uniqueness of matrix square roots and an application abstract let a m n c let a denote the spectrum of a and f a the field of values of a it is shown that if a 0 then a has a unique square root b m n c with b in the open right complex half plane this result and lyapunov s theorem are then applied to prove that if f a 0 then a has a unique square root with positive definite hermitian part we will also answer affirmatively an open question about the existence of a real square root b m n r for a m n r with f a 0 where the field of values of b is in the open right half plane

an algorithm to compute averages on matrix lie groups averaging is a common way to alleviate errors and random fluctuations in measurements and to smooth out data averaging also provides a way to merge structured data in a smooth manner the present paper describes an algorithm to compute averages on matrix lie groups in particular we discuss the case of averaging over the special orthogonal group of matrices the unitary group of matrices and the group of symmetric positive definite matrices

variants of newton s method using fifth order quadrature formulas abstract some variants of newton s method are developed in this work in order to solve nonlinear equations depending on one or several variables based in rules of quadrature of fifth order we prove the third or fifth order of convergence of these methods for dimension one and the second or third order in several variables depending on the behaviour of the second derivative moreover different numeric tests confirm or improve theoretic results and allow us to compare these variants with newton s classical method

covariance subspace and intrinsic crame spl acute r rao bounds crame spl acute r rao bounds on estimation accuracy are established for estimation problems on arbitrary manifolds in which no set of intrinsic coordinates exists the frequently encountered examples of estimating either an unknown subspace or a covariance matrix are examined in detail the set of subspaces called the grassmann manifold and the set of covariance positive definite hermitian matrices have no fixed coordinate system associated with them and do not possess a vector space structure both of which are required for deriving classical crame spl acute r rao bounds intrinsic versions of the crame spl acute r rao bound on manifolds utilizing an arbitrary affine connection with arbitrary geodesics are derived for both biased and unbiased estimators in the example of covariance matrix estimation closed form expressions for both the intrinsic and flat bounds are derived and compared with the root mean square error rmse of the sample covariance matrix scm estimator for varying sample support k the accuracy bound on unbiased covariance matrix estimators is shown to be about 10 log 10 n k sup 1 2 db where n is the matrix order remarkably it is shown that from an intrinsic perspective the scm is a biased and inefficient estimator and that the bias term reveals the dependency of estimation accuracy on sample support observed in theory and practice the rmse of the standard method of estimating subspaces using the singular value decomposition svd is compared with the intrinsic subspace crame spl acute r rao bound derived in closed form by varying both the signal to noise ratio snr of the unknown p dimensional subspace and the sample support in the simplest case the crame spl acute r rao bound on subspace estimation accuracy is shown to be about p n p sup 1 2 1 2 snr sup 1 2 rad for p dimensional subspaces it is seen that the svd based method yields accuracies very close to the crame spl acute r rao bound establishing that the principal invariant subspace of a random sample provides an excellent estimator of an unknown subspace the analysis approach developed is directly applicable to many other estimation problems on manifolds encountered in signal processing and elsewhere such as estimating rotation matrices in computer vision and estimating subspace basis vectors in blind source separation

stability of an elastic rod buckling into a soft wall the conjugate point theory of the calculus of variations is extended to apply to the buckling of an elastic rod in an external field we show that the operator approach presented by manning mannin

ostrowski type methods for solving systems of nonlinear equations abstract four generalized algorithms builded up from ostrowski s method for solving systems of nonlinear equations are written and analyzed a development of an inverse first order divided difference operator for functions of several variables is presented as well as a direct computation of the local order of convergence for these variants of ostrowski s method furthermore a sequence that approximates the order of convergence is generated for the examples and it confirms in a numerical way that the order of the methods is well deduced

variants of newton s method for functions of several variables some variants of newton s method are developed in this work in order to solve systems of nonlinear equations based in trapezoidal and midpoint rules of quadrature we prove the quadratic convergence of one of these methods moreover different numeric tests confirm theoretic results and allow us to compare these variants with newton s classical method

tangent bundle maps on the grassmann manifold application to empirical arithmetic averaging the present paper elaborates on tangent bundle maps on the grassmann manifold with application to subspace arithmetic averaging in particular the present contribution elaborates on the work about retraction lifting maps devised for the stiefel manifold in the recently published paper t kaneko s fiori and t tanaka empirical arithmetic averaging over the compact stiefel manifold ieee trans signal process vol 61 no 4 pp 883 894 february 2013 and discusses the extension of such maps to the grassmann manifold tangent bundle maps are devised on the basis of the thin qr matrix decomposition the polar matrix decomposition and the exponential map also tangent bundle pseudo maps based on the matrix cayley transform are devised theoretical and numerical comparisons about the devised tangent bundle maps are performed in order to get an insight into their relative merits and demerits with special emphasis to their computational burden the averaging algorithm based on the thin qr decomposition maps stands out as it exhibits the best trade off between numerical precision and computational burden such algorithm is further compared with two grassmann averaging algorithms drawn from the scientific literature on an handwritten digits recognition data set the thin qr tangent bundle maps based algorithm exhibits again numerical features that make it preferable over such algorithms

kalman filtering with intermittent observations convergence for semi markov chains and an intrinsic performance measure this technical note shows that the stationary distribution for the covariance of kalman filtering with intermittent observations exists under mild conditions for a very general class of packet dropping models semi markov chain these results are proved using the geometric properties of riccati recursions with respect to a particular riemannian distance moreover the riemannian mean induced by that distance is always bounded therefore it can be used for characterizing the performance of the system for regimes where the moments of the covariance do not exist other interesting properties of that mean include the symmetry between covariance and information matrices averaging covariances or their inverse gives the same result and its interpretation in information geometry as the natural mean for the manifold of gaussian distributions

on newton type methods with cubic convergence recently there has been some progress on newton type methods with cubic convergence that do not require the computation of second derivatives weerakoon and fernando appl math lett 13 2000 87 derived the newton method and a cubically convergent variant by rectangular and trapezoidal approximations to newton s theorem while frontini and sormani j comput appl math 156 2003 345 140 2003 419 derived further cubically convergent variants by using different approximations to newton s theorem homeier j comput appl math 157 2003 227 169 2004 161 independently derived one of the latter variants and extended it to the multivariate case here we show that one can modify the werrakoon fernando approach by using newton s theorem for the inverse function and derive a new class of cubically convergent newton type methods

a new class of methods with higher order of convergence for solving systems of nonlinear equations by studying the commonness of some fifth order methods modified from third order ones for solving systems of nonlinear equations we propose a new class of three step methods of convergence order five by modifying a class of two step methods with cubic convergence next for a given method of order p 2 which uses the extended newton iteration yk xk af xk 1f xk as a predictor a new method of order p 2 is proposed for example we construct a class of m 2 step methods of convergence order 2m 3 by introducing only one evaluation of the function to each of the last m steps for any positive integer m in this paper we mainly focus on the class of fifth order methods when m 1 computational efficiency in the general form is considered several examples for numerical tests are given to show the asymptotic behavior and the computational efficiency of these higher order methods

a third order newton type method to solve systems of nonlinear equations in this paper we present a third order newton type method to solve systems of nonlinear equations in the first we present theoretical preliminaries of the method secondly we solve some systems of nonlinear equations all test problems show the third order convergence of our method

a theory for learning by weight flow on stiefel grassman manifold recently we introduced the concept of neural network learning on stiefel grassman manifold for multilayer perceptron like networks contributions of other authors have also appeared in the scientific literature about this topic this article presents a general theory for it and illustrates how existing theories may be explained within the general framework proposed here

matrix exponentiated gradient updates for on line learning and bregman projection we address the problem of learning a symmetric positive definite matrix the central issue is to design parameter updates that preserve positive definiteness our updates are motivated with the von neumann divergence rather than treating the most general case we focus on two key applications that exemplify our methods on line learning with a simple square loss and finding a symmetric positive definite matrix subject to linear constraints the updates generalize the exponentiated gradient eg update and adaboost respectively the parameter is now a symmetric positive definite matrix of trace one instead of a probability vector which in this context is a diagonal positive definite matrix with trace one the generalized updates use matrix logarithms and exponentials to preserve positive definiteness most importantly we show how the derivation and the analyses of the original eg update and adaboost generalize to the non diagonal case we apply the resulting matrix exponentiated gradient meg update and definiteboost to the problem of learning a kernel matrix from distance measurements

kolmogoroff nagumo mean over the affine symplectic group of matrices the present work shows that harris exponential mean log averaging rule over the space of optical transference matrices may be regarded as an instance of the kolmogoroff nagumo averaging rule over the affine symplectic group as such harris averaging rule may be generalized to a mean 1 rule that can be implemented by different maps the present work also shows that the involved maps may be computed in closed form by low degree polynomial expressions

some variant of newton s method with third order convergence we present a new modification of the newton s method which produces iterative methods with order of convergence three a general error analysis providing the higher order of convergence is given and the best efficiency in term of function evaluations of two of this new methods is provided

third order methods from quadrature formulae for solving systems of nonlinear equations we extend to p dimensional problems a modification of the newton method based on quadrature formulas of order at least one which produces iterative methods with order of convergence three a general error analysis providing the higher order of convergence is given these new methods may be more efficient then other third order methods as they do not require the use of the second order frechet derivative

some modifications of newton s method with fifth order convergence in this paper we present some new modifications of newton s method for solving non linear equations analysis of convergence shows that these methods have order of convergence five numerical tests verifying the theory are given and based on these methods a class of new multistep iterations is developed

achieving higher order of convergence for solving systems of nonlinear equations abstract in this paper we develop a class of third order methods which is a generalization of the existing ones and a method of fourth order method then introduce a technique that improves the order of convergence of any given iterative method for solving systems of nonlinear equations based on a given iterative method of order p 2 which uses the extended newton iteration as a predictor a new method of order p 2 is proposed with only one additional evaluation of the function moreover if the given iterative method of order p 3 uses the newton iteration as a predictor then a new method of order p 3 can be developed applying this procedure we obtain some new methods with higher order of convergence moreover computational efficiency is analyzed and comparisons are made between these new methods and the ones from which have been derived finally several numerical tests are performed to show the asymptotic behaviors which confirm the theoretical results

geometric constructions of iterative functions to solve nonlinear equations in this paper we present the geometrical interpretation of several iterative methods to solve a nonlinear scalar equation in addition we also review the extension to general banach spaces and some computational aspects of these methods

fast fixed point neural blind deconvolution algorithm the aim of this letter is to introduce a new blind deconvolution algorithm based on fixed point optimization of a bussgang type cost function the cost function relies on approximate bayesian estimation achieved by an adaptive neuron the main feature of the presented algorithm is fast convergence that guarantees good deconvolution performances with limited computational demand as compared with algorithms of the same class

a fourth order method from quadrature formulae to solve systems of nonlinear equations in this paper we obtain a fourth order convergence method to solve systems of nonlinear equations this method is based on a quadrature formulae a general error analysis providing the fourth order of convergence is given numerical examples show the fourth order convergence this method does not use the second order frechet derivative

on the computational efficiency index and some iterative methods for solving systems of nonlinear equations in this paper two new iterative methods are built up and analyzed a generalization of the efficiency index used in the scalar case to several variables in iterative methods for solving systems of nonlinear equations is revisited analytic proofs of the local order of convergence based on developments of multilineal functions and numerical concepts that will be used to illustrate the analytic results are given an approximation of the computational order of convergence is computed independently of the knowledge of the root and the necessary time to get one correct decimal is studied in our examples

iterative methods of order four and five for systems of nonlinear equations the adomian decomposition is used in order to obtain a family of methods to solve systems of nonlinear equations the order of convergence of these methods is proved to be p 2 under the same conditions as the classical newton method also numerical examples will confirm the theoretical results

equivariant adaptive source separation source separation consists of recovering a set of independent signals when only mixtures with unknown coefficients are observed this paper introduces a class of adaptive algorithms for source separation that implements an adaptive version of equivariant estimation and is henceforth called equivariant adaptive separation via independence easi the easi algorithms are based on the idea of serial updating this specific form of matrix updates systematically yields algorithms with a simple structure for both real and complex mixtures most importantly the performance of an easi algorithm does not depend on the mixing matrix in particular convergence rates stability conditions and interference rejection levels depend only on the normalized distributions of the source signals closed form expressions of these quantities are given via an asymptotic performance analysis the theme of equivariance is stressed throughout the paper the source separation problem has an underlying multiplicative structure the parameter space forms a matrix multiplicative group we explore the favorable consequences of this fact on implementation performance and optimization of easi algorithms

an efficient fifth order method for solving systems of nonlinear equations in this paper we present a three step iterative method of convergence order five for solving systems of nonlinear equations the methodology is based on the two step homeier s method with cubic convergence homeier 2004 computational efficiency in its general form is discussed and a comparison between the efficiency of proposed technique and existing ones is made the performance is tested through numerical examples moreover theoretical results concerning order of convergence and computational efficiency are verified in the examples it is shown that the present method has an edge over existing methods particularly when applied to large systems of equations

minimum mean square distance estimation of a subspace we consider the problem of subspace estimation in a bayesian setting since we are operating in the grassmann manifold the usual approach which consists of minimizing the mean square error mse between the true subspace u and its estimate u may not be adequate as the mse is not the natural metric in the grassmann manifold gn p i e the set of p dimensional subspaces in rn as an alternative we propose to carry out subspace estimation by minimizing the mean square distance between u and its estimate where the considered distance is a natural metric in the grassmann manifold viz the distance between the projection matrices we show that the resulting estimator is no longer the posterior mean of u but entails computing the principal eigenvectors of the posterior mean of uut derivation of the minimum mean square distance mmsd estimator is carried out in a few illustrative examples including a linear gaussian model for the data and bingham or von mises fisher prior distributions for u in all scenarios posterior distributions are derived and the mmsd estimator is obtained either analytically or implemented via a markov chain monte carlo simulation method the method is shown to provide accurate estimates even when the number of samples is lower than the dimension of u an application to hyperspectral imagery is anally investigated

strong local minimizers in optimal control problems with state constraints second order necessary conditions this paper is devoted to second order necessary optimality conditions for strong local minima for a mayer type optimal control problem with a general control constraint u subset mathbb r m a

a study on neural learning on manifold foliations the case of the lie group su 3 learning on differential manifolds may involve the optimization of a function of many parameters in this letter we deal with riemannian gradient based optimization on a lie group namely the group of unitary unimodular matrices su 3 in this special case subalgebras of the associated lie algebra su 3 may be individuated by computing pair wise gell mann matrices commutators subalgebras generate subgroups of a lie group as well as manifold foliation we show that the riemannian gradient may be projected over tangent structures to foliation giving rise to foliation gradients exponentiations of foliation gradients may be computed in closed forms which closely resemble rodriguez forms for the special orthogonal group so 3 we thus compare optimization by riemannian gradient and foliation gradients

feature extraction across individual time series observations with spikes using wavelet principal component analysis clinical studies frequently include repeated measurements of individuals often for long periods we present a methodology for extracting common temporal features across a set of individual time series observations in particular the methodology explores extreme observations within the time series such as spikes as a possible common temporal phenomenon wavelet basis functions are attractive in this sense as they are localized in both time and frequency domains simultaneously allowing for localized feature extraction from a time varying signal we apply wavelet basis function decomposition of individual time series with corresponding wavelet shrinkage to remove noise we then extract common temporal features using linear principal component analysis on the wavelet coefficients before inverse transformation back to the time domain for clinical interpretation we demonstrate the methodology on a subset of a large fetal activity study aiming to identify temporal patterns in fetal movement fm count data in order to explore formal fm counting as a screening tool for identifying fetal compromise and thus preventing adverse birth outcomes copyright 2013 john wiley sons ltd

some iterative methods for solving a system of nonlinear equations in this paper we suggest and analyze two new two step iterative methods for solving the system of nonlinear equations using quadrature formulas we prove that these new methods have cubic convergence several numerical examples are given to illustrate the efficiency and the performance of the new iterative methods these new iterative methods may be viewed as an extension and generalizations of the existing methods for solving the system of nonlinear equations

a modified newton method with cubic convergence the multivariate case recently a modification of the newton method for finding a zero of a univariate function with local cubic convergence has been introduced here we extend this modification to the multi dimensional case i e we introduce a modified newton method for vector functions that converges locally cubically without the need to compute higher derivatives the case of multiple roots is not treated per iteration the method requires one evaluation of the function vector and solving two linear systems with the jacobian as coefficient matrix where the jacobian has to be evaluated twice since the additional computational effort is nearly that of an additional newton step the proposed method is useful especially in difficult cases where the number of iterations can be reduced by a factor of two in comparison to the newton method this much better convergence is indeed possible as shown by a numerical example also the modified newton method can be advantageous in cases where the evaluation of the function is more expensive than solving a linear system with the jacobian as coefficient matrix an example for this is given where numerical quadrature is involved finally we discuss shortly possible extensions of the method to make it globally convergent

a class of intrinsic schemes for orthogonal integration numerical integration of odes on the orthogonal stiefel manifold is considered points on this manifold are represented as n k matrices with orthonormal columns of particular interest is the case when n gg k mainly two requirements are imposed on the integration schemes first they should have arithmetic complexity of order nk2 second they should be intrinsic in the sense that they require only the ode vector field to be defined on the stiefel manifold as opposed to for instance projection methods the design of the methods makes use of retractions maps two algorithms are proposed one where the retraction map is based on the qr decomposition of a matrix and one where it is based on the polar decomposition numerical experiments show that the new methods are superior to standard lie group methods with respect to arithmetic complexity and may be more reliable than projection methods owing to their intrinsic nature