stepsize restrictions for stability in the numerical solution of ordinary and partial differential equations abstract this paper deals with the stability analysis of one step methods for the numerical solution of initial value problems both stiff ordinary and partial differential equations are included the problem is considered how to restrict the stepsize in the methods in order that they behave stable we review and extend some recent results on this problem that are based on the use of stability regions in the complex plane we focus on differential equations that are essentially more general than the classical test equation u t u t with a complex constant further the emphasis is on stability of the methods with respect to the maximum norm

the sharpness of stability estimates corresponding to a general resolvent condition abstract in the stability analysis of numerical processes the problem arises of establishing moderate upper bounds for the norm of the n th power of given s s matrices recently new upper bounds were derived under generalized versions of the resolvent condition occurring in the kreiss matrix theorem in the present paper the question is answered of whether these new bounds are sharp with respect to their behaviour as n and s

a generalization of joint diagonalization criteria for source separation in the field of blind source separation joint diagonalization based approaches constitute an important framework leading to useful algorithms such as the popular joint approximate diagonalization of eigenmatrices jade and simultaneous third order tensor diagonalization stotd algorithms however they are often restricted to the case of cumulants of order four in this paper we extend the results leading to jade and stotd to cumulants of any order greater than or equal to three by exhibiting a new family of contrast functions that constitutes then a unified framework for the above known results this also leads us to generalize some links between contrast functions and joint diagonalization criteria on which these algorithms are based in turn one contrast of the new family allows us to show that a function previously proposed as a separation criterion is also a contrast moreover for the two generalized jade and stotd contrasts the analytical optimal solution in the case of two sources is derived and shown to keep the same simple expression whatever the cumulant order finally some computer simulations illustrate the potential advantage one can take by considering statistics of different orders for the joint diagonalization of cumulant matrices

weak ill posedness of spatial discretizations of absorbing boundary conditions for schrodinger type equations when we wish to solve numerically a differential problem defined on an infinite domain it is necessary to consider a finite subdomain and to use artificial boundary conditions in such a way that the solutions in the finite subdomain approximate the original solution these boundary conditions are called absorbing when small reflections to the interior domain are allowed in this paper we develop a general class of absorbing boundary conditions for schrodinger type equations by using rational approximations to the transparent boundary conditions with this approach previous absorbing boundary conditions in the literature are included in this class we use the method of lines for the discretization of the initial boundary value problems obtained this way we show that the ordinary differential systems that arise after the spatial discretization are weakly ill posed explaining a previous conjecture of fevens and jiang the time discretization is carried out with a stable runge kutta methods where the high order ones may be used to compensate for the possible troubles present in the problems semidiscretized in space

a generalization of the numerical range of a matrix abstract a new generalization of the classical numerical range of a matrix is introduced various properties and basic characterizations of this generalized numerical range are established one of these characterizations is used in proving a resolvent inequality for a discretized version of the differential operator d 2 dx 2

separation of independent sources from correlated inputs the characterization of independent stationary stochastic components sources is generally achieved by using the spectral matrix of partially correlated measurements which are linearly related to the components of interest in the general case where no assumptions are made concerning the way the sources are mixed on the measurements the spectral matrix is not able to extract the true sources while spectral analysis only uses second order properties of independent stochastic sources a procedure based on higher order analysis fourth order cross cumulants is developed this approach leads to a complete identification of the sources

a comparison of computational color constancy algorithms ii experiments with image data for pt i see ibid vol 11 no 9 p 972 84 2002 we test a number of the leading computational color constancy algorithms using a comprehensive set of images these were of 33 different scenes under 11 different sources representative of common illumination conditions the algorithms studied include two gray world methods a version of the retinex method several variants of forsyth s 1990 gamut mapping method cardei et al s 2000 neural net method and finlayson et al s color by correlation method finlayson et al 1997 2001 hubel and finlayson 2000 we discuss a number of issues in applying color constancy ideas to image data and study in depth the effect of different preprocessing strategies we compare the performance of the algorithms on image data with their performance on synthesized data all data used for this study are available online at http www cs sfu ca spl sim color data and implementations for most of the algorithms are also available http www cs sfu ca spl sim color code experiments with synthesized data part one of this paper suggested that the methods which emphasize the use of the input data statistics specifically color by correlation and the neural net algorithm are potentially the most effective at estimating the chromaticity of the scene illuminant unfortunately we were unable to realize comparable performance on real images here exploiting pixel intensity proved to be more beneficial than exploiting the details of image chromaticity statistics and the three dimensional 3 d gamut mapping algorithms gave the best performance

a comparison of computational color constancy algorithms i methodology and experiments with synthesized data we introduce a context for testing computational color constancy specify our approach to the implementation of a number of the leading algorithms and report the results of three experiments using synthesized data experiments using synthesized data are important because the ground truth is known possible confounds due to camera characterization and pre processing are absent and various factors affecting color constancy can be efficiently investigated because they can be manipulated individually and precisely the algorithms chosen for close study include two gray world methods a limiting case of a version of the retinex method a number of variants of forsyth s 1990 gamut mapping method cardei et al s 2000 neural net method and finlayson et al s color by correlation method finlayson et al 1997 2001 hubel and finlayson 2000 we investigate the ability of these algorithms to make estimates of three different color constancy quantities the chromaticity of the scene illuminant the overall magnitude of that illuminant and a corrected illumination invariant image we consider algorithm performance as a function of the number of surfaces in scenes generated from reflectance spectra the relative effect on the algorithms of added specularities and the effect of subsequent clipping of the data all data is available on line at http www cs sfu ca spl sim color data and implementations for most of the algorithms are also available http www cs sfu ca spl sim color code

on the 2d bioheat equation with convective boundary conditions and its numerical realization via a highly accurate approach abstract a series solution for a 2d bioheat pennes conduction model with convective boundary conditions is established by using the classical fourier method to validate and compare results a numerical method for constructing highly accurate numerical solutions is also derived and numerically illustrated both the series solution and the proposed numerical method open interesting possibilities of application in inverse perfusion coefficient identification problems

an instability phenomenon in spectral methods the eigenvalues of chebyshev and legendre spectral differentiation matrices which determine the allowable time step in an explicit time integration are extraordinarily sensitive to rounding errors and other perturbations on a grid of n points per space dimension machine rounding leads to errors in the eigenvalues of size o n 2 this phenomenon may lead to inconsistency between predicted and observed time step restrictions one consequence of it is that spectral differentiation by interpolation in legendre points which has a favorable o n 1 time step restriction for the model problem u t u x in theory is subject to an o n 2 restriction in practice the same effect occurs with chebyshev points for the model problem u t xu x another consequence is that a spectral calculation with a fixed time step may be stable in double precision but unstable in single precision we know of no other examples in numerical computation of this kind of precision dependent stability

several concepts to investigate strongly nonnormal eigenvalue problems eigenvalue analysis plays an important role in understanding physical phenomena however if one deals with strongly nonnormal matrices or operators the eigenvalues alone may not tell the full story a well known tool which can be useful to get more insight in the reliability or sensitivity of eigenvalues is varepsilon pseudospectra apart from varepsilon pseudospectra we consider other tools which might help to learn more about the eigenvalue problem viz condition numbers of the eigenvalues condition numbers of sets of eigenvectors and angles between invariant subspaces all these concepts will be studied and compared for both standard and generalized eigenvalue problems the tools can be used to analyze large eigenvalue problems we apply the different concepts to a generalized eigenvalue problem obtained from magnetohydrodynamics in this problem one is interested in an interior part of the spectrum called the alfven spectrum

stable spectral collocation solutions to a class of benjamin bona mahony initial value problems we are concerned with stable spectral collocation solutions to non periodic benjamin bona mahony bbm modified bbm and benjamin bona mahony burgers bbm b initial value problems on the real axis the spectral collocation is based alternatively on the scaled hermite and sinc functions in order to march in time we use several one step and linear multistep finite difference schemes such that the method of lines mol involved is stable in sense of lax the method based on hermite functions ensures the correct behavior of the solutions at large spatial distances and in long time periods in order to prove the stability we use the pseudospectra of the linearized spatial discretization operators the extent at which the energy integral of bbm model is conserved over time is analyzed for hermite collocation along with various finite difference schemes this analysis has been fairly useful in optimizing the scaling parameter the effectiveness of our approach has been confirmed by some numerical experiments

stability of rational multistep approximations of holomorphic semigroups in this paper we prove the stability of semidiscretizations in time of holomorphic semigroups in banach spaces by means of a stable rational multistep methods no assumptions on the method other than a stability are required our result is applicable in the maximum norm analysis of parabolic problems

bounding partial sums of fourier series in weighted l 2 norms with applications to matrix analysis for integrable functions f defined on the interval we denote the partial sums of the corresponding fourier series by sn f n 0 1 2 in this paper we deal with the problem of bounding supn sn where denotes an operator norm induced by a weighted l2 norm for functions f on for weight functions w belonging to a class introduced by helson and szego ann mat pura appl 51 1960 107 we present explicit upper bounds for supn sn in terms of w the authors were led to the problem of deriving explicit upper bounds for supn sn by the need for such bounds in an analysis related to the kreiss matrix theorem a famous result in the fields of linear algebra and numerical analysis accordingly the present paper highlights the relevance of bounds on supn sn to these fields

a novel algorithm for color constancy color constancy is the skill by which it is possible to tell the color of an object even under a colored light i interpret the color of an object as its color under a fixed canonical light rather than as a surface reflectance function this leads to an analysis that shows two distinct sets of circumstances under which color constancy is possible in this framework color constancy requires estimating the illuminant under which the image was taken the estimate is then used to choose one of a set of linear maps which is applied to the image to yield a color descriptor at each point this set of maps is computed in advance the illuminant can be estimated using image measurements alone because given a number of weak assumptions detailed in the text the color of the illuminant is constrained by the colors observed in the image this constraint arises from the fact that surfaces can reflect no more light than is cast on them for example if one observes a patch that excites the red receptor strongly the illuminant cannot have been deep blue two algorithms are possible using this constraint corresponding to different assumptions about the world the first algorithm crule will work for any surface reflectance crule corresponds to a form of coefficient rule but obtains the coefficients by using constraints on illuminant color the set of illuminants for which crule will be successful depends strongly on the choice of photoreceptors for narrowband photoreceptors crule will work in an unrestricted world the second algorithm mwext requires that both surface reflectances and illuminants be chosen from finite dimensional spaces but under these restrictive conditions it can recover a large number of parameters in the illuminant and is not an attractive model of human color constancy crule has been tested on real images of mondriaans and works well i show results for crule and for the retinex algorithm of land land 1971 land 1983 land 1985 operating on a number of real images the experimental work shows that for good constancy a color constancy system will need to adjust the gain of the receptors it employs in a fashion analagous to adaptation in humans

waveform preserving blind estimation of multiple independent sources the problem of blind estimation of source signals is to estimate the source signals without knowing the characteristics of the transmission channel it is shown that the minimum variance unbiased estimates can be obtained if and only if the transmission channel can be identified blindly it is shown that the channel can be blindly identified if and only if there is not more than one gaussian source this condition suggests that waveform preserving blind estimation can be achieved over a wide range of signal processing applications including those cases in which the source signals have identical nongaussian distributions the constructive proof of the necessary and sufficient condition serves as a foundation for the development of waveform preserving blind signal estimation algorithms examples are presented to demonstrate the applications of the theoretical results

on the use of stability regions in the numerical analysis of initial value problems this paper deals with the stability analysis of one step methods in the numerical solution of initial boundary value problems for linear ordinary and partial differential equations restrictions on the stepsize are derived which guarantee the rate of error growth in these methods to be of moderate size these restrictions are related to the stability region of the method and to numerical ranges of matrices stemming from the differential equation under consideration the errors in the one step methods are measured in arbitrary norms not necessarily generated by an inner product the theory is illustrated in the numerical solution of the heat equation and some other differential equations where the error growth is measured in the maximum norm

the equivalence of l2 stability the resolvent condition and strict h stability abstract the kreiss matrix theorem asserts that a family of n n matrices is l 2 stable if and only if either a resolvent condition r or a hermitian norm condition h is satisfied we give a direct considerably shorter proof of the power boundedness of an n n matrix satisfying r sharpening former results by showing that power boundedness depends at most linearly on the dimension n we also show that l 2 stability is characterized by an h condition employing a general h numerical radius instead of the usual h norm thus generalizing a sufficient stability criterion due to lax and wendroff

the stability of difference formulas for delay differential equations a new simple geometric technique is presented for analyzing the stability of difference formulas for the model delay differential equation y t py t qy t 8 where p and q are complex constants and the delay 8 is a positive constant the technique is based on the argument principle and directly relates the region of absolute stability for ordinary differential equations corresponding to the py t term with the region corresponding to the delay term qy t 8 a sufficient condition for stability is that these regions be disjoint the technique is used to show that for each a stable a a stable or stiffly stable linear multistep formula for ordinary differential equations there is a correspond ing linear multistep formula for delay differential equations with analogous stability properties the analogy does not extend however to a stable one step formulas

stepsize restrictions for stability of one step methods in the numerical solution of initial value problems this paper deals with the analysis of general one step methods for the numerical solution of initial boundary value problems for stiff ordinary and partial differential equations restrictions on the stepsize are derived that are necessary and sufficient for the rate of error growth in these methods to be of moderate size these restrictions are related to disks contained in the stability region of the method and the errors are measured with arbitrary norms not necessarily generated by an inner product the theory is illustrated in the numerical solution of a diffusion convection problem where the error growth is measured with the maximum norm

nonsymmetrical contrasts for sources separation in this paper the problem of the blind separation of independent sources is considered our approach relies on high order inverse criteria after generalizing the definition of classical contrast functions we exhibit a wide class of generally nonsymmetrical functions that will be called generalized contrasts and whose maximization is proved to be a sufficient condition for source separation we also establish a connection with cumulant matching showing an equivalence between the two approaches then in the case of two sources a statistical study of the estimated parameter based on one of these new contrasts is presented finally computer simulations illustrate the results and demonstrate all the interest we can find in considering a nonsymmetrical contrast

eigenvalues and pseudo eigenvalues of toeplitz matrices the eigenvalues of a nonhermitian toeplitz matrix a are usually highly sensitive to perturbations having condition numbers that increase exponentially with the dimension n an equivalent statement is that the resolvent zz a of a toeplitz matrix may be much larger in norm than the eigenvalues alone would suggest exponentially large as a function of n even when z is far from the spectrum because of these facts the meaningfulness of the eigenvalues of nonhermitian toeplitz matrices for any but the most theoretical purposes should be considered suspect in many applications it is more meaningful to investigate the e pseudo eigenvalues the complex numbers z with ll zz a 11 l this paper analyzes the pseudospectra of toeplitz matrices and in particular relates them to the symbols of the matrices and thereby to the spectra of the associated toeplitz and laurent operators our results are reasonably complete in the triangular case and preliminary in the cases of nontriangular toeplitz matrices block toeplitz matrices and toeplitz like matrices with smoothly varying coefficients computed examples of pseudospectra are presented throughout and applications in numerical analysis are mentioned

self adaptive source separation i convergence analysis of a direct linear network controlled by the herault jutten algorithm it is known that self adaptive separation of a linear mixture of non gaussian independent sources can be achieved with a feedback linear neural network that is adapted by the herault jutten 1991 algorithm yet realizability of the feedback requires implementation constraints an equivalent direct without feedback network is considered that is free of these constraints while the self adaptive rule is kept unchanged the separating states are shown to be equilibrium points their stability status is studied in the case of two sources then we show that the algorithm is convergent in the quasi quadratic mean sense toward a separating state for a small enough step size

cumulant based independence measures for linear mixtures this paper deals with independence measures for linear mixtures of mutually independent random variables such measures also known as contrasts constitute useful criteria in solving blind source separation problems by making use of the schur convexity properties we show that it is possible to define a wide ranging class of contrast functions based on the auto cumulants of the components of the random vector being considered among the most appealing characteristics of these new contrast functions is that they can be used to combine cumulants of different orders in a flexible way furthermore extensions of existing cross cumulant base contrasts are proposed finally some particularization of our approach to measures of decorrelation is considered a general characterization of these decorrelation measures using strictly schur convex functions is provided

stability of the method of lines it is well known that a necessary condition for the lax stability of the method of lines is that the eigenvalues of the spatial discretization operator scaled by the time stepk lie within a distanceo k of the stability region of the time integration formula ask 0 in this paper we show that a necessary and sufficient condition for stability except for an algebraic factor is that the pseudo eigenvalues of the same operator lie within a distanceo o k of the stability region ask 0 our results generalize those of an earlier paper by considering a runge kutta and other one step formulas b implicit as well as explicit linear multistep formulas c weighted norms d algebraic stability e finite and infinite time intervals and f stability regions with cusps in summary the theory presented in this paper amounts to a transplantation of the kreiss matrix theorem from the unit disk for simple power iterations to an arbitrary stability region for method of lines calculations

on a generalization of the resolvent condition in the kreiss matrix theorem this paper deals with a condition on the resolvent of s x s matrices a in one of the equivalent assertions of the kreiss matrix theorem the spectral norm of the resolvent of a at 4 must satisfy an inequality for all 4 lying outside the unit disk in c we consider a generalization in which domains different from the unit disk and more general norms are allowed under this generalized resolvent condition an upper bound is derived for the norms of the nth powers of s x s matrices b here b depends on a via a relation b 0 a where p is an arbitrary rational function the upper bound grows linearly with s 1 and is independent of n 1 this generalizes an upper bound occurring in the kreiss theorem where b a like the classical kreiss theorem the upper bound derived in this paper can be used in the stability analysis of numerical methods for solving differential equations

about the sharpness of the stability estimates in the kreiss matrix theorem one of the conditions in the kreiss matrix theorem involves the resolvent of the matrices a under consideration this so called resolvent condition is known to imply for all n 1 the upper bounds an ek n 1 and an ek n 1 here is the spectral norm k is the constant occurring in the resolvent condition and the order of a is equal to n 1 1 it is a long standing problem whether these upper bounds can be sharpened for all fixed k 1 to bounds in which the right hand members grow much slower than linearly with n 1 and with n 1 respectively in this paper it is shown that such a sharpening is impossible the following result is proved for each e 0 there are fixed values c 0 k 1 and a sequence of n 1 n 1 matrices an satisfyirig the resolvent condition such that an n c n 1 1 e c n 1 1 e for n n 1 2 3 the result proved in this paper is also relevant to matrices a whose e pseudospectra lie at a distance not exceeding ke from the unit disk for all e 0

rapid prolate pseudospectral differentiation and interpolation with the fast multipole method pseudospectral methods utilizing prolate spheroidal wave functions as basis functions have been shown to possess advantages over the conventional pseudospectral methods based on trigonometric and orthogonal polynomials however the spectral differentiation and interpolation steps of the prolate pseudospectral method involve matrix vector products which if evaluated directly entail o n2 memory requirement and computational complexity where n is the number of unknowns utilized for discretization and interpolation in this work we show that the fast multipole method fmm can be used to reduce the memory requirement and computational complexity of the prolate pseudospectral method to o n example simulation results demonstrate the speed and accuracy of the resulting fast prolate pseudospectral solver

resolvent conditions and bounds on the powers of matrices with relevance to numerical stability of initial value problems we deal with the problem of establishing upper bounds for the norm of the nth power of square matrices this problem is of central importance in the stability analysis of numerical methods for solving linear initial value problems for ordinary partial or delay differential equations a review is presented of upper bounds which were obtained in the literature under the resolvent condition occurring in the kreiss matrix theorem as well as under variants of that condition moreover we prove new bounds under resolvent conditions which generalize some of the reviewed ones the paper concludes by applying one of the new upper bounds in a stability analysis of the trapezoidal rule for delay differential equations

multichannel arma processes parametric modeling of multichannel time series is accomplished by using higher than second order statistics hos of the observed nongaussian data cumulants of vector processes are defined using a kronecker product formulation and consistency of their sample estimators is addressed identifiability results in connection with the hos based parameter estimation of causal and noncausal multivariate arma processes are established estimates of the parameters of causal arma models are obtained as the solution to a set of linear equations whereas those of noncausal arma models are obtained as the solution to a cumulant matching algorithm conventional approaches based on second order statistics can identify a multichannel system only to within post multiplication by a unimodular matrix hos based methods yield solutions that are unique to within post multiplication by an extended permutation matrix additionally the multiminimum phase assumption can be relaxed and the observations may be contaminated with colored gaussian noise frequency domain methods for nonparametric system identification are discussed briefly simulations results validating the multichannel parameter estimation algorithms are provided

eigenvalue stability of radial basis function discretizations for time dependent problems differentiation matrices obtained with infinitely smooth radial basis function rbf collocation methods have under many conditions eigenvalues with positive real part preventing the use of such methods for time dependent problems we explore this difficulty at theoretical and practical levels theoretically we prove that differentiation matrices for conditionally positive definite rbfs are stable for periodic domains we also show that for gaussian rbfs special node distributions can achieve stability in 1 d and tensor product nonperiodic domains as a more practical approach for bounded domains we consider differentiation matrices based on least squares rbf approximations and show that such schemes can lead to stable methods on less regular nodes by separating centers and nodes least squares techniques open the possibility of the separation of accuracy and stability characteristics

p stability properties of runge kutta methods for delay differential equations recently the author and others have studied a class of runge kutta methods for ddes these methods make use of certain constrained meshes which allow to get optimal order results in this paper we study their asymptotic stability properties using the test equation begin gathered y t ay t by t tau t 0 hfill y t g t for tau leqq t leqq 0 hfill end gathered wherea b 0 andg t is continuous and complex valued it is known thaty t 0 ast if b re a in particular we show that any astable one step collocation method for odes inherits the same property when it is applied to ddes with a constrained mesh i e it is p stable

chebyshev pseudospectral method for computing numerical solution of convection diffusion equation abstract a method for computing highly accurate numerical solutions of 1d convection diffusion equations is proposed in this method the equation is first discretized with respect to the spatial variable transforming the original problem into a set of ordinary differential equations and then the resulting system is integrated in time by the fourth order runge kutta method spatial discretization is done by using the chebyshev pseudospectral collocation method before describing the method we review a finite difference based method by salkuyeh d khojasteh salkuyeh on the finite difference approximation to the convection diffusion equation appl math comput 179 2006 79 86 and contrary to the proposal of the author we show that this method is not suitable for problems involving time dependent boundary conditions which calls for revision stability analysis based on pseudoeigenvalues to determine the maximum time step for the proposed method is also carried out superiority of the proposed method over a revised version of salkuyeh s method is verified by numerical examples

color by correlation a simple unifying framework for color constancy the paper considers the problem of illuminant estimation how given an image of a scene recorded under an unknown light we can recover an estimate of that light obtaining such an estimate is a central part of solving the color constancy problem thus the work presented will have applications in fields such as color based object recognition and digital photography rather than attempting to recover a single estimate of the illuminant we instead set out to recover a measure of the likelihood that each of a set of possible illuminants was the scene illuminant we begin by determining which image colors can occur and how these colors are distributed under each of a set of possible lights we discuss how for a given camera we can obtain this knowledge we then correlate this information with the colors in a particular image to obtain a measure of the likelihood that each of the possible lights was the scene illuminant finally we use this likelihood information to choose a single light as an estimate of the scene illuminant computation is expressed and performed in a generic correlation framework which we develop we propose a new probabilistic instantiation of this correlation framework and show that it delivers very good color constancy on both synthetic and real images we further show that the proposed framework is rich enough to allow many existing algorithms to be expressed within it the gray world and gamut mapping algorithms are presented in this framework and we also explore the relationship of these algorithms to other probabilistic and neural network approaches to color constancy

stability of a chebychev pseudospectral solution of the wave equation with absorbing boundaries abstract stability of the pseudospectral chebychev collocation solution of the two dimensional acoustic wave problem with absorbing boundary conditions is investigated the continuous one dimensional problem with one absorbing boundary and one dirichlet boundary has previously been shown to be far from normal consequently the spectrum of that problem says little about the stability behavior of the solution our analysis proves that the discrete formulation with dirichlet boundaries at all boundaries is near normal and hence the formulation with absorbing boundaries at all boundaries either for one dimensional or two dimensional wave propagation is not far from normal the near normality follows from the near normality of the second order derivative pseudospectral differential operator further the nearness to normality is independent of the boundary discretization stability limits on the timestep are however dependent on the boundary operator with an explicit euler method having the most restrictive condition the crank nicolson implementation has a stability limit the same as the dirichlet formulation furthermore in this case the restriction scales by 1 2 in moving from one dimension to two dimensions exactly as in the central finite difference approximation numerical results confirm the predicted values on allowable timesteps obtained from a spectral analysis for both chebychev and modified chebychev implementations we conclude that the spectrum of the evolution operator is informative for predicting the behavior of the numerical solution

a multilinear constraint on dichromatic planes for illumination estimation a new multilinear constraint on the color of the scene illuminant based on the dichromatic reflection model is proposed the formulation avoids the problem common to previous dichromatic methods of having to first identify pixels corresponding to the same surface material once pixels from two or more materials have been identified their corresponding dichromatic planes can be intersected to yield the illuminant color however it is not always easy to determine which pixels from an arbitrary region of an image belong to which dichromatic plane the image region may cover an area of the scene encompassing several different materials and hence pixels from several different dichromatic planes the new multilinear constraint accounts for this multiplicity of materials and provides a mechanism for choosing the most plausible illuminant from a finite set of candidate illuminants the performance of this new method is tested on a database of real images

stability estimates for families of matrices of nonuniformly bounded order this paper addresses the problem of establishing upper bounds for the norm of the nth power of square matrices the focus is on upper bounds that are valid for families of matrices of finite but nonuniformly bounded order such upper bounds are relevant to the stability analysis of numerical methods for solving differential equations in the famous kreiss matrix theorem a condition on the resolvent of matrices occurs which implies that the norm of the nth power does not grow faster than linearly with n in this paper a slightly stronger version of this resolvent condition is studied which is often satisfied in cases of practical interest we prove that this version implies growth at a rate that is essentially lower than in the case of the classical kreiss resolvent condition we consider also a condition on the iterated resolvent of matrices which is related to one of the assumptions occurring in the hille yosida theorem this condition is known to imply growth at a rate b n we prove that when a slightly stronger version of the iterated resolvent condition is in force there is growth at a rate that is at most n 1 1 with 0 q p i

color constancy from mutual reflection mutual reflection occurs when light reflected from one surface illuminates a second surface in this situation the color of one or both surfaces can be modified by a color bleeding effect in this article we examine how sensor values e g rgb values are modified in the mutual reflection region and show that a good approximation of the surface spectral reflectance function for each surface can be recovered by using the extra information from mutual reflection thus color constancy results from an examination of mutual reflection use is made of finite dimensional linear models for ambient illumination and for surface spectral reflectance if m and n are the number of basis functions required to model illumination and surface spectral reflectance respectively then we find that the number of different sensor classes p must satisfy the condition p 2 n m 3 if we use three basis functions to model illumination and three basis functions to model surface spectral reflectance then only three classes of sensors are required to carry out the algorithm results are presented showing a small increase in error over the error inherent in the underlying finite dimension models

pseudospectra of waveform relaxation operators abstract the performance of the waveform relaxation method for solving systems of odes depends largely on the choices that are made for splitting size of time window and preconditioning although it is known that superlinear convergence is obtained on finite time windows the convergence may be slow in the first few iterations we propose the use of pseudospectra to analyze the convergence ratio of the first few iterations when waveform relaxation is applied to linear systems of odes through pseudospectral radii we can examine the effect of preconditioning and overlapping on the rate of convergence we may also use this to estimate a suitable size of the time window numerical experiments performed on a system of odes arising from the discretization of an advection diffusion equation confirm the validity of the obtained estimates

on the uniform power boundedness of a family of matrices and the applications to one leg and linear multistep methods we study the difference equations obtained when a linear multistep method is applied to the scalar test equationdy dt y and constant stepsizeh lets be the region of the absolute stability of the method and letd be a closed subset ofs on the riemann sphere mathbb c it is shown that the solutions of these difference equations are bounded forn 0 uniformly for h d s is itself closed in mathbb c iff s is free of cusps the question is studed by means of contractivity analysis and a matrix theorem derived from the matrix theorem of kreiss

globally convergent blind source separation based on a multiuser kurtosis maximization criterion we consider the problem of recovering blindly i e without the use of training sequences a number of independent and identically distributed source user signals that are transmitted simultaneously through a linear instantaneous mixing channel the received signals are hence corrupted by interuser interference iui and we can model them as the outputs of a linear multiple input multiple output mimo memoryless system assuming the transmitted signals to be mutually independent i i d and to share the same non gaussian distribution a set of necessary and sufficient conditions for the perfect blind recovery up to scalar phase ambiguities of all the signals exists and involves the kurtosis as well as the covariance of the output signals we focus on a straightforward blind constrained criterion stemming from these conditions from this criterion we derive an adaptive algorithm for blind source separation which we call the multiuser kurtosis muk algorithm at each iteration the algorithm combines a stochastic gradient update and a gram schmidt orthogonalization procedure in order to satisfy the criterion s whiteness constraints a performance analysis of its stationary points reveals that the muk algorithm is free of any stable undesired local stationary points for any number of sources hence it is globally convergent to a setting that recovers them all

superefficiency in blind source separation blind source separation is the problem of extracting independent signals from their mixtures without knowing the mixing coefficients nor the probability distributions of source signals and may be applied to eeg and meg imaging of the brain it is already known that certain algorithms work well for the extraction of independent components the present paper is concerned with superefficiency of these based on the statistical and dynamical analysis in a statistical estimation using t examples the covariance of any two extracted independent signals converges to 0 of the order of 1 t on line dynamics shows that the covariance is of the order of spl eta when the learning rate spl eta is fixed to a small constant in contrast with the above general properties a surprising superefficiency holds in blind source separation under certain conditions where superefficiency implies that covariance decreases in the order of 1 t sup 2 or of spl eta sup 2 the paper uses the natural gradient learning algorithm and method of estimating functions to obtain superefficient procedures for both batch estimation and on line learning a standardized estimating function is introduced to this end superefficiency does not imply that the error variances of the extracted signals decrease in the order of 1 t sup 2 or spl eta sup 2 but implies that their covariances and independencies do