arXiv subjects

Ilgis Ibragimov

Publications and source records attributed to Ilgis Ibragimov.

3 recordsLinked to original sources

arXiv2023

Application of Structured Matrices for Solving Hartree-Fock Equations

This work was originally published by the author in 1999 in a book [1] and later became part of the author's doctoral thesis in 1999 [2]. Since the original language of these works is not English, the author provides a translation of the key ideas of these publications in this work. In addition, the chapter related to numerical experiments was recalculated on modern computers and using contemporary benchmark datasets. This article presents a novel approach to solving Hartree-Fock equations using Toeplitz and tensor matrices and bases based on regular finite elements. The issues discussed include the choice of basis, the dependence of data volume and number of arithmetic operations on the number of basis functions, as well as the arithmetic complexity and accuracy of computing two- and four-center integrals. The approach has been implemented in a software package, and results have been obtained that are in good agreement with theory.

math.SP↗

arXiv2017

The Multi-Dimensional Decomposition with Constraints

We search for the best fit in Frobenius norm of $A \in {\mathbb C}^{m \times n}$ by a matrix product $B C^*$, where $B \in {\mathbb C}^{m \times r}$ and $C \in {\mathbb C}^{n \times r}$, $r \le m$ so $B = \{b_{ij}\}$, ($i=1, \dots, m$,~ $j=1, \dots, r$) definite by some unknown parameters $σ_1, \dots, σ_k$, $k << mr$ and all partial derivatives of $\displaystyle \frac{δb_{ij}}{δσ_l}$ are definite, bounded and can be computed analytically. We show that this problem transforms to a new minimization problem with only $k$ unknowns, with analytical computation of gradient of minimized function by all $σ$. The complexity of computation of gradient is only 4 times bigger than the complexity of computation of the function, and this new algorithm needs only $3mr$ additional memory. We apply this approach for solution of the three-way decomposition problem and obtain good results of convergence of Broyden algorithm.

math.SP↗

arXiv2017

The ELEGANT NMR Spectrometer

Compact and portable in-situ NMR spectrometers which can be dipped in the liquid to be measured, and are easily maintained, with affordable coil constructions and electronics, together with an apparatus to recover depleted magnets are presented, that provide a new real-time processing method for NMR spectrum acquisition, that remains stable despite magnetic field fluctuations.

physics.ins-det↗