Effective partitioning method for computing weighted Moore-Penrose inverse

We introduce a method and an algorithm for computing the weighted Moore-Penrose inverse of multiple-variable polynomial matrix and the related algorithm which is appropriated for sparse polynomial matrices. These methods and algorithms are generalizations of algorithms developed in [M.B. Tasic, P.S. Stanimirovic, M.D. Petkovic, Symbolic computation of weighted Moore-Penrose inverse using partitioning method, Appl. Math. Comput. 189 (2007) 615-640] to multiple-variable rational and polynomial matrices and improvements of these algorithms on sparse matrices. Also, these methods are generalizations of the partitioning method for computing the Moore-Penrose inverse of rational and polynomial matrices introduced in [P.S. Stanimirovic, M.B. Tasic, Partitioning method for rational and polynomial matrices, Appl. Math. Comput. 155 (2004) 137-163; M.D. Petkovic, P.S. Stanimirovic, Symbolic computation of the Moore-Penrose inverse using partitioning method, Internat. J. Comput. Math. 82 (2005) 355-367] to the case of weighted Moore-Penrose inverse. Algorithms are implemented in the symbolic computational package MATHEMATICA

Inverting a matrix function around a singularity via local rank factorization

This paper proposes a recursive procedure, called the extended local rank factorization (elrf), that characterizes the order of the pole and the coefficients of the Laurent series representation of the inverse of a regular analytic matrix function around a given point. The elrf consists in performing a finite sequence of rank factorizations of matrices of nonincreasing dimension, at most equal to the dimension of the original matrix function. Each step of the sequence is associated with a reduced rank condition, while the termination of the elrf corresponds to a full rank condition; this last step reveals the order of the pole. The Laurent coefficients B n are calculated recursively as B_n = C n + sum_{k=1}^n D_k B_{n−k} , where C_n , D_k have simple closed form expressions in terms of the quantities generated by the elrf. It is also shown that the elrf characterizes the structure of Jordan pairs, Jordan chains, and the local Smith form. The procedure is easily cast in an algorithmic form, and a MATLAB implementation script is provided. It is further found that the elrf coincides with the complete reduction process (crp) in Avrachenkov, Haviv, and Howlett [SIAM J. Matrix Anal. Appl., 22 (2001), pp. 1175–1189]. Using this connection, the results on the elrf provide both an explicit recursive formula for B n implied by the crp, and the link between the crp and the structure of the local Smith form

Completely Recursive Least Squares and Its Applications

The matrix-inversion-lemma based recursive least squares (RLS) approach is of a recursive form and free of matrix inversion, and has excellent performance regarding computation and memory in solving the classic least-squares (LS) problem. It is important to generalize RLS for generalized LS (GLS) problem. It is also of value to develop an efficient initialization for any RLS algorithm. In Chapter 2, we develop a unified RLS procedure to solve the unconstrained/linear-equality (LE) constrained GLS. We also show that the LE constraint is in essence a set of special error-free observations and further consider the GLS with implicit LE constraint in observations (ILE-constrained GLS). Chapter 3 treats the RLS initialization-related issues, including rank check, a convenient method to compute the involved matrix inverse/pseudoinverse, and resolution of underdetermined systems. Based on auxiliary-observations, the RLS recursion can start from the first real observation and possible LE constraints are also imposed recursively. The rank of the system is checked implicitly. If the rank is deficient, a set of refined non-redundant observations is determined alternatively. In Chapter 4, base on [Li07], we show that the linear minimum mean square error (LMMSE) estimator, as well as the optimal Kalman filter (KF) considering various correlations, can be calculated from solving an equivalent GLS using the unified RLS. In Chapters 5 & 6, an approach of joint state-and-parameter estimation (JSPE) in power system monitored by synchrophasors is adopted, where the original nonlinear parameter problem is reformulated as two loosely-coupled linear subproblems: state tracking and parameter tracking. Chapter 5 deals with the state tracking which determines the voltages in JSPE, where dynamic behavior of voltages under possible abrupt changes is studied. Chapter 6 focuses on the subproblem of parameter tracking in JSPE, where a new prediction model for parameters with moving means is introduced. Adaptive filters are developed for the above two subproblems, respectively, and both filters are based on the optimal KF accounting for various correlations. Simulations indicate that the proposed approach yields accurate parameter estimates and improves the accuracy of the state estimation, compared with existing methods

On disjoint range operators in a Hilbert space

AbstractFor a bounded linear operator M in a Hilbert space H, various relations among the ranges R(M),R(M∗), R(M+M∗) and the null spaces N(M),N(M∗) are considered from the point of view of their relations to the known classes of operators, such as EP, co-EP, weak-EP, GP, DR, or SR. Particular attention is paid to the range projectors of the operators M, M∗ and some further characteristics of these projectors are derived as well

Random vibration of linear and nonlinear structural systems with singular matrices: A frequency domain approach

A frequency domain methodology is developed for stochastic response determination of multi-degree-of-freedom (MDOF) linear and nonlinear structural systems with singular matrices. This system modeling can arise when a greater than the minimum number of coordinates/DOFs is utilized, and can be advantageous, for instance, in cases of complex multibody systems where the explicit formulation of the equations of motion can be a nontrivial task. In such cases, the introduction of additional/redundant DOFs can facilitate the formulation of the equations of motion in a less labor intensive manner. Specifically, relying on the generalized matrix inverse theory, a Moore-Penrose (M-P) based frequency response function (FRF) is determined for a linear structural system with singular matrices. Next, relying on the M-P FRF a spectral input-output (excitation-response) relationship is derived in the frequency domain for determining the linear system response power spectrum. Further, the above methodology is extended via statistical linearization to account for nonlinear systems. This leads to an iterative determination of the system response mean vector and covariance matrix. Furthermore, to account for singular matrices, the generalization of a widely utilized formula that facilitates the application of statistical linearization is proved as well. The formula relates to the expectation of the derivatives of the system nonlinear function and is based on a Gaussian response assumption. Several linear and nonlinear MDOF structural systems with singular matrices are considered as numerical examples for demonstrating the validity and applicability of the developed frequency domain methodology

A Bayesian analysis of simultaneous equation models by combining recursive analytical and numerical approaches

Economics

Some aspects of the theory, application, and computation of generalised inverses of matrices

The idea of generalising the classical notion of the inverse of a non-singular matrix arose as far back as in 1920, but it was not until the late fifties that the development of the theory gained any impetus. Since then , as is the case in the development of many new concepts , work done in parallel in various parts of the world has resulted in a great deal of untidiness in the literature : confusion over terminology , and even duplication of theory. More recently, however, some attempts have been made to bring together people active in the field of generalised inverses, in order to reach consensus on some aspects of definition and terminology, and to publish more general works on the subject. Towards this purpose, a symposium on the theory and application of generalised inverses of matrices was held in Lubbock, Texas, and its proceedings published in 1968 (see [25] ). A few other works of this nature (see [4], (19a] ) have appeared , but the bulk of the literature still comprises numerous diverse papers offering further ideas on the theoretical properties which these matrices have , and drawing attention to their application in areas of statistics , numerical analysis , filtering , modern control and estimation theory, pattern recognition and many others. This essay offers a look at generalised inverses in the following way: firstly a broad basis and background is established in the first three chapters to provide greater understanding of the motivation for the remaining chapters, where the approach then changes to become far more detailed. Within this general framework, Chapter 1 offers a brief glimpse of the history and development of work in the field. In Chapter 2 some of the most significant properties of these inverses are described, while in Chapters 3 and 4 and 5 attention is given to interesting and remarkable computational algorithms relating to generalised inverses (some well suited to machine processing). The material of Chapters 4 and 5 is largely due to Decell, Stallings and Boullion, and Tanabe, in [6], [24] and [27], respectively, while the source of material for the first three chapters is the literature generally, with Penrose's two papers providing a rough framework for Chapters 1 and 2 (see [17])

On Quasi-Newton Forward--Backward Splitting: Proximal Calculus and Convergence

We introduce a framework for quasi-Newton forward--backward splitting algorithms (proximal quasi-Newton methods) with a metric induced by diagonal $\pm$ rank-$r$ symmetric positive definite matrices. This special type of metric allows for a highly efficient evaluation of the proximal mapping. The key to this efficiency is a general proximal calculus in the new metric. By using duality, formulas are derived that relate the proximal mapping in a rank-$r$ modified metric to the original metric. We also describe efficient implementations of the proximity calculation for a large class of functions; the implementations exploit the piece-wise linear nature of the dual problem. Then, we apply these results to acceleration of composite convex minimization problems, which leads to elegant quasi-Newton methods for which we prove convergence. The algorithm is tested on several numerical examples and compared to a comprehensive list of alternatives in the literature. Our quasi-Newton splitting algorithm with the prescribed metric compares favorably against state-of-the-art. The algorithm has extensive applications including signal processing, sparse recovery, machine learning and classification to name a few.Comment: arXiv admin note: text overlap with arXiv:1206.115