4 research outputs found
Accurate computation of the Moore-Penrose inverse of strictly totally positive matrices
The computation of the Moore-Penrose inverse of structured strictly totally positive matrices is addressed. Since these matrices are usually very ill-conditioned, standard algorithms fail to provide accurate results. An algorithm based on the factorization and which takes advantage of the special structure and the totally positive character of these matrices is presented. The first stage of the algorithm consists of the accurate computation of the bidiagonal decomposition of the matrix. Numerical experiments illustrating the good behavior of our approach are included.Numerical experiments illustrating the good behavior of our approach are included
Computation of Moore-Penrose generalized inverses of matrices with meromorphic function entries
J.R. Sendra is member of the Research Group ASYNACS (Ref.CT-CE2019/683)In this paper, given a field with an involutory automorphism, we introduce the notion of Moore-Penrose field by requiring that all matrices over the field have Moore-Penrose inverse. We prove that only characteristic zero fields can be Moore-Penrose, and that the field of rational functions over a Moore-Penrose field is also Moore-Penrose. In addition, for a matrix with rational functions entries with coefficients in a field K, we find sufficient conditions for the elements in K to ensure that the specialization of the Moore-Penrose inverse is the Moore-Penrose inverse of the specialization of the matrix. As a consequence, we provide a symbolic algorithm that, given a matrix whose entries are rational expression over C of finitely many meromeorphic functions being invariant by the involutory automorphism, computes its Moore-Penrose inverve by replacing the functions by new variables, and hence reducing the problem to the case of matrices with complex rational function entries.Ministerio de Economía y CompetitividadEuropean Regional Development Fun
Recurrent neural networks for solving matrix algebra problems
The aim of this dissertation is the application of recurrent neural
networks (RNNs) to solving some problems from a matrix algebra
with particular reference to the computations of the generalized
inverses as well as solving the matrix equations of constant (timeinvariant)
matrices. We examine the ability to exploit the correlation
between the dynamic state equations of recurrent neural networks for
computing generalized inverses and integral representations of these
generalized inverses. Recurrent neural networks are composed of
independent parts (sub-networks). These sub-networks can work
simultaneously, so parallel and distributed processing can be
accomplished. In this way, the computational advantages over the
existing sequential algorithms can be attained in real-time
applications. We investigate and exploit an analogy between the
scaled hyperpower family (SHPI family) of iterative methods for
computing the matrix inverse and the discretization of Zhang Neural
Network (ZNN) models. A class of ZNN models corresponding to the
family of hyperpower iterative methods for computing the generalized
inverses on the basis of the discovered analogy is defined. The Matlab
Simulink implementation of the introduced ZNN models is described
in the case of scaled hyperpower methods of the order 2 and 3. We
present the Matlab Simulink model of a hybrid recursive neural
implicit dynamics and give a simulation and comparison to the
existing Zhang dynamics for real-time matrix inversion. Simulation
results confirm a superior convergence of the hybrid model compared
to Zhang model
Distance measures and whitening procedures for high dimensional data
The need to effectively analyse high dimensional data is increasingly crucial to many fields as data collection and storage capabilities continue to grow. Working with high
dimensional data is fraught with difficulties, making many data analysis methods inadvisable, unstable or entirely unavailable.
The Mahalanobis distance and data whitening are two methods that are integral to multivariate data analysis. These methods are reliant on the inverse of the covariance matrix,
which is often non-existent or unstable in high dimensions. The methods that are currently used to circumvent singularity in the covariance matrix often impose structural
assumptions on the data, which are not always appropriate or known.
In this thesis, three novel methods are proposed. Two of these methods are distance measures which measure the proximity of a point x to a set of points X. The simplicial distances find the average volume of all k-dimensional simplices between x and vertices of X.
The minimal-variance distances aim to minimize the variance of the distances produced, while adhering to a constraint ensuring similar behaviour to the Mahalanobis distance.
Finally, the minimal-variance whitening method is detailed. This is a method of data whitening, and is constructed by minimizing the total variation of the transformed data
subject to a constraint.
All of these novel methods are shown to behave similarly to the Mahalanobis distances and data whitening methods that are used for full-rank data. Furthermore, unlike the methods
that rely on the inverse covariance matrix, these new methods are well-defined for degenerate data and do not impose structural assumptions. This thesis explores the aims, constructions and limitations of these new methods, and offers many empirical examples and comparisons of their performances when used with high dimensional data