arXiv subjects

Abbas Salemi

Publications and source records attributed to Abbas Salemi.

4 recordsLinked to original sources

arXiv2025

Error Estimates for Gauss--Christoffel Quadrature under Reduced Regularity Conditions

Gauss--Christoffel quadrature is a fundamental method for numerical integration, and its convergence analysis is closely related to the decay of Chebyshev expansion coefficients. Classical estimates, including those due to Trefethen, are based on weighted bounded variation assumptions involving the singular weight $(1-x^{2})^{-1/2}$, which may be too restrictive for functions with limited regularity at the endpoints. In this paper, we establish a new error bound for Gauss--Christoffel quadrature under weakened regularity assumptions. The analysis relies on a new identity for higher-order derivatives of Chebyshev polynomials. As a consequence, we obtain an improved decay estimate for Chebyshev coefficients, where the classical weighted condition \[ V_{r}=\int_{-1}^{1}\frac{|f^{(r+1)}(x)|}{\sqrt{1-x^{2}}}\,dx \] is replaced by the weaker condition \[ U_{r}=\int_{-1}^{1}|f^{(r+1)}(x)|\,dx. \] This result leads to a corresponding error estimate for the Gauss--Christoffel quadrature rule, which is less restrictive than previous bounds. The approach is also extended to the Gauss--Gegenbauer case. Numerical experiments are provided to illustrate the theoretical results.

math.NA↗

arXiv2025

When is the Resolvent Like a Rank One Matrix?

For a square matrix $A$, the resolvent of $A$ at a point $z \in \mathbb{C}$ is defined as $(A-zI )^{-1}$. We consider the set of points $z \in \mathbb{C}$ where the relative difference in 2-norm between the resolvent and the nearest rank one matrix is less than a given number $\epsilon \in (0,1)$. We establish a relationship between this set and the $\epsilon$-pseudospectrum of $A$, and we derive specific results about this set for Jordan blocks and for a class of large Toeplitz matrices. We also derive disks about the eigenvalues of $A$ that are contained in this set, and this leads to some new results on disks about the eigenvalues that are contained in the $\epsilon$-pseudospectrum of $A$. In addition, we consider the set of points $z \in \mathbb{C}$ where the absolute value of the inner product of the left and right singular vectors corresponding to the largest singular value of the resolvent is less than $\epsilon$. We demonstrate numerically that this set can be almost as large as the one where the relative difference between the resolvent and the nearest rank one matrix is less than $\epsilon$ and we give a partial explanation for this. Some possible applications are discussed.

math.NA↗

arXiv2024

Properties of core-EP matrices and binary relationships

In this paper, various properties of core-EP matrices are investigated. We introduce the MPDMP matrix associated with $A$ and by means of it, some properties and equivalent conditions of core-EP matrices can be obtained. Also, properties of MPD, DMP, and CMP inverses are studied and we prove that in the class of core-EP matrices, DMP, MPD, and Drazin inverses are the same. Moreover, DMP and MPD binary relation orders are introduced and the relationship between these orders and other binary relation orders are considered.

math.NA↗

arXiv2020

DISCERN: Diversity-based Selection of Centroids for k-Estimation and Rapid Non-stochastic Clustering

One of the applications of center-based clustering algorithms such as K-Means is partitioning data points into K clusters. In some examples, the feature space relates to the underlying problem we are trying to solve, and sometimes we can obtain a suitable feature space. Nevertheless, while K-Means is one of the most efficient offline clustering algorithms, it is not equipped to estimate the number of clusters, which is useful in some practical cases. Other practical methods which do are simply too complex, as they require at least one run of K-Means for each possible K. In order to address this issue, we propose a K-Means initialization similar to K-Means++, which would be able to estimate K based on the feature space while finding suitable initial centroids for K-Means in a deterministic manner. Then we compare the proposed method, DISCERN, with a few of the most practical K estimation methods, while also comparing clustering results of K-Means when initialized randomly, using K-Means++ and using DISCERN. The results show improvement in both the estimation and final clustering performance.

cs.LG↗