The generalized superoptimal preconditioner

Convergence analysis of superoptimal PCG algorithm for Toeplitz systems with a Fisher–Hartwig singularity

We define a new generalized inverse (named the gDMP inverse) for a Hilbert space operator using its generalized Drazin inverse and its Moore–Penrose inverse. Thus, we extend the DMP inverse for a square matrix to more general case. Also, we introduce two new classes of operators, g -EP and g -normal operators which include, respectively, EP operators and normal operators. A new binary relation is associated with the gDMP inverse is presented and studied. The notion of core-EP inverse for matrices is extended to generalized Drazin invertible operators on Hilbert space.

This paper is a comprehensive report on test matrices for the generalized inversion of matrices. Two principles are described how to construct singular square or arbitrary rectangular test matrices and their Moore-Penrose inverses. By prescribing the singular values of the matrices or by suitably choosing the free parameters test matrices with condition numbers of any size can be obtained. We also deal with test matrices which are equal to their Moore-Penrose inverse. In addition to many advices how to construct test matrices the paper presents many test matrices explicitly, in particular singular square matrices of ordern, sets of 7×6 and 7×5 matrices of different rank, a set of 5×5 matrices which are equal to their Moore-Penrose inverse and some special test matrices known from literature. For the set of 7×6 parameter matrices also the singular values corresponding to six values of the parameter are listed. For three simple parameter matrices of order 5×4 and 6×5 even test results obtained by eight different algorithms are quoted.

This study presents analytical and numerical‐analytical decomposition methods for determining complex one‐parameter generalized inverse Moore–Penrose matrices. The analytical approach is based on the third Moore–Penrose condition, offering three solution options. The first option employs complex decompositions of the given matrix and its Moore–Penrose inverse. The second option combines the first and third Moore–Penrose conditions, while the third option integrates the second and third conditions. For the first and third options, if any derived iterative procedure converges, the Moore–Penrose inverse matrix can be constructed using the corresponding matrix blocks. In contrast, the second option provides simplified relations, enabling the direct computation of the Moore–Penrose inverse matrix. Numerical‐analytical methods build on the second analytical solution, utilizing differential Pukhov transformations as the primary mathematical tool. A model example featuring a rectangular complex matrix is analyzed. A numerical‐analytical solution is derived using three matrix discretes, from which corresponding matrix blocks are reconstructed. The Moore–Penrose inverse matrix is then obtained through its complex decomposition.

In this article we show that, contrary to finite matrices (with real or complex entries) an invertible infinite matrix V could have a Moore–Penrose inverse that is not a classical inverse of V. This also answers a recent open problem on infinite matrices.

Linear regression (LR) is a core model in supervised machine learning performing a regression task. One can fit this model using either an analytic/closed-form formula or an iterative algorithm. Fitting it via the analytic formula becomes a problem when the number of predictors is greater than the number of samples because the closed-form solution contains a matrix inverse that is not defined when having more predictors than samples. The standard approach to solve this issue is using the Moore–Penrose inverse or the L2 regularization. We propose another solution starting from a machine learning model that, this time, is used in unsupervised learning performing a dimensionality reduction task or just a density estimation one—factor analysis (FA)—with one-dimensional latent space. The density estimation task represents our focus since, in this case, it can fit a Gaussian distribution even if the dimensionality of the data is greater than the number of samples; hence, we obtain this advantage when creating the supervised counterpart of factor analysis, which is linked to linear regression. We also create its semisupervised counterpart and then extend it to be usable with missing data. We prove an equivalence to linear regression and create experiments for each extension of the factor analysis model. The resulting algorithms are either a closed-form solution or an expectation–maximization (EM) algorithm. The latter is linked to information theory by optimizing a function containing a Kullback–Leibler (KL) divergence or the entropy of a random variable.

On the Moore-Penrose Pseudo Inverse of the Incidence Matrix for Weighted Undirected Graph

Moving least‐squares approximation with discontinuous derivative basis functions (MLSA‐DBF) is introduced for analysis of shell structures with slope discontinuities. To deal with shells with arbitrary slope discontinuities, the Cartesian coordinate is introduced in the construction of MLSA on the shell surface. The possible causes of singularity in the moment matrix of MLSA on the shell surface with slope discontinuities are identified, and the Moore–Penrose pseudoinverse is used to obtain the generalized inverse of the singular moment matrix resulting from linear dependency and insufficient influence nodes in the MLSA. Following the proposed formulations for shear deformable shell structures with slope discontinuities in the Cartesian coordinates, several numerical examples are analyzed to demonstrate the performance, validity, accuracy, and convergence properties of the proposed MLSA‐DBF approach. Copyright © 2007 John Wiley & Sons, Ltd.

Pseudo-inverse multivariate/matrix-variate distributions

On level-2 condition number for the weighted Moore–Penrose inverse

Approximation and continuity of Moore–Penrose inverses of orthogonal row block matrices

A comparative study of covariance selection models for the inference of gene regulatory networks

It is well known that every $n \times n$ Stieltjes matrix has an inverse that is an $n \times n$ nonsingular symmetric matrix with nonnegative entries, and it is also easily seen that the converse of this statement fails in general to be true for $n > 2$. In the preceding paper by Martinez, Michon, and San Martin [SIAM J. Matrix Anal. Appl., 15 (1994), pp. 98--106], such a converse result is in fact shown to be true for the new class of strictly ultrametric matrices. A simpler proof of this basic result is given here, using more familiar tools from linear algebra.

This paper presents a linear and quadratic residual bound for eigenvalues of an indefinite possible singular Hermitian matrix. These bounds are a generalization of results on a semidefinite Hermitian matrix [Z. Drmač and V. Hari, SIAM J. Matrix Anal. Appl., 18 (1997), pp. 21–29]. The bounds here contain an extra factor which depends on the norm of a J‐unitary matrix, where J is diagonal matrix with $\pm 1$ on its diagonal.

The method of moments for the simplified scattering and radiation on thin wire models can lead to the large-scale linear systems with non-Hermitian Toeplitz-like coefficient matrices. Since such reduced linear systems are typically dense and ill conditioned, iterative solvers are rarely considered to deal with the discretized linear systems in these electromagnetic model simulations. However, utilizing the Toeplitz-like structure of coefficient matrices, we can use the fast Toeplitz matrix-vector product (MVP) to refresh iterative solvers for the discretized linear systems. Compared with the traditional solvers, the fast numerical method can reduce the computational cost from to and the storage from to without using any lossy compression. Due to the Toeplitz-like structure of coefficient matrices, circulant preconditioners are considered to further accelerate the convergence of iterative solvers. Since the circulant matrices can be diagonalized via fast Fourier transforms (FFTs), the fast Toeplitz MVP is utilized to retain the total computational complexity of the proposed methods at , where is the number of spatial grid nodes. Theoretical analyses of the spectra of the preconditioned linear systems are also preliminarily investigated. Finally, extensive numerical experiments involving both scattering and radiation issues are employed to demonstrate the effectiveness of the proposed methods in aspects of the number of iterations and CPU time elapsed.