Approximating the inverse and the Moore‐Penrose inverse of complex matrices

A parametric family of fourth‐order schemes for computing the inverse and the Moore‐Penrose inverse of a complex matrix is designed. A particular value of the parameter allows us to obtain a fifth‐order method. Convergence analysis of the different methods is studied. Every iteration of the proposed schemes involves four matrix multiplications. A numerical comparison with other known methods, in terms of the average number of matrix multiplications and the mean of CPU time, is presented.     

Eigenvalues of functions of orthogonal projectors

By representing two orthogonal projectors in a finite dimensional vector space as partitioned matrices, several characterizations concerning eigenvalues of various functions of the pair are obtained. These results substantially extend the ones already available in the literature. Additionally, some related results dealing with the functions of a pair of orthogonal projectors are provided, with the emphasis laying on the problem of invertibility.     

Smoothed analysis of some condition numbers

In this note we do a smoothed analysis, in the sense of ( http://www‐math.mit.edu/～spielman/SmoothedAnalysis/ ), of the condition number for the Moore–Penrose inverse. Usual average analysis follows in a trivial manner as follow similar analyses for the condition number of the polar factorization. Copyright © 2005 John Wiley & Sons, Ltd.     

A fast convergent iterative solver for approximate inverse of matrices

In this paper, a rapid iterative algorithm is proposed to find robust approximations for the inverse of nonsingular matrices. The analysis of convergence reveals that this high‐order method possesses eighth‐order convergence. The interesting point is that, this rate is attained using less number of matrix‐by‐matrix multiplications in contrast to the existing methods of the same type in the literature. The extension of the method for finding Moore–Penrose inverse of singular or rectangular matrices is also presented. Numerical comparisons will be given to show the applicability, stability and consistency of the new scheme by paying special attention on the computational time. Copyright © 2013 John Wiley & Sons, Ltd.     

An Alternative Proof of the Greville Formula

A simple proof of the Greville formula for the recursive computation of the Moore–Penrose (MP) inverse of a matrix is presented. The proof utilizes no more than the elementary properties of the MP inverse.     

Newton-Like Iteration Based on a Cubic Polynomial for Structured Matrices

We recall Newtons iteration for computing the inverse or Moore–Penrose generalized inverse of a matrix. Then we specialize this approach to the case of structured matrices where all input, output and intermediate auxiliary matrices are represented in a compressed form, via their short displacement generators. We design a new Newton-like iteration based on a cubic polynomial and show its effectiveness by some numerical experiments for matrices from the Toeplitz-like class and the Cauchy-like class.     

Removal of blur in images based on least squares solutions

We propose an image restoration method. The method generalizes image restoration algorithms that are based on the Moore–Penrose solution of certain matrix equations that define the linear motion blur. Our approach is based on the usage of least squares solutions of these matrix equations, wherein an arbitrary matrix of appropriate dimensions is included besides the Moore–Penrose inverse. In addition, the method is a useful tool for improving results obtained by other image restoration methods. Toward that direction, we investigate the case where the arbitrary matrix is replaced by the matrix obtained by the Haar basis reconstructed image. The method has been tested by reconstructing an image after the removal of blur caused by the uniform linear motion and filtering the noise that is corrupted with the image pixels. The quality of the restoration is observable by a human eye. Benefits of using the method are illustrated by the values of the improvement in signal‐to‐noise ratio and in the values of peak signal‐to‐noise ratio. Copyright © 2013 John Wiley & Sons, Ltd.     

Recursive Determination of the Generalized Moore–Penrose <Emphasis Type="Italic">M</Emphasis>-Inverse of a Matrix

In this paper, we obtain recursive relations for the determination of the generalized Moore–Penrose M-inverse of a matrix. We develop separate relations for situations when a rectangular matrix is augmented by a row vector and when such a matrix is augmented by a column vector.     

A note on the Drazin inverses with Banachiewicz-Schur forms

Let H be a Hilbert space, M the closed subspace of H with orthocomplement M^⊥. According to the orthogonal decomposition H=M⊕M^⊥, every operator M∈B(H) can be written in a block-form . In this note, we give necessary and sufficient conditions for a partitioned operator matrix M to have the Drazin inverse with Banachiewicz-Schur form. In addition, this paper investigates the relations among the Drazin inverse, the Moore-Penrose inverse and the group inverse when they can be expressed in the Banachiewicz-Schur forms.     

Least‐squares solutions and least‐rank solutions of the matrix equation AXA* = B and their relations

A Hermitian matrix X is called a least‐squares solution of the inconsistent matrix equation AXA^* = B, where B is Hermitian. A^* denotes the conjugate transpose of A if it minimizes the F‐norm of B ? AXA^*; it is called a least‐rank solution of AXA^* = B if it minimizes the rank of B ? AXA^*. In this paper, we study these two types of solutions by using generalized inverses of matrices and some matrix decompositions. In particular, we derive necessary and sufficient conditions for the two types of solutions to coincide. Copyright © 2012 John Wiley & Sons, Ltd.     

Tight upper bounds for the convergence of the randomized extended Kaczmarz and Gauss–Seidel algorithms

The randomized extended Kaczmarz and Gauss–Seidel algorithms have attracted much attention because of their ability to treat all types of linear systems (consistent or inconsistent, full rank or rank deficient). In this paper, we present tight upper bounds for the convergence of the randomized extended Kaczmarz and Gauss–Seidel algorithms. Numerical experiments are given to illustrate the theoretical results.     

Fast enclosure for the minimum norm least squares solution of the matrix equation AXB = C

Fast algorithms for enclosing the minimum norm least squares solution of the matrix equation AXB = C are proposed. To develop these algorithms, theories for obtaining error bounds of numerical solutions are established. The error bounds obtained by these algorithms are verified in the sense that all the possible rounding errors have been taken into account. Techniques for accelerating the enclosure and obtaining smaller error bounds are introduced. Numerical results show the properties of the proposed algorithms. Copyright © 2015 John Wiley & Sons, Ltd.     

Singular Normal Extremals and Conjugate Points for Bolza Functionals

In this paper, we study the problem of the optimal control for Bolza functionals by investigating extremals containing singular arcs. We use the Moore–Penrose generalized inverse, which allows one to determine normality criteria and sufficient conditions for the nonexistence of conjugate points.     

On Frobenius normwise condition numbers for Moore–Penrose inverse and linear least‐squares problems

Condition numbers play an important role in numerical analysis. Classical condition numbers are normwise: they measure the size of both input perturbations and output errors using norms. In this paper, we give explicit, computable expressions depending on the data, for the normwise condition numbers for the computation of the Moore–Penrose inverse as well as for the solutions of linear least‐squares problems with full‐column rank. Copyright © 2007 John Wiley & Sons, Ltd.     

On condition numbers for Moore–Penrose inverse and linear least squares problem involving Kronecker products

In this paper, we investigate the normwise, mixed, and componentwise condition numbers and their upper bounds for the Moore–Penrose inverse of the Kronecker product and more general matrix function compositions involving Kronecker products. We also present the condition numbers and their upper bounds for the associated Kronecker product linear least squares solution with full column rank. In practice, the derived upper bounds for the mixed and componentwise condition numbers for Kronecker product linear least squares solution can be efficiently estimated using the Hager–Higham Algorithm. Copyright © 2012 John Wiley & Sons, Ltd.     

On the Moore–Penrose inverse in solving saddle‐point systems with singular diagonal blocks

This paper deals with the role of the generalized inverses in solving saddle‐point systems arising naturally in the solution of many scientific and engineering problems when finite‐element tearing and interconnecting based domain decomposition methods are used to the numerical solution. It was shown that the Moore–Penrose inverse may be obtained in this case at negligible cost by projecting an arbitrary generalized inverse using orthogonal projectors. Applying an eigenvalue analysis based on the Moore–Penrose inverse, we proved that for simple model problems, the number of conjugate gradient iterations required for the solution of associate dual systems does not depend on discretization norms. The theoretical results were confirmed by numerical experiments with linear elasticity problems. Copyright © 2011 John Wiley & Sons, Ltd.     

On the invertibility of the operator A–XB

For a given pair of (A, B) and an arbitrary operator X, expressions for the inverse, the Moore–Penrose inverse and the generalized Drazin inverse of the operator A–XB are derived under some conditions. Copyright © 2009 John Wiley & Sons, Ltd.     

The Boyer–Moore–Horspool heuristic with Markovian input

The Boyer–Moore–Horspool string‐matching heuristic is an algorithm for locating occurrences of a fixed pattern in a random text. Under the assumption that the text is an independently and identically distributed sequence of characters, the probabilistic behavior of this algorithm was investigated by Mahmoud, Smythe, and Régnier [Random Struct Alg 10 (1997), 169–186]. Here, we obtain similar results under the assumption that the text is generated by an irreducible Markov chain. A natural Markov renewal process structure is exploited to obtain the asymptotic behavior of the number of comparisons. Under suitable normalization, it is shown that a central limit theorem holds for the number of comparisons. The analysis is completely probabilistic and does not use the shift generating function. © 2001 John Wiley & Sons, Inc. Random Struct. Alg., 18: 153–163, 2001     

Explicit solution of the operator equation

In this paper we find the explicit solution of the equation

A^*X+X^*A=B