critical statistics for non hermitian matrices we introduce a generalized ensemble of non hermitian matrices interpolating between the gaussian unitary ensemble the ginibre ensemble and the poisson ensemble the joint eigenvalue distribution of this model is obtained by means of an extension of the itzykson zuber formula to general complex matrices its correlation functions are studied both in the case of weak non hermiticity and in the case of strong non hermiticity in the weak non hermiticity limit we show that the spectral correlations in the bulk of the spectrum display critical statistics the asymptotic linear behavior of the number variance is already approached for energy differences of the order of the eigenvalue spacing to lowest order its slope does not depend on the degree of non hermiticity close the edge the spectral correlations are similar to the hermitian case in the strong non hermiticity limit the crossover behavior from the ginibre ensemble to the poisson ensemble first appears close to the surface of the spectrum our model may be relevant for the description of the spectral correlations of an open disordered system close to an anderson transition

correlations of eigenvectors for non hermitian random matrix models we establish a general relation between the diagonal correlator of eigenvectors and the spectral green close quote s function for non hermitian random matrix models in the large n limit we apply this result to a number of non hermitian random matrix models and show that the outcome is in good agreement with numerical results copyright ital 1999 ital the american physical society

eigenvalue statistics of random real matrices completing ginibre s work we determine the joint probability density of eigenvalues in a gaussian ensemble of real asymmetric matrices which is invariant under orthogonal transformations the symmetry parameter r may vary from 1 antisymmetric ensemble through 0 completely asymmetric ensemble to 1 symmetric ensemble the elliptic law for the average density of eigenvalues in the limit of large dimension is recovered matrices of the type considered appear in models for neural network dynamics and dissipative quantum dynamics

universal eigenvector correlations in quaternionic ginibre ensembles non hermitian random matrices enjoy non trivial correlations in the statistics of their eigenvectors we study the overlap among left and right eigenvectors in ginibre ensembles with quaternion val

statistics of multiple avoided crossings the author considers the energy levels of a generic quantum mechanical system with hamiltonian h depending on a parameter x if the energy levels en are plotted as a function of x the curves do not cross the points of closest approach are called avoided crossings these have a distinctive geometry and are important because they determine the limits of applicability of the adiabatic theorem he describes some theoretical results on the parameter space density of avoided crossings for systems with the spectral statistics of the gaussian orthogonal ensemble goe these results are in good agreement with numerical experiments

point estimation of simultaneous methods for solving polynomial equations a survey ii the construction of computationally verifiable initial conditions which provide both the guaranteed and fast convergence of the numerical root finding algorithm is one of the most important problems in solving nonlinear equations smale s point estimation theory from 1981 was a great advance in this topic it treats convergence conditions and the domain of convergence in solving an equation f z 0 using only the information of f at the initial point z 0 the study of a general problem of the construction of initial conditions of practical interest providing guaranteed convergence is very difficult even in the case of algebraic polynomials in the light of smale s point estimation theory an efficient approach based on some results concerning localization of polynomial zeros and convergent sequences is applied in this paper to iterative methods for the simultaneous determination of simple zeros of polynomials we state new improved initial conditions which provide the guaranteed convergence of frequently used simultaneous methods for solving algebraic equations ehrlich aberth s method ehrlich aberth s method with newton s correction borsch supan s method with weierstrass correction and halley like or wang zheng method the introduced concept offers not only a clear insight into the convergence analysis of sequences generated by the considered methods but also explicitly gives their order of convergence the stated initial conditions are of significant practical importance since they are computationally verifiable they depend only on the coefficients of a given polynomial its degree n and initial approximations to polynomial zeros

complete diagrammatics of the single ring theorem using diagrammatic techniques we provide explicit functional relations between the cumulant generating functions for the biunitarily invariant ensembles in the limit of large size of matrices the formalism allows us to map two distinct areas of free random variables hermitian positive definite operators and non normal r diagonal operators we also rederive the haagerup larsen theorem and show how its recent extension to the eigenvector correlation function appears naturally within this approach

spectral theory of sparse non hermitian random matrices sparse non hermitian random matrices arise in the study of disordereda 13 physical systems with asymmetric local interactions and have applications ranginga 13 from neural networks to ecosystem dynamics the spectral characteristics of thesea 13 matrices provide crucial information on system stability and susceptibility however a 13 their study is greatly complicated by the twin challenges of a lack of symmetry anda 13 a sparse interaction structure in this review we provide a concise and systematica 13 introduction to the main tools and results in this field we show how the spectraa 13 of sparse non hermitian matrices can be computed via an analogy with infinitea 13 dimensional operators obeying certain recursion relations with reference to threea 13 illustrative examples adjacency matrices of regular oriented graphs adjacencya 13 matrices of oriented erd s renyi graphs and adjacency matrices of weighted orienteda 13 erd s renyi graphs we demonstrate the use of these methods to obtain both analytica 13 and numerical results for the spectrum the spectral distribution the location of outliera 13 eigenvalues and the statistical properties of eigenvectors a 13

the empirical mode decomposition and the hilbert spectrum for nonlinear and non stationary time series analysis a new method for analysing nonlinear and non stationary data has been developed the key part of the method is the empirical mode decomposition method with which any complicated data set can be dec

statistical properties of eigenvectors in non hermitian gaussian random matrix ensembles statistical properties of eigenvectors in non hermitian random matrix ensembles are discussed with an emphasis on correlations between left and right eigenvectors two approaches are described one is an exact calculation for ginibre s ensemble in which each matrix element is an independent identically distributed gaussian complex random variable the other is a simpler calculation using n 1 as an expansion parameter where n is the rank of the random matrix this is applied to girko s ensemble consequences of eigenvector correlations which may be of physical importance in applications are also discussed it is shown that eigenvalues are much more sensitive to perturbations than in the corresponding hermitian random matrix ensembles it is also shown that in problems with time evolution governed by a non hermitian random matrix transients are controlled by eigenvector correlations

spectrum of the fokker planck operator representing diffusion in a random velocity field we study spectral properties of the fokker planck operator that represents particles moving via a combination of diffusion and advection in a time independent random velocity field presenting in detail work outlined elsewhere j t chalker and z j wang phys rev lett 79 1797 1997 we calculate analytically the ensemble averaged one particle green function and the eigenvalue density for this fokker planck operator using a diagrammatic expansion developed for resolvents of non hermitian random operators together with a mean field approximation the self consistent born approximation which is well controlled in the weak disorder regime for dimension dg2 the eigenvalue density in the complex plane is nonzero within a wedge that encloses the negative real axis particle motion is diffusive at long times but for short times we find a novel time dependence of the mean square displacement r 2 ensuremath sim t 2 d in dimension dg2 associated with the imaginary parts of eigenvalues

the distribution of overlaps between eigenvectors of ginibre matrices we study the overlaps between eigenvectors of nonnormal matrices they quantify the stability of the spectrum and characterize the joint eigenvalues increments under dyson type dynamics well known work by chalker and mehlig calculated the expectation of these overlaps for complex ginibre matrices for the same model we extend their results by deriving the distribution of diagonal overlaps the condition numbers and their correlations we prove i convergence of condition numbers for bulk eigenvalues to an inverse gamma distribution more generally we decompose the quenched overlap i e conditioned on eigenvalues as a product of independent random variables ii asymptotic expectation of off diagonal overlaps both for microscopic or mesoscopic separation of the corresponding eigenvalues iii decorrelation of condition numbers associated to eigenvalues at mesoscopic distance at polynomial speed in the dimension iv second moment asymptotics to identify the fluctuations order for off diagonal overlaps when the related eigenvalues are separated by any mesoscopic scale v a new formula for the correlation between overlaps for eigenvalues at microscopic distance both diagonal and off diagonal these results imply estimates on the extreme condition numbers the volume of the pseudospectrum and the diffusive evolution of eigenvalues under dyson type dynamics at equilibrium

on the guaranteed convergence of the fourth order simultaneous method for polynomial zeros one of the most important and challenging problems in solving nonlinear equations is the construction of computationally verifiable initial conditions that provide both the guaranteed and fast convergence of the considered numerical method a suitable convergence procedure based partially on smale s point estimation theory from 1981 is applied in this paper to the fourth order iterative method for the simultaneous approximation of simple zeros of polynomials proposed by zheng and sun in 1999 we have stated initial conditions which ensure the guaranteed convergence of this method these conditions are of significant practical importance since they depend only on available data the coefficients of a given polynomial its degree n and initial approximations to polynomial zeros

on the new fourth order methods for the simultaneous approximation of polynomial zeros a new iterative method of the fourth order for the simultaneous determination of polynomial zeros is proposed this method is based on a suitable zero relation derived from the fourth order method for a single zero belonging to the schroder basic sequence one of the most important problems in solving polynomial equations the construction of initial conditions that enable both guaranteed and fast convergence is studied in detail for the proposed method these conditions are computationally verifiable since they depend only on initial approximations the polynomial coefficients and the polynomial degree which is of practical importance the construction of improved methods in ordinary complex arithmetic and complex circular arithmetic is discussed finally numerical examples and the comparison with existing fourth order methods are given

on the detection of determinism in a time series the nonlinearly scaled distributions of the strengths of the orthogonal modes in the data of a time series are compared with that derived from its surrogate counterpart to assess its chaoticity or the stochastic nature chaoticity manifests in the relatively decreasing strengths of the weaker modes with increasing dimension of the orthogonal spaces mapping the process the distributions of the strengths of the modes for the data and the surrogate are compared statistically singular value decomposition is used for the implementation the ability of the proposed method to distinguish between chaotic and stochastic dynamics is demonstrated through a number of simulated series including linear and nonlinear stochastic noise and processes described by lorenz rossler and mackey glass equations henon map and ecological map and some real life examples

summing free unitary random matrices i use quaternion free probability calculus an extension of free probability to non hermitian matrices which is introduced in a succinct but self contained way to derive in the large size limit the mean densities of the eigenvalues and singular values of sums of independent unitary random matrices weighted by complex numbers in the case of circular unitary ensemble summands i write them in terms of two master equations which i then solve and numerically test in four specific cases i conjecture a finite size extension of these results exploiting the complementary error function i prove a central limit theorem and its first subleading correction for independent identically distributed zero drift unitary random matrices

fuzzy entropy and its application for enhanced subspace filtering fuzzy entropy fuzzyen which employs the fuzzy probability to characterize the similarity between vectors is a robust nonlinear statistic to quantify the complexity or regularity of nonlinear time series the aim of this study is to investigate the statistical properties of fuzzyen and improve the subspace denoising technique using fuzzyen we first show the asymptotic normality of fuzzyen and derive its variance for finite sample behavior we then analyze the two pending and fundamental issues in subspace denoising i e depending on the so called noise floor and the unaltered noise existing in signal subspace from the point of view of fuzzy logic a fuzzyen assisted subspace iterative soft threshold fesist denoising method which can effectively overcome the deficiency in the existing subspace filtering ssf techniques is presented the effectiveness of the method is first demonstrated on two synthetic chaotic series and then tested on real biological signals the results demonstrate the superiority of the proposed method over existing ssf techniques as well as the empirical mode decomposition and wavelet decomposition approaches

the guaranteed convergence of laguerre like method abstract the construction of initial conditions that provide a guaranteed convergence of zero finding methods has attracted a great deal of attention for many years in this paper we consider convergent properties of the laguerre like method of the fourth order for the simultaneous approximation of polynomial zeros using a procedure based on smale s point estimation theory and some recent results concerned with localization of complex polynomial zeros we state initial conditions which enable both the guaranteed and fast convergence of this method these conditions are computationally verifiable since they depend only on initial approximations polynomial coefficients and polynomial degree which is of practical importance

gaussian ensemble of 2 2 pseudo hermitian random matrices we present a random matrix theory for systems invariant under the joint action of parity and time reversal and more generally for pseudo hermitian systems this brings out the appearance of the metric in a systematic way so that consistency with the postulates of quantum mechanics is maintained here we specialize only to 2 2 matrices and we construct a pseudo unitary group with explicit examples nearest neighbour level spacing distributions for various classes of ensembles are found to exhibit a degree of level repulsion different from those hitherto known this work is not only relevant to quantum chaos but also to two dimensional statistical mechanics and consistent non local relativistic theories

symplectic geometry spectrum analysis of nonlinear time series various time series decomposition techniques including wavelet transform singular spectrum analysis empirical mode decomposition and independent component analysis have been developed for non linear dynamic system analysis in this paper we describe a symplectic geometry spectrum analysis sgsa method to decompose a time series into a set of independent additive components sgsa is performed in four steps embedding symplectic qr decomposition grouping and diagonal averaging the obtained components can be used for de noising prediction control and synchronization we demonstrate the effectiveness of sgsa in reconstructing and predicting two noisy benchmark nonlinear dynamic systems the lorenz and mackey glass attractors examples of prediction of a decadal average sunspot number time series and a mechanomyographic signal recorded from human skeletal muscle further demonstrate the applicability of the sgsa method in real life applications

singular value decomposition in attractor reconstruction pitfalls and precautions applicability of singular value decomposition for reconstructing the strange attractor from one dimensional chaotic time series proposed by broomhead and king is extensively tested and discussed previously published doubts about its reliability are confirmed singular value decomposition by nature a linear method can bring distorted and misleading results when nonlinear structures are studied

pseudospectra of linear operators if a matrix or linear operator a is far from normal its eigenvalues or more generally its spectrum may have little to do with its behavior as measured by quantities such as an or exp ta more may be learned by examining the sets in the complex plane known as the pseudospectra of a defined by level curves of the norm of the resolvent zi a 1 five years ago the author published a paper that presented computed pseudospectra of thirteen highly nonnormal matrices arising in various applications since that time analogous computations have been carried out for differential and integral operators this paper a companion to the earlier one presents ten examples each chosen to illustrate one or more mathematical or physical principles

a note on the convergence of the weierstrass sor method for polynomial roots convergence properties of the sor weierstrass method for the simultaneous approximation of polynomial roots are considered the choice of acceleration parameter is discussed

non hermitian localization and population biology the time evolution of spatial fluctuations in inhomogeneous d dimensional biological systems is analyzed a single species continuous growth model in which the population disperses via diffusion and convection is considered time independent environmental heterogeneities such as a random distribution of nutrients or sunlight are modeled by quenched disorder in the growth rate linearization of this model of population dynamics shows that the fastest growing localized state dominates in a time proportional to a power of the logarithm of the system size using an analogy with a schr odinger equation subject to a constant imaginary vector potential we propose a delocalization transition for the steady state of the nonlinear problem at a critical convection threshold separating localized and extended states in the limit of high convection velocity the linearized growth problem in d dimensions exhibits singular scaling behavior described by a d ensuremath 1 dimensional generalization of the noisy burgers equation with universal singularities in the density of states associated with disorder averaged eigenvalues near the band edge in the complex plane the burgers mapping leads to unusual transverse spreading of convecting delocalized populations

the behaviour of approximations of the sor weierstrass method in this paper the behaviour of iterative methods of weierstrass type with acceleration parameter h 0 1 is considered the main result is concerned with initial conditions under which these methods fail it is shown that for a given polynomial and acceleration parameter h 1 there exists the unique divergent point from cn

fast and robust fixed point algorithms for independent component analysis independent component analysis ica is a statistical method for transforming an observed multidimensional random vector into components that are statistically as independent from each other as possible we use a combination of two different approaches for linear ica comon s information theoretic approach and the projection pursuit approach using maximum entropy approximations of differential entropy we introduce a family of new contrast functions for ica these contrast functions enable both the estimation of the whole decomposition by minimizing mutual information and estimation of individual independent components as projection pursuit directions the statistical properties of the estimators based on such contrast functions are analyzed under the assumption of the linear mixture model and it is shown how to choose contrast functions that are robust and or of minimum variance finally we introduce simple fixed point algorithms for practical optimization of the contrast functions

singular spectrum analysis a toolkit for short noisy chaotic signals abstract singular spectrum analysis ssa is developed further based on experience with applications to geophysical time series it is shown that ssa provides a crude but robust approximation of strange attractors by tori in the presence of noise the method works well for short noisy time series the lagged covariance matrix of the processes studied is the basis of ssa we select subsets of eigenelements and associated principal components pcs in order to provide i a noise reduction algorithm ii a detrending algorithm and iii an algorithm for the identification of oscillatory components reconstructed components rcs are developed to provide optimal reconstruction of a dynamic process at precise epochs rather than averaged over the window length of the analysis ssa is combined with advanced spectral analysis methods the maximum entropy method mem and the multi taper method mtm to refine the interpretation of oscillatory behavior a combined ssa mem method is also used for the prediction of selected subsets of rcs the entire toolkit is validated against a set of four prescribed time series generated by known processes quasi periodic or chaotic it is also applied to a time series of global surface air temperatures 130 years long which has attracted considerable attention in the context of the global warming issue and provides a severe test for noise reduction and prediction

enclosing clusters of zeros of polynomials lagrange interpolation and partial fraction expansion can be used to derive a gerschgorin type theorem that gives simple and powerful a posteriori error bounds for the zeros of a polynomial if approximations to all zeros are available compared to bounds from a corresponding eigenvalue problem a factor of at least two is gained the accuracy of the bounds is analyzed and special attention is given to ensure that the bounds work well not only for single zeros but also for multiple zeros and clusters of close zeros a rouche type theorem is also given that in many cases reduces the bound even further

symplectic rotational geometry in human biomechanics in this paper we present rotational symplectic geometry for use in modern biomechanics of human motion generalized hamiltonian formalism has been formulated on the configuration manifold consisting of rotational lie groups the proposed symplectic biodynamics includes conservative dissipative and driving forces and spinal reflex like servo controls reduction of the angular configuration manifold enables topological brain like control as well as a subsequent topological analysis

random matrices close to hermitian or unitary overview of methods and results the paper discusses recent progress in understanding statistical properties of eigenvalues of weakly non hermitian and non unitary random matrices the first type of ensembles is of the form i with being a large random n n hermitian matrix with independent entries deformed by a certain anti hermitian n n matrix i satisfying in the limit of large dimension n the condition tr 2 n tr 2 here can be either a random or just a fixed given hermitian matrix ensembles of such a type with 0 emerge naturally when describing quantum scattering in systems with chaotic dynamics and serve to describe resonance statistics related models are used to mimic complex spectra of the dirac operator with chemical potential in the context of quantum chromodynamics ensembles of the second type arising naturally in scattering theory of discrete time systems are formed by n n matrices with complex entries such that i for 0 this coincides with the circular unitary ensemble and 0 i describes deviation from unitarity our result amounts to answering statistically the following old question given the singular values of a matrix describe the locus of its eigenvalues we systematically show that the obtained expressions for the correlation functions of complex eigenvalues describe a non trivial crossover from wigner dyson statistics of real unimodular eigenvalues typical of hermitian unitary matrices to ginibre statistics in the complex plane typical of ensembles with strong non hermiticity tr 2 tr 2 when n finally we discuss scarce results available on eigenvector statistics for weakly non hermitian random matrices

the theory of newton s method we review the most important theoretical results on newton s method concerning the convergence properties the error estimates the numerical stability and the computational complexity of the algorithm we deal with the convergence for smooth and nonsmooth equations underdetermined equations and equations with singular jacobians only those extensions of the newton method are investigated where a generalized derivative and or a generalized inverse is used

the theory of smale s point estimation and its applications abstract the main result of this paper is that we exact smale s point estimation theory i e without assuming k p z 1 p k z k k 2 being bounded by the point estimation convergence theorem of the ne method is set up by making use of the majorizing method the proof of the theorem is simple and precise while the required point estimation conditions are weaker than all those of known point estimation convergence theorems another result of this paper is an application of the above new theory to the durand kerner method we compare the point estimation conditions for the durand kerner method with other known point estimation conditions numerical results show that our results have evident advantages

distribution of eigenvalues in non hermitian anderson models we develop a theory which describes the behavior of eigenvalues of a class of one dimensional random non hermitian operators introduced recently by hatano and nelson we prove that the eigenvalues are distributed along a curve in the complex plane an equation for the curve is derived and the density of complex eigenvalues is found in terms of spectral characteristics of a reference hermitian disordered system the generic properties of the eigenvalue distribution are discussed

an efficient higher order family of root finders a one parameter family of iterative methods for the simultaneous approximation of simple complex zeros of a polynomial based on a cubically convergent hansen patrick s family is studied we show that the convergence of the basic family of the fourth order can be increased to five and six using newton s and halley s corrections respectively since these corrections use the already calculated values the computational efficiency of the accelerated methods is significantly increased further acceleration is achieved by applying the gauss seidel approach single step mode one of the most important problems in solving nonlinear equations the construction of initial conditions which provide both the guaranteed and fast convergence is considered for the proposed accelerated family these conditions are computationally verifiable they depend only on the polynomial coefficients its degree and initial approximations which is of practical importance some modifications of the considered family providing the computation of multiple zeros of polynomials and simple zeros of a wide class of analytic functions are also studied numerical examples demonstrate the convergence properties of the presented family of root finding methods

almost hermitian random matrices crossover from wigner dyson to ginibre eigenvalue statistics by using the method of orthogonal polynomials we analyze the statistical properties of complex eigenvalues of random matrices describing a crossover from hermitian matrices characterized by the wigner dyson statistics of real eigenvalues to strongly non hermitian ones whose complex eigenvalues were studied by ginibre two point statistical measures as e g spectral form factor number variance and small distance behavior of the nearest neighbor distance distribution p s are studied in more detail in particular we found that the latter function may exhibit unusual behavior p s ensuremath propto s 5 2 for some parameter values

non hermitian random matrix models free random variable approach using the standard concepts of free random variables we show that for a large class of non hermitian random matrix models the support of the eigenvalue distribution follows from their hermitian analogs using a conformal transformation we also extend the concepts of free random variables to the class of non hermitian matrices and apply them to the models discussed by ginibre girko elliptic ensemble j ginibre j math phys bold 6 1440 1965 v l girko ital spectral theory of random matrices in russian nauka moscow 1988 and mahaux weidenm umlt u ller chaotic resonance scattering c mahaux and h a weidenm umlt u ller ital shell model approach to nuclear reactions north holland amsterdam 1969 copyright ital 1997 ital the american physical society

on statistics of bi orthogonal eigenvectors in real and complex ginibre ensembles combining partial schur decomposition with supersymmetry we suggest a method of studying the joint probability density jpd of an eigenvalue and the associated non orthogonality overlap factor also known as the eigenvalue condition number of the left and right eigenvectors for non selfadjoint gaussian random matrices of size n times n first we derive the general finite n expression for the jpd of a real eigenvalue lambda and the associated non orthogonality factor in the real ginibre ensemble and then analyze its bulk and edge scaling limits the ensuing distribution is maximally heavy tailed so that all integer moments beyond normalization are divergent a similar calculation for a complex eigenvalue z and the associated non orthogonality factor in the complex ginibre ensemble is presented as well and yields a distribution with the finite first moment its bulk scaling limit yields a distribution whose first moment reproduces the well known result of chalker and mehlig phys rev lett 81 16 3367 3370 1998 and we provide the edge scaling distribution for this case as well our method involves evaluating the ensemble average of products and ratios of integer and half integer powers of characteristic polynomials for ginibre matrices which we perform in the framework of a supersymmetry approach our paper complements recent studies by bourgade and dubach the distribution of overlaps between eigenvectors of ginibre matrices 2018 arxiv 1801 01219

expanding the applicability of newton s method using smale s α theory we present a tighter convergence analysis than earlier studies such as in cianciaruso 2007 guo 2007 shen and li 2010 smale 1986 1987 wang and zhao 1995 wang 1999 wang and han 1990 of newton s method using smale s a theory by introducing the notion of the center c 0 condition in particular in the semilocal convergence case we show that if the center c 0 condition is smaller than the c condition then the new majorizing sequence is tighter than the old majorizing sequence the new convergence criteria are weaker than the older convergence criteria furthermore in the local convergence case we obtain a larger radius of convergence and tighter error estimates on the distances involved these improvements are obtained under the same computational cost numerical examples and applications are also provided in this study to show that the older results cannot apply but the new results apply to solve equations

on the guaranteed convergence of the square root iteration method the construction of initial conditions which guarantee the convergence of the applied iterative method is one of the most important problems in solving nonlinear equations which attracted the great attention for many years in this paper we give a precise convergent analysis of the ostrowski like method of the fourth order for the simultaneous determination of polynomial zeros using a procedure based on smale s point estimation theory and some recent results related to the localization of complex polynomial zeros we state initial conditions which enable both the guaranteed and fast convergence of this method these conditions are computationally verifiable since they depend only on polynomial coefficients its degree and initial approximations which is of practical importance

a posteriori error bound methods for the inclusion of polynomial zeros using carstensen s results from 1991 we state a theorem concerning the localization of polynomial zeros and derive two a posteriori error bound methods with the convergence order 3 and 4 these methods possess useful property of inclusion methods to produce disks containing all simple zeros of a polynomial we establish computationally verifiable initial conditions that guarantee the convergence of these methods some computational aspects and the possibility of implementation on parallel computers are considered including two numerical examples a comparison of a posteriori error bound methods with the corresponding circular interval methods regarding the computational costs and sizes of produced inclusion disks were given

non hermitian euclidean random matrix theory we develop a theory for the eigenvalue density of arbitrary non hermitian euclidean matrices closed equations for the resolvent and the eigenvector correlator are derived the theory is applied to the random green s matrix relevant to wave propagation in an ensemble of pointlike scattering centers this opens a new perspective in the study of wave diffusion anderson localization and random lasing

a new fourth order family of simultaneous methods for finding polynomial zeros using a suitable fixed point relation with a complex parameter a new one parameter family of simultaneous methods of the fourth order for finding complex zeros of a polynomial is derived in ordinary complex arithmetic convergence analysis of the presented family is performed under computationally verifiable initial conditions which depend only on polynomial coefficients and initial approximations further improvements of the proposed family of methods are discussed and a modification for finding multiple zeros is presented some numerical results for various values of the parameter are given

the convergence of a family of parallelzero finding methods abstract one of the crucial problems in solving polynomial equations is to state some kind of quantitative conditions that provide both safe and fast convergence in the spirit of smale s point estimation theory introduced in 1981 we construct in this paper computationally verifiable initial conditions which guarantee the convergence of the recently proposed one parameter family of methods of the fourth order for the simultaneous finding polynomial zeros these conditions are computationally verifiable since they depend only on initial approximations polynomial coefficients and polynomial degree which is of significant practical importance

point estimation of a family of simultaneous zero finding methods abstract the construction of initial conditions which provide the safe convergence of iterative processes is one of the most important problems in solving equations in this paper we give initial conditions for a one parameter family of iterative methods for the simultaneous determination of all simple zeros of a polynomial zeros this family has the order of convergence four and generates some new methods with good convergence properties the presented initial conditions are of practical importance since depend only of available data coefficients of a polynomial and initial approximations to the zeros

squared eigenvalue condition numbers and eigenvector correlations from the single ring theorem we extend the so called single ring theorem feinberg and zee 1997 nucl phys b 504 579 also known as the haagerup larsen theorem haagerup and larsen 2000 j funct anal 176 331 we do this by showing that in the limit when the size of the matrix goes to infinity a particular correlator between left and right eigenvectors of the relevant non hermitian matrix x being the spectral density weighted by the squared eigenvalue condition number is given by a simple formula involving only the radial spectral cumulative distribution function of x we show that this object allows the calculation of the conditional expectation of the squared eigenvalue condition number we give examples and provide a cross check of the analytic prediction by the large scale numerics

integrable structure of ginibre s ensemble of real random matrices and a pfaffian integration theorem in the recent publication e kanzieper and g akemann in phys rev lett 95 230201 2005 an exact solution was reported for the probability pn k to find exactly k real eigenvalues in the spectrum of an n n real asymmetric matrix drawn at random from ginibre s orthogonal ensemble ginoe in the present paper we offer a detailed derivation of the above result by concentrating on the proof of the pfaffian integration theorem the key ingredient of our analysis of the statistics of real eigenvalues in the ginoe we also initiate a study of the correlations of complex eigenvalues and derive a formula for the joint probability density function of all complex eigenvalues of a ginoe matrix restricted to have exactly k real eigenvalues in the particular case of k 0 all correlation functions of complex eigenvalues are determined

a note on mixed matrix moments for the complex ginibre ensemble we consider the mixed matrix moments for the complex ginibre ensemble these are well known we consider the relation to the expected overlap functions of chalker and mehlig this leads to new asymptotic problems for the overlap we obtain some results but we also state some remaining open problems

weak convergence conditions for the newton s method in banach space using general majorizing sequences using the majorizing principle we develop sufficient convergence conditions for the semilocal convergence of newton s method in a banach space setting the involved majorizing sequences are very general and as special cases include most relevant sequences from the published literature our convergence conditions can be weaker and our error analysis can be finer than before

a symplectic method for approximating all the eigenvalues of a hamiltonian matrix a fast method for computing all the eigenvalues of a hamiltonian matrix m is given the method relies on orthogonal symplectic similarity transformations which preserve structure and have desirable numerical properties the algorithm is about four times faster than the standard q r algorithm the computed eigenvalues are shown to be the exact eigenvalues of a matrix m e where vert e vert depends on the square root of the machine precision the accuracy of a computed eigenvalue depends on its condition and its magnitude larger eigenvalues typically being more accurate

extended plefka expansion for stochastic dynamics we propose an extension of the plefka expansion which is well known for the dynamics of discrete spins to stochastic differential equations with continuous degrees of freedom and exhibiting generic nonlinearities the scenario is sufficiently general to allow application to e g biochemical networks involved in metabolism and regulation the main feature of our approach is to constrain in the plefka expansion not just first moments akin to magnetizations but also second moments specifically two time correlations and responses for each degree of freedom the end result is an effective equation of motion for each single degree of freedom where couplings to other variables appear as a self coupling to the past i e memory term and a coloured noise this constitutes a new mean field approximation that should become exact in the thermodynamic limit of a large network for suitably long ranged couplings for the analytically tractable case of linear dynamics we establish this exactness explicitly by appeal to spectral methods of random matrix theory for gaussian couplings with arbitrary degree of symmetry

delocalization of eigenvectors of random matrices with independent entries author s rudelson mark vershynin roman abstract we prove that an n by n random matrix g with independent entries is completely delocalized suppose the entries of g have zero means variances uniformly bounded below and a uniform tail decay of exponential type then with high probability all unit eigenvectors of g have all coordinates of magnitude o n 1 2 modulo logarithmic corrections this comes a consequence of a new geometric approach to delocalization for random matrices

eigenvector statistics of the product of ginibre matrices we develop a method to calculate left right eigenvector correlations of the product of m independent n n complex ginibre matrices for illustration we present explicit analytical results for the vector overlap for a couple of examples for small m and n we conjecture that the integrated overlap between left and right eigenvectors is given by the formula o 1 m 2 n 1 and support this conjecture by analytical and numerical calculations we derive an analytical expression for the limiting correlation density as n for the product of ginibre matrices as well as for the product of elliptic matrices in the latter case we find that the correlation function is independent of the eccentricities of the elliptic laws

on the convergence of wang zheng s method the choice of initial conditions ensuring safe convergence of the implemented iterative method is one of the most important problems in solving polynomial equations these conditions should depend only on the coefficients of a given polynomial p and initial approximations to the zeros of p in this paper we state initial conditions with the described properties for the wang zheng method for the simultaneous approximation of all zeros of p the safe convergence and the fourth order convergence of this method are proved