Related papers: On a perturbation analysis of Higham squared maxim…
Gaussian elimination (GE) is the most used dense linear solver. Error analysis of GE with selected pivoting strategies on well-conditioned systems can focus on studying the behavior of growth factors. Although exponential growth is possible…
The Higham matrix is a complex symmetric matrix A=B+iC, where both B and C are real, symmetric and positive definite and $\mathrm{i}=\sqrt{-1}$ is the imaginary unit. For any Higham matrix A, Ikramov et al. showed that the growth factor in…
This paper combines modern numerical computation with theoretical results to improve our understanding of the growth factor problem for Gaussian elimination. On the computational side we obtain lower bounds for the maximum growth for…
The growth factor in Gaussian elimination measures how large the entries of an LU factorization can be relative to the entries of the original matrix. It is a key parameter in error estimates, and one of the most fundamental topics in…
The Gaussian Elimination with Partial Pivoting (GEPP) is a classical algorithm for solving systems of linear equations. Although in specific cases the loss of precision in GEPP due to roundoff errors can be very significant, empirical…
We propose a novel approach to estimating the precision matrix of multivariate Gaussian data that relies on decomposing them into a low-rank and a diagonal component. Such decompositions are very popular for modeling large covariance…
Flexible systems are linear systems of inclusions in which the elements of the coefficient matrix are external numbers in the sense of nonstandard analysis. External numbers represent real numbers with small, individual error terms. Using…
A Gaussian elimination algorithm is presented that reveals the numerical rank of a matrix by yielding small entries in the Schur complement. The algorithm uses the maximum volume concept to find a square nonsingular submatrix of maximum…
Gaussian elimination (GE) is the archetypal direct algorithm for solving linear systems of equations and this has been its primary application for thousands of years. In the last decade, GE has found another major use as an iterative…
We consider the problem of finding a dense submatrix of a matrix with i.i.d. Gaussian entries, where density is measured by average value. This problem arose from practical applications in biology and social sciences…
Variational approximation methods have proven to be useful for scaling Bayesian computations to large data sets and highly parametrized models. Applying variational methods involves solving an optimization problem, and recent research in…
In recent years, several algorithms, which approximate matrix decomposition, have been developed. These algorithms are based on metric conservation features for linear spaces of random projection types. We show that an i.i.d sub-Gaussian…
Gaussian Graphical Models (GGMs) are popular tools for studying network structures. However, many modern applications such as gene network discovery and social interactions analysis often involve high-dimensional noisy data with outliers or…
This paper is devoted to the problem of sampling Gaussian fields in high dimension. Solutions exist for two specific structures of inverse covariance : sparse and circulant. The proposed approach is valid in a more general case and…
Many matrices that arise in the solution of signal processing problems have a special displacement structure. For example, adaptive filtering and direction-of-arrival estimation yield matrices of Toeplitz type. A recent method of Gohberg,…
In Gaussian graphical models, the likelihood equations must typically be solved iteratively. We investigate two algorithms: A version of iterative proportional scaling which avoids inversion of large matrices, and an algorithm based on…
The Gaussian expansion has been developed since early 80s as a powerful analytical method, which enables nonperturbative studies of various systems using `perturbative' calculations. Recently the method has been used to suggest that 4d…
The growth problem in Gaussian elimination (GE) remains a foundational question in numerical analysis and numerical linear algebra. Wilkinson resolved the growth problem in GE with partial pivoting (GEPP) in his initial analysis from the…
Let $\orig{A}$ be any matrix and let $A$ be a slight random perturbation of $\orig{A}$. We prove that it is unlikely that $A$ has large condition number. Using this result, we prove it is unlikely that $A$ has large growth factor under…
We consider the problem of recovering an unknown matching between a set of $n$ randomly placed points in $\mathbb{R}^d$ and random perturbations of these points. This can be seen as a model for particle tracking and more generally, entity…