and we reconstruct it by using replica calculations we then consider the full model its low t properties with the help of number theory and a hartree fock resummation of the high temperature series we show that replica theory allows us to solve the model in the high t phase our solution is based on one link integral techniques and is based on substituting a fourier transform with a generic unitary transformation we discuss this approach as a powerful tool to describe systems with a complex landscape in the absence of quenched disorder

iterative solutions to the kalman yakubovich conjugate matrix equation abstract two operations are introduced for complex matrices in terms of these two operations an infinite series expression is obtained for the unique solution of the kalman yakubovich conjugate matrix equation based on the obtained explicit solution some iterative algorithms are given for solving this class of matrix equations convergence properties of the proposed algorithms are also analyzed by using some properties of the proposed operations for complex matrices

graph bipartitioning and spin glasses on a random network of fixed finite valence the authors study the problem of bipartitioning a random graph of fixed finite valence using a mean field replica symmetric theory of an ising ferromagnet with zero magnetisation constraint the thermodynamics is determined by the probability distribution of an auxiliary field the expression for the ground state energy agrees with that proposed by mezard and parisi 1986 using a cavity field method but their expression for the fraction of crazy spins is reinterpreted

random graph coloring statistical physics approach the problem of vertex coloring in random graphs is studied using methods of statistical physics and probability our analytical results are compared to those obtained by exact enumeration and monte carlo simulations we critically discuss the merits and shortcomings of the various methods and interpret the results obtained we present an exact analytical expression for the two coloring problem as well as general replica symmetric approximated solutions for the thermodynamics of the graph coloring problem with p colors and k body edges

iterative method to solve the generalized coupled sylvester transpose linear matrix equations over reflexive or anti reflexive matrix abstract the iterative method of generalized coupled sylvester transpose linear matrix equations a x b c y t d s 1 e x t f g y h s 2 over reflexive or anti reflexive matrix pair x y is presented on the condition that the coupled matrix equations are consistent we show that the solution pair x y proposed by the iterative method can be obtained within finite iterative steps in the absence of roundoff error for any initial value given a reflexive or anti reflexive matrix moreover the optimal approximation reflexive or anti reflexive matrix solution pair to an arbitrary given reflexive or anti reflexive matrix pair can be derived by searching the least frobenius norm solution pair of the new generalized coupled sylvester transpose linear matrix equations finally some numerical examples are given which illustrate that the introduced iterative algorithm is quite efficient

iterative least squares solutions of coupled sylvester matrix equations abstract in this paper we present a general family of iterative methods to solve linear equations which includes the well known jacobi and gauss seidel iterations as its special cases the methods are extended to solve coupled sylvester matrix equations in our approach we regard the unknown matrices to be solved as the system parameters to be identified and propose a least squares iterative algorithm by applying a hierarchical identification principle and by introducing the block matrix inner product the star product for short we prove that the iterative solution consistently converges to the exact solution for any initial value the algorithms proposed require less storage capacity than the existing numerical ones finally the algorithms are tested on computer and the results verify the theoretical findings

the modified conjugate gradient methods for solving a class of generalized coupled sylvester transpose matrix equations in this paper we consider the iteration solutions of generalized coupled sylvester transpose matrix equations a1xb1 c1ytd1 f1 a2yb2 c2xtd2 f2 when the coupled matrix equations are consistent we propose a modified conjugate gradient method to solve the equations and prove that a solution x y can be obtained within finite iterative steps in the absence of roundoff error for any initial value furthermore we show that the minimum norm solution can be got by choosing a special kind of initial matrices when the coupled matrix equations are inconsistent we present another modified conjugate gradient method to find the least squares solution with the minimum norm finally some numerical examples are given to show the behavior of the considered algorithms

weighted least squares solutions to general coupled sylvester matrix equations this paper is concerned with weighted least squares solutions to general coupled sylvester matrix equations gradient based iterative algorithms are proposed to solve this problem this type of iterative algorithm includes a wide class of iterative algorithms and two special cases of them are studied in detail in this paper necessary and sufficient conditions guaranteeing the convergence of the proposed algorithms are presented sufficient conditions that are easy to compute are also given the optimal step sizes such that the convergence rates of the algorithms which are properly defined in this paper are maximized and established several special cases of the weighted least squares problem such as a least squares solution to the coupled sylvester matrix equations problem solutions to the general coupled sylvester matrix equations problem and a weighted least squares solution to the linear matrix equation problem are simultaneously solved several numerical examples are given to illustrate the effectiveness of the proposed algorithms

least squares solution with the minimum norm to general matrix equations via iteration two iterative algorithms are presented in this paper to solve the minimal norm least squares solution to a general linear matrix equations including the well known sylvester matrix equation and lyapunov matrix equation as special cases the first algorithm is based on the gradient based searching principle and the other one can be viewed as its dual form necessary and sufficient conditions for the step sizes in these two algorithms are proposed to guarantee the convergence of the algorithms for arbitrary initial conditions sufficient condition that is easy to compute is also given moreover two methods are proposed to choose the optimal step sizes such that the convergence speeds of the algorithms are maximized between these two methods the first one is to minimize the spectral radius of the iteration matrix and explicit expression for the optimal step size is obtained the second method is to minimize the square sum of the f norm of the error matrices produced by the algorithm and it is shown that the optimal step size exits uniquely and lies in an interval several numerical examples are given to illustrate the efficiency of the proposed approach

convergence of gradient based iterative solution of coupled markovian jump lyapunov equations the solution of coupled discrete time markovian jump lyapunov matrix equations cdmjlmes is important in stability analysis and controller design for markovian jump linear systems this paper presents a simple and effective iterative method to produce numerical solutions to this class of matrix equations the gradient based algorithm is developed from an optimization point of view a necessary and sufficient condition guaranteeing the convergence of the algorithm is established this condition shows that the algorithm always converges provided the cdmjlmes have unique solutions which is evidently different from the existing results that converge conditionally a simple sufficient condition which is easy to test is also provided the optimal step size in the algorithm such that the convergence rate of the algorithm is maximized is given explicitly it turns out that an upper bound of the convergence rate is bounded by a function of the condition number of the augmented coefficient matrix of the cdmjlmes some parameters are introduced to the algorithm that will potentially reduce the condition number and thus increase the convergence rate of the algorithm a numerical example is used to illustrate the efficiency of the proposed approach

on solutions of the matrix equations x axb c and x axb c this paper studies the solutions of complex matrix equations x axb c and x axb c and obtains explicit solutions of the equations by the method of characteristic polynomial and a method of real representation of a complex matrix respectively