halley s method for the matrix sector function the matrix n sector function is a generalization of the matrix sign function it can be used to determine the number of eigenvalues of a matrix in a specific sector of the complex plane and to extract the eigenpairs belonging to this sector without explicitly computing the eigenvalues it is known that newton s method which can be used for computing the matrix sign function is not globally convergent for the matrix sector function the only existing algorithm for computing the matrix sector function is based on the continued fraction expansion approximation to the principal nth root of an arbitrary complex matrix in this paper we introduce a new algorithm based on halley s generalized iteration formula for solving nonlinear equations it is shown that the iteration has good error propagation properties and high accuracy finally we give two application examples and summarize the results of our numerical experiments comparing newton s the continued fraction and halley s method

the matrix sign function a survey of the matrix sign function is presented including some historical background definitions and properties approximation theory and computational methods and condition theory and estimation procedures applications to areas such as control theory eigendecompositions and roots of matrices are outlined and some new theoretical results are also given

image estimation using doubly stochastic gaussian random field models the two dimensional 2 d doubly stochastic gaussian dsg model was introduced by one of the authors to provide a complete model for spatial filters which adapt to the local structure in an image signal here we present the optimal estimator and 2 d fixed lag smoother for this dsg model extending earlier work of ackerson and fu as the optimal estimator has an exponentially growing state space we investigate suboptimal estimators using both a tree and a decision directed method experimental results are presented

invariant imbedding scattering processes and projectors in systems abstract this paper explores various methods of solving problems in modern control theory with particular emphasis on the optimal control problem the concepts of invariant imbedding scattering and projectors are given and then applied to the optimal control problem it is shown that the scattering process is the general theory from which invariant imbedding method can be derived the method of block diagonalization of matrices and the derivation of projectors is given for a general time varying system these two methods are applied to the optimal control problem where it is shown that the general algorithm follows directly from the procedures

fast and stable algorithms for computing the principal nth root of a complex matrix and the matrix sector function this paper presents rapidly convergent and more stable recursive algorithms for finding the principal nth root of a complex matrix and the associated matrix sector function the developed algorithms significantly improve the computational aspects of finding the principal nth root of a matrix and the matrix sector function thus the developed algorithms will enhance the capabilities of the existing computational algorithms such as the principal nth root algorithm the matrix sign algorithm and the matrix sector algorithm for developing applications to control system problems

determining continuous time state equations from discrete time state equations via the principal q th root method fast computational methods are developed for finding the equivalent continuous time state equations from discrete time state equations the computational methods utilize the direct truncation method the matrix continued fraction method and the geometric series method in conjunction with the principal q th root of the discrete time system matrix for quick determination of the approximants of a matrix logarithm function it is shown that the use of the principal q th root of a matrix enables us to enlarge the convergence region of the expansion of a matrix logarithm function and to improve the accuracy of the approximants of the matrix logarithm function

scale based diffusive image filtering preserving boundary sharpness and fine structures image acquisition techniques often suffer from low signal to noise ratio snr and or contrast to noise ratio cnr although many acquisition techniques are available to minimize these post acquisition filtering is a major off line image processing technique commonly used to improve the snr and cnr a major drawback of filtering is that it often diffuses blurs important structures along with noise here the authors introduce two scale based filtering methods that use local structure size or object scale information to arrest smoothing around fine structures and across even low gradient boundaries the first of these methods uses a weighted average over a scale dependent neighborhood while the other employs scale dependent diffusion conductance to perform filtering both methods adaptively modify the degree of filtering at any image location depending on local object scale object scale allows the authors to accurately use a restricted homogeneity parameter for filtering in regions with fine details and in the vicinity of boundaries while a generous parameter in the interiors of homogeneous regions qualitative experiments based on both phantoms and patient magnetic resonance images show significant improvements using the scale based methods over the extant anisotropic diffusive filtering method in preserving fine details and sharpness of object boundaries quantitative analyses utilizing 25 phantom images generated under a range of conditions of blurring noise and background variation confirm the superiority of the new scale based approaches

roots of real matrices abstract an algorithm for computing the roots of a matrix with real elements and the real part of the eigenvalues not zero will be described the algorithm is based on the sign matrix given in an earlier paper and the projectors of the matrix

on the matrix equation f x a abstract this paper deals with the matrix equation f x a where a c n n is a given matrix and is a complex holomorphic function defined on an open subset d of c it furnishes information concerning the set s of all solutions x c n n of x a the subset sp s of all solutions that are polynomials in a and the subset sd s of all diagonalizable solutions a necessary and sufficient condition for s to be nonempty is established relations among all solutions of x a lying in the same similarity class are established in the particular case where a r n n d is symmetric with respect to the real line r c and z f z for every z d a necessary and sufficient condition for s r n n to be nonempty is established necessary and sufficient conditions for sp and sd to be empty finite infinite and commutative are established explicit computations of all the elements of sp and sd are furnished and the numbers of elements of these two sets are determined necessary and sufficient conditions for the relations s sp s sd or sp sd to be satisfied are established necessary and sufficient conditions for the existence of normal hermitian skew hermitian positive definite and unitary solutions are established it is shown that the set p a of all polynomials p c x such that x p a is a solution of x a depends only on the minimal polynomial of a and not on the size of a an example is furnished of a matrix equation x a that has no solution in f n n for any field f but has a solution in r n n when r is a certain ring these results are applied to the particular equations xm a ex a and the fixed point equation x x

normalized cuts and image segmentation we propose a novel approach for solving the perceptual grouping problem in vision rather than focusing on local features and their consistencies in the image data our approach aims at extracting the global impression of an image we treat image segmentation as a graph partitioning problem and propose a novel global criterion the normalized cut for segmenting the graph the normalized cut criterion measures both the total dissimilarity between the different groups as well as the total similarity within the groups we show that an efficient computational technique based on a generalized eigenvalue problem can be used to optimize this criterion we applied this approach to segmenting static images as well as motion sequences and found the results to be very encouraging

separating eigenvalues using the matrix sign function abstract howland has used the matrix sign function to separate the eigenvalues of a given matrix the derived method required complex arithmetic even in the case of real matrices a modification of the method is presented that does not suffer from this drawback moreover it is shown that the problem may be decomposed into smaller subproblems which leads to considerable savings in computational work

hybrid dimensionality reduction method based on support vector machine and independent component analysis this paper presents a new hybrid dimensionality reduction method to seek projection through optimization of both structural risk supervised criterion and data independence unsupervised criterion classification accuracy is used as a metric to evaluate the performance of the method by minimizing the structural risk projection originated from the decision boundaries directly improves the classification performance from a supervised perspective from an unsupervised perspective projection can also be obtained based on maximum independence among features or attributes in data to indirectly achieve better classification accuracy over more intrinsic representation of the data orthogonality interrelates the two sets of projections such that minimum redundancy exists between the projections leading to more effective dimensionality reduction experimental results show that the proposed hybrid dimensionality reduction method that satisfies both criteria simultaneously provides higher classification performance especially for noisy data sets in relatively lower dimensional space than various existing methods

a fast method for computing the principal nth roots of complex matrices abstract based on the generalized continued fraction method for finding the nth roots of real numbers this paper presents a fast computation method for finding the principal nth roots of complex matrices computation algorithms with high convergence rates are developed and their global convergence properties are investigated from the viewpoint of systems theory

computation of the principal nth roots of complex matrices a new computational algorithm for finding the principal n th roots and associated roots of complex matrices is developed via a recursive solution of a block discrete state equation

block power method for computing solvents and spectral factors of matrix polynomials abstract this paper is concerned with the extension of the power method used for finding the largest eigenvalue and associated eigenvector of a matrix to its block from for computing the largest block eigenvalue and associated block eigenvector of a non symmetric matrix based on the developed block power method several algorithms are developed for solving the complete set of solvents and spectral factors of a matrix polynomial without prior knowledge of the latent roots of the matrix polynomial moreover when any right left solvent of a matrix polynomial is given the proposed method can be used to determine the corresponding left right solvent such that both right and left solvents have the same eigenspectra the matrix polynomial of interest must have distinct block solvents and a corresponding non singular polynomial matrix the established algorithms can be applied in the analysis and or design of systems described by high degree vector differential equations and or matrix fraction descriptions

a note on mode decoupling of linear time invariant systems using the generalized sign matrix it is shown how to decouple a selected set of modes of linear time invariant systems the decoupling procedure is based on a spectral decomposition of the system matrix using the generalized sign function this decomposition does not require knowledge of eigenvalues and eigenvectors the resulting projectors act as decoupling matrices for a spectral domain of the solution this decomposition is particularly useful in linear time invariant system equations the computational procedure is given within

a fast kalman filter for images degraded by both blur and noise in this paper a fast kalman filter is derived for the nearly optimal recursive restoration of images degraded in a deterministic way by blur and in a stochastic way by additive white noise straightforwardly implemented optimal restoration schemes for two dimensional images degraded by both blur and noise create dimensionality problems which in turn lead to large storage and computational requirements when the band toeplitz structure of the model matrices and of the distortion matrices in the matrix vector formulations of the original image and of the noisy blurred observation are approximated by circulant matrices these matrices can be diagonalized by means of the fft consequently a parallel set of n dynamical models suitable for the derivation of n low order vector kalman filters in the transform domain is obtained in this way the number of computations is reduced from the order of o n4 to that of o n 2 log 2 n for n n images

nonlinear component analysis as a kernel eigenvalue problem a new method for performing a nonlinear form of principal component analysis is proposed by the use of integral operator kernel functions one can efficiently compute principal components in high dimensional feature spaces related to input space by some nonlinear map for instance the space of all possible five pixel products in 16 16 images we give the derivation of the method and present experimental results on polynomial feature extraction for pattern recognition

kernel entropy component analysis we introduce kernel entropy component analysis kernel eca as a new method for data transformation and dimensionality reduction kernel eca reveals structure relating to the renyi entropy of the input space data set estimated via a kernel matrix using parzen windowing this is achieved by projections onto a subset of entropy preserving kernel principal component analysis kernel pca axes this subset does not need in general to correspond to the top eigenvalues of the kernel matrix in contrast to the dimensionality reduction using kernel pca we show that kernel eca may produce strikingly different transformed data sets compared to kernel pca with a distinct angle based structure a new spectral clustering algorithm utilizing this structure is developed with positive results furthermore kernel eca is shown to be an useful alternative for pattern denoising

edge adaptive kalman filtering for image restoration with ringing suppression the authors extend the two dimensional 2 d linear space invariant lsi reduced update kalman filter rukf to edge adaptive space invariant restoration of noisy and blurred images using a decision directed approach the edge adaptive rukf was motivated by the need to suppress the ringing artifacts caused by lsi processing the authors show that ringing artifacts can be suppressed to a great extent by using multiple image models that provide a better match to local edge orientation the maximum a posteriori probability decision procedure developed for model selection at each pixel can be used with other 2 d kalman filtering algorithms as well

extraction of mth roots in matrix rings over fields abstract a matrix a m n f f an arbitrary field with characteristic p not necessarily positive is an m th power in m n f if and only if each of its p x primary components is where p x runs through the irreducible factors of the minimal or characteristic polynomial of a this paper establishes necessary and sufficient criteria for determining when such a p x primary matrix is an m th power the criteria fall into three cases 1 p x x if e 1 e 2 is the sequence of exponents of p x which form the elementary divisors of a extended by adding infinitely many 0 terms the criterion states that for all i 1 e i 1 m 1 e im 0 or 1 2 p x x m not divisible by p if is a root of the separable core q x of p x and e f then the multiplicity of each elementary divisor of a must be representable as a sum of integers not necessarily distinct each of which is the degree of some irreducible factor over e of the polynomial x m 3 p x x m a power of p in this case the criterion combines the criteria in the first two cases if e 1 e 2 is the sequence of exponents of p x which form the elementary divisors of a extended by adding infinitely many 0 terms then for all i 1 e i 1 m 1 e im 0 or 1 and the multiplicity of each e i must be representable as a sum of integers not necessarily distinct each of which is the degree of some irreducible factor over e of the polynomial x m

spectral decomposition of a matrix using the generalized sign matrix an algorithm for spectral decomposition is presented which does not require knowledge of eigenvalues and eigenvectors a set of eigenprojectors are defined which covers the entire spectrum of a matrix and special attention is given to the projection on the zero eigenvalue some useful applications are discussed in the paper

newton s method for the matrix square root one approach to computing a square root of a matrix a is to apply newton s method to the quadratic matrix equation f x x2 a 0 two widely quoted matrix square root iterations obtained by rewriting this newton iteration are shown to have excellent mathematical convergence properties however by means of a perturbation analysis and supportive numerical examples it is shown that these simplified iterations are numerically unstable a further variant of newton s method for the matrix square root recently proposed in the literature is shown to be for practical purposes numerically stable

residual methods for the large scale matrix pth root and some related problems abstract the problem of finding the p th root of a matrix has received special attention in the last few years standard approaches for this problem include and combine some variations of newton s method which in turn involve matrix factorizations that in general are not suitable for large scale problems motivated by some recently developed low cost iterative schemes for nonlinear problems we consider and analyze specialized residual methods that only require a few matrix matrix products per iteration and hence are suitable for the large scale case as a by product we also discuss the advantages of residual methods for general nonlinear problems whose variables separate preliminary and encouraging numerical results are presented for computing p th roots of large scale symmetric and positive definite matrices for different values of p

computing real square roots of a real matrix bjorck and hammarling 1 describe a fast stable schur method for computing a square root x of a matrix a x2 a we present an extension of their method which enables real arithmetic to be used throughout when computing a real square root of a real matrix for a nonsingular real matrix a conditions are given for the existence of a real square root and for the existence of a real square root which is a polynomial in a the number of square roots of the latter type is determined the conditioning of matrix square roots is investigated and an algorithm is given for the computation of a well conditioned square root

a unified self stabilizing neural network algorithm for principal and minor components extraction recently many unified learning algorithms have been developed for principal component analysis and minor component analysis these unified algorithms can be used to extract principal components and if altered simply by the sign can also serve as a minor component extractor this is of practical significance in the implementations of algorithms this paper proposes a unified self stabilizing neural network learning algorithm for principal and minor components extraction and studies the stability of the proposed unified algorithm via the fixed point analysis method the proposed unified self stabilizing algorithm for principal and minor components extraction is extended for tracking the principal subspace ps and minor subspace ms the averaging differential equation and the energy function associated with the unified algorithm for tracking ps and ms are given it is shown that the averaging differential equation will globally asymptotically converge to an invariance set and the corresponding energy function exhibit a unique global minimum attained if and only if its state matrices span the ps or ms of the autocorrelation matrix of a vector data stream it is concluded that the proposed unified algorithm for tracking ps and ms can efficiently track an orthonormal basis of the ps or ms simulations are carried out to further illustrate the theoretical results achieved

a parallel algorithm for principal n th roots of matrices an iterative algorithm for computing the principal nth root of a positive definite matrix is presented the algorithm is based on the gauss legendre approximation of a definite integral we present a parallelization in which we use as many processors as the order of the approximation an analysis of the error introduced at each step of the iteration indicates that the algorithm converges more rapidly as the order of the approximation and thus the number of processors increases we describe the results of our implementation of an eight processor meiko cs 2 comparing the parallel algorithm to the fastest sequential algorithm which is the hoskins walton method

mean vector component analysis for visualization and clustering of nonnegative data mean vector component analysis mvca is introduced as a new method for visualization and clustering of nonnegative data the method is based on dimensionality reduction by preserving the squared length and implicitly also the direction of the mean vector of the original data the optimal mean vector preserving basis is obtained from the spectral decomposition of the inner product matrix and it is shown to capture clustering structure mvca corresponds to certain uncentered principal component analysis pca axes unlike traditional pca these axes are in general not corresponding to the top eigenvalues mvca is shown to produce different visualizations and sometimes considerably improved clustering results for nonnegative data compared with pca

data spectroscopy eigenspaces of convolution operators and clustering this paper focuses on obtaining clustering information about a distribution from its i i d samples we develop theoretical results to understand and use clustering information contained in the eigenvectors of data adjacency matrices based on a radial kernel function with a sufficiently fast tail decay in particular we providepopulation analyses to gain insights into which eigenvectors should be used and when the clustering information for the distribution can be recovered from the sample we learn that a fixed number of top eigenvectors might at the same time contain redundant clustering information and miss relevant clustering information we use this insight to design the data spectroscopic clustering daspec algorithm that utilizes properly selected eigenvectors to determine the number of clusters automatically and to group the data accordingly our findings extend the intuitions underlying existing spectral techniques such as spectral clustering and kernel principal components analysis and provide new understanding into their usability and modes of failure simulation studies and experiments on real world data are conducted to show the potential of our algorithm in particular daspec is found to handle unbalanced groups and recover clusters of different shapes better than the competing methods

the matrix sign decomposition and its relation to the polar decomposition the sign function of a square matrix was introduced by roberts in 1971 we show that it is useful to regard s sign a as being part of a matrix sign decomposition a sn where n a2 1 2 this decomposition leads to the new representation sign a a a2 1 2 most results for the matrix sign decomposition have a counterpart for the polar decomposition a uh and vice versa to illustrate this we derive best approximation properties of the factors u h and s determine bounds for a s and a u and describe integral formulas for s and u we also derive explicit expressions for the condition numbers of the factors s and n an important equation expresses the sign of a block 2 2 matrix involving a in terms of the polar factor u of a we apply this equation to a family of iterations for computing s by pandey kenney and laub to obtain a new family of iterations for computing u the iterations have some attractive properties including suitability for parallel computation

interpolation theory and λ matrices abstract this paper shows how interpolation theory can be used in the structural analysis of matrices which often occur in the description of the dynamics of multivariable systems using scalar interpolatory polynomials for the linear matrix case and interpolatory matrices for the general case a set of spectral projectors is derived the fundamental properties of the interpolatory polynomials and matrices are derived and used to develop a suitable form for the resolvent

on statistical identification of a class of linear space invariant image blurs using non minimum phase arma models an optimal statistical parameter estimation technique was previously presented by the authors ibid vol assp 34 p 963 72 for the identification of a class of unknown linear systems that can be modeled by a zero phase finite impulse response filter this technique is now extended to a more general case using a spectrally equivalent minimum phase semp representation of the system the original zero phase system parameters are computed from the semp model parameters

the algebraic theory of latent projectors in lambda matrices multivariable systems and controls are often formulated in terms of differential equations which give rise to lambda matrices of the form a l a email protected n a email protected n 1 a n the inverses of regular lambda matrices can be represented by the latent projectors or matrix residues that have very specific properties this paper describes the general theory of latent roots latent vectors and latent projectors and gives the relationships to eigenvalues eigenvectors and eigenprojectors of the companion form

local variance controlled forward and backward diffusion for image enhancement and noise reduction in order to improve signal to noise ratio snr and contrast to noise ratio this paper introduces a local variance controlled forward and backward lvcfab diffusion algorithm for edge enhancement and noise reduction in our algorithm an alternative fab diffusion algorithm is proposed the results for the alternative fab algorithm show better algorithm behavior than other existing diffusion fab approaches furthermore two distinct discontinuity measures and the alternative fab diffusion are incorporated into a lvcfab diffusion algorithm where the joint use of the two measures leads to a complementary effect for preserving edge features in digital images this lvc mechanism adaptively modifies the degree of diffusion at any image location and is dependent on both local gradient and inhomogeneity qualitative experiments based on general digital images and magnetic resonance images show significant improvements when the lvcfab diffusion algorithm is used versus the existing anisotropic diffusion and the previous fab diffusion algorithms for enhancing edge features and improving image contrast quantitative analyses based on peak snr confirm the superiority of the proposed lvcfab diffusion algorithm

approximation errors of online sparsification criteria many machine learning frameworks such as resource allocating networks kernel based methods gaussian processes and radial basis function networks require a sparsification scheme in order to address the online learning paradigm for this purpose several online sparsification criteria have been proposed to restrict the model definition on a subset of samples the most known criterion is the linear approximation criterion which discards any sample that can be well represented by the already contributing samples an operation with excessive computational complexity several computationally efficient sparsification criteria have been introduced in the literature such as the distance the coherence and the babel criteria in this paper we provide a framework that connects these sparsification criteria to the issue of approximating samples by deriving theoretical bounds on the approximation errors moreover we investigate the error of approximating any feature by proposing upper bounds on the approximation error for each of the aforementioned sparsification criteria two classes of features are described in detail the empirical mean and the principal axes in the kernel principal component analysis

regions of convergence of a pade family of iterations for the matrix sector function and the matrix pth root in this paper we prove a conjecture on a common region of a convergence of pade iterations for the matrix sector function for this purpose we show that all pade approximants to a special case of hypergeometric function have a power series expansion with positive coefficients using a sharpened version of schwarz s lemma we also demonstrate a better estimate of the convergence speed our results are also applicable to a family of rational iterations for computing the matrix pth root

an adaptive recursive 2 d filter for removal of gaussian noise in images a 2d recursive low pass filter with adaptive coefficients for restoring images degraded by gaussian noise is proposed some of the ideas developed are also submitted for nongaussian noise the adaptation is performed with respect to three local image features edges spots and flat regions for which detectors are developed by extending some existing methods it is demonstrated that the filter can easily be extended so that simultaneous noise removal and edge enhancement is possible a comparison with other approaches is made some examples illustrate the performance of the filter

detection localization and estimation of edges a method to detect locate and estimate edges in a one dimensional signal is presented it is inherently more accurate than all previous schemes as it explicitly models and corrects interaction between nearby edges the method is iterative with initial estimation of edges provided by the zero crossings of the signal convolved with laplacian of gaussian log filter the necessary computations necessitate knowledge of this convolved output only in a neighborhood around each zero crossing and in most cases could be performed locally by independent parallel processors results on one dimensional slices extracted from real images and on images which have been proposed independently in the row and column directions are shown an analysis of the method is provided including issues of complexity and convergence and directions of future research are outlined

adaptive nonlinear image restoration by a modified kalman filtering approach an adaptive nonlinear kalman type filter is presented for the restoration of two dimensional images degraded by general image formation system degradations and additive white noise a vector difference equation model is used to model the degradation process the object plane distribution function is partitioned into disjoint regions based on the amount of spatial activity in the image and difference equation models are used to characterize this nonstationary object plane distribution function features of the restoration filter include the ability to account for the response of the human visual system to additive noise in an image a two dimensional interpolation scheme to improve the estimates of the initial states in each region and a nearest neighbor algorithm to choose the previous state of vector for the state of pixel i j

a power method for computing square roots of complex matrices abstract in this paper higher order convergent methods for computing square roots of nonsingular complex matrices are derived these methods are globally convergent and are based on eigenvalue shifting and powering specifically it is shown for each positive integerr 2 a convergent method of orderrcan be developed these algorithms can be used to compute square roots of general nonsingular complex matrices such as computing square roots of matrices with negative eigenvalues

efficient online subspace learning with an indefinite kernel for visual tracking and recognition we propose an exact framework for online learning with a family of indefinite not positive kernels as we study the case of nonpositive kernels we first show how to extend kernel principal component analysis kpca from a reproducing kernel hilbert space to krein space we then formulate an incremental kpca in krein space that does not require the calculation of preimages and therefore is both efficient and exact our approach has been motivated by the application of visual tracking for which we wish to employ a robust gradient based kernel we use the proposed nonlinear appearance model learned online via kpca in krein space for visual tracking in many popular and difficult tracking scenarios we also show applications of our kernel framework for the problem of face recognition

laplacian eigenmaps for dimensionality reduction and data representation one of the central problems in machine learning and pattern recognition is to develop appropriate representations for complex data we consider the problem of constructing a representation for data lying on a low dimensional manifold embedded in a high dimensional space drawing on the correspondence between the graph laplacian the laplace beltrami operator on the manifold and the connections to the heat equation we propose a geometrically motivated algorithm for representing the high dimensional data the algorithm provides a computationally efficient approach to nonlinear dimensionality reduction that has locality preserving properties and a natural connection to clustering some potential applications and illustrative examples are discussed

determination of a matrix function using the divided difference method of newton and the interpolation technique of hermite computing a function f a of an n by n matrix a is a frequently occurring problem in control theory and other applications in this paper we introduce an effective approach for the determination of matrix function f a we propose a new technique which is based on the extension of newton divided difference and the interpolation technique of hermite and using the eigenvalues of the given matrix a the new algorithm is tested on several problems to show the efficiency of the presented method finally the application of this method in control theory is highlighted

kalman filtering in two dimensions further results the two dimensional reduced update kalman filter was recently introduced the corresponding scalar filtering equations were derived for the case of estimating a gaussian signal in white gaussian noise and were shown to constitute a general nonsymmetric half plane recursive filter this paper extends the method to the deconvolution problem of image restoration this paper also provides a more thorough treatment of the uniquely two dimensional boundary condition problems numerical and subjective examples are presented

computation of roots of real and complex matrices an algorithm is presented in this paper by which the rth root of real or complex matrices can be found without the computation of the eigenvalues and eigenvectors of the matrix all required computations are in the real domain the method is based on the newton raphson algorithm and is capable of finding roots even when the matrix is defective computing the root of a matrix from eigenvalues and eigenvectors would be the preferred method if these data were available

extraction of nth roots of 2 2 matrices abstract this paper is concerned with the determination of algebraic formulae giving all the solutions of the matrix equation x n a where n is a positive integer greater than 2 and a is a 2 2 matrix with real or complex elements if a is a 2 2 scalar matrix the equation x n a has infinitely many solutions and we obtain explicit formulae giving all the solutions if a is a non scalar 2 2 matrix the equation x n a has a finite number of solutions and we give a formula expressing all solutions in terms of a and the roots of a suitably defined n th degree polynomial in a single variable this leads to very simple formulae for all the solutions when a is either a singular matrix or a non singular matrix with two coincident eigenvalues similarly when n 3 or 4 we get explicit algebraic formulae for all the solutions we also determine the precise number of solutions in various cases

fast and stable algorithms for computing the principal square root of a complex matrix this note presents recursive algorithms that are rapidly convergent and more stable for finding the principal square root of a complex matrix also the developed algorithms are utilized to derive the fast and stable matrix sign algorithms which are useful in developing applications to control system problems

identification of image and blur parameters for the restoration of noncausal blurs an optimal statistical parameter estimation technique is presented for the identification of unknown image and blur model parameters the development leads to an autoregressive moving average arma model identification problem where the image model coefficients define the ar part and the blur parameters define the ma part conditional maximum likelihood estimates of the unknown parameters are derived both in the absence and in the presence of observation noise the proposed algorithms constitute a generalization of previous work on blur identification in that they are able to locate the zero loci of the blurred image spectrum on the entire z 1 z 2 plane simulation results as well as photographically blurred images processed with the proposed algorithms are shown as examples

two stage multiple kernel learning for supervised dimensionality reduction in supervised dimensionality reduction methods for pattern recognition tasks the information of the class labels is considered through the process of reducing the input dimensionality to improve the classification accuracy using nonlinear mappings for this purpose makes these models more appropriate for nonlinearly distributed data in this paper a new nonlinear supervised dimensionality reduction model is introduced the dimensionality reduction process in this model is performed through a multiple kernel learning paradigm in two stages in the first stage three suitable criteria for supervised dimensionality reduction containing fisher homoscedasticity and between class distance criteria are used to find the kernel weights with these weights a linear combination of several valid kernels is computed to make a new suitable kernel function in the second stage the kernel discriminant analysis method is employed for nonlinear supervised dimensionality reduction using the kernel computed in the first stage many experiments on a variety of real world datasets including handwritten digits images objects images and other datasets show that the proposed approach among a number of well known related techniques results in accurate and fast classifications a novel approach for non linear supervised dimensionality reduction is proposed the proposed method uses kernel discriminant analysis method with new kernels a multiple kernel learning mkl paradigm produces the new kernel function proper criteria for supervised dimensionality reduction are optimized in the mkl part the proposed method is successfully evaluated using many real world datasets

a parallelizable eigensolver for real diagonalizable matrices with real eigenvalues in this paper preliminary research results on a new algorithm for finding all the eigenvalues and eigenvectors of a real diagonalizable matrix with real eigenvalues are presented the basic mathematical theory behind this approach is reviewed and is followed by a discussion of the numerical considerations of the actual implementation the numerical algorithm has been tested on thousands of matrices on both a cray 2 and an ibm rs 6000 model 580 workstation the results of these tests are presented finally issues concerning the parallel implementation of the algorithm are discussed the algorithm s heavy reliance on matrix matrix multiplication coupled with the divide and conquer nature of this algorithm should yield a highly parallelizable algorithm

a computer aided method for solvents and spectral factors of matrix polynomials a new approach for determining the complete sets of solvents and spectral factors of a monic matrix polynomial is proposed a systematic method for determining the initial guess for the extended multidimensional newton raphson method is first proposed such that the eigenspectrum corresponding to each solvent of the matrix polynomial can be determined with the evaluated eigenspectra complete sets of solvents and spectral factors of a monic matrix polynomial can be obtained by utilizing the applications and the advantages of the principal nth root method the matrix sign function and the block power method the established algorithms can be applied in the analysis and or design of systems described by high degree vector differential equations and or matrix fractions

a newton squaring algorithm for computing the negative invariant subspace of a matrix by combining newton s method for the matrix sign function with a squaring procedure a basis for the negative invariant subspace of a matrix can be computed efficiently the algorithm presented is a variant of multiplication rich schemes for computing the matrix sign function such as the well known inversion free schulz method which requires two matrix multiplications per step however by avoiding a complete computation of the matrix sign and instead concentrating only on the negative invariant subspace the final newton steps can be replaced by steps which require only one matrix squaring each this efficiency is attained without sacrificing the quadratic convergence of newton s method

solving the algebraic riccati equation with the matrix sign function abstract this paper presents some improvements to the matrix sign function algorithm for the algebraic riccati equation a simple reorganization changes nonsymmetric matrix inversions into symmetric matrix inversions scaling accelerates convergence of the basic iteration and yields a new quadratic formula for certain 2 by 2 algebraic riccati equations numerical experience suggests the algorithm be supplemented with a refinement strategy similar to iterative refinement for systems of linear equations refinement also produces an error estimate the resulting procedure is numerically stable it compares favorably with current schur vector based algorithms

numerical approximation of the product of the square root of a matrix with a vector abstract given an n n symmetric positive definite matrix a and a vector c two numerical methods for approximating a 1 2 c are developed analyzed and computationally tested the first method applies a newton iteration to a specific nonlinear system to approximate a 1 2 c while the second method applies a step control method to numerically solve a specific initial value problem to approximate a 1 2 c assuming that a is first reduced to tridiagonal form the first method requires o n 2 operations per iteration while the second method requires o n operations per iteration in contrast numerical methods that first approximate a 1 2 and then compute a 1 2 c generally require o n 3 operations per iteration