# Registered Data

Contents

- 1 [CT095]
- 1.1 [01675] Solution of Linear Systems of Equations using a Gauss-Seidel-based Method
- 1.2 [02325] A Hybrid Method for Solving Linear KKT Systems
- 1.3 [00325] Constrained Spectral Dilation of a Matrix
- 1.4 [00337] Extending Matrix-less Methods for Eigenvalues and Eigenvectors
- 1.5 [00823] Weighted Trace-Penalty Minimization for Full Configuration Interaction

# [CT095]

## [01675] Solution of Linear Systems of Equations using a Gauss-Seidel-based Method

**Session Date & Time**: 4D (Aug.24, 15:30-17:10)**Type**: Contributed Talk**Abstract**: Linear Systems of Equations (LSsEs) often arise from applications in real-life. Because of the importance of such applications, some repository, such as the SuiteSparse Matrix Collection at http://www.cise.ufl.edu/research/sparse/matrices, has been created to document them, and solicit assistance for solving them. The direct and indirect methods commonly applied to solve LSsEs have been complemented by other schemes so as to take care of some peculiarities in the linear systems. An adaptation based on the salient features of the Gauss-Seidel method is herein proposed for the solution of any LSEs. Exploiting one of the features led to the development of a solver for LSsEs, which has been computationally validated as accurate, efficient and robust.**Classification**:__65F10__,__65F05__,__Numerical linear algebra__**Author(s)**:**Olabode Matthias Bamigbola**(University of Ilorin)- Montaz Ali (University of Witwatersand)
- Adebisi Ibrahim (Oduduwa University, Ile-Ife)
- Amos Ezeh (University of Ilorin)

## [02325] A Hybrid Method for Solving Linear KKT Systems

**Session Date & Time**: 4D (Aug.24, 15:30-17:10)**Type**: Contributed Talk**Abstract**: We propose an iterative method for solving linear systems arising from optimization problems with a separable objective function and dense constraints and suitable preconditioner for quick convergence. The method is implemented in Julia using Krylov.jl for the iterative solution, CUDA.jl to enable GPU capabilities, and a custom kernel for construction of the preconditioner. The method attains faster solution times than direct methods common in solvers such as Mosek and Gurobi in theory and in practice.**Classification**:__65F10__,__15A29__,__90C05__,__65K05__,__90C25__**Author(s)**:**Shaked Regev**(Gridmatic)- Shaked Regev (Gridmatic)

## [00325] Constrained Spectral Dilation of a Matrix

**Session Date & Time**: 4D (Aug.24, 15:30-17:10)**Type**: Contributed Talk**Abstract**: Let $ A \in \mathbb{C}^{n \times n}$ and $ B \in \mathbb{C}^{m \times m}.$ Then $A$ is said to be a spectral dilation of $B$ and written as $ B \subset A$ if $ m< n$ and there exist matrices $C$ and $D$ such that $$ A= S\left[\begin{array}{c|c} B & C \\ \hline 0 & D \end{array}\right] S^{-1}$$ for some nonsingular matrix $S \in \mathbb{C}^{n\times n}. $ Obviously, if $A$ is a spectral dilation of $B$ then $A$ has a partially specified eigen-structure as given by the eigen-structure of $B.$ Now, given an $n\times n$ matrix $A$ and an $m\times m$ matrix $B$ with $m < n,$ consider the constrained spectral dilation problem $($CSDP$)$ $$ \widehat{A} = \mathrm{arg}\min \{ \| X- A \|_2 : X \in \mathbb{C}^{n\times n}, \; B \subset X \}.$$ If $\widehat{A}$ exists then it is a spectral dilation of $B$ nearest to $A$ and $$\mathrm{d_S}(A, B) := \|\widehat{A}- A\|_2 = \min\{ \|X - A\|_2 : X \in \mathbb{C}^{n\times n}, \; B \subset X\} $$ is the shortest distance from $A$ to a spectral dilation of $B.$ The main aim of this work is to present a concise framework for solving the constrained spectral dilation problem and undertake an in-depth analysis of various issues that influence existence of a solution.**Classification**:__65F15__,__15A18__,__65F55__**Author(s)**:**Rafikul Alam**(Indian Institute of Technology Guwahati)

## [00337] Extending Matrix-less Methods for Eigenvalues and Eigenvectors

**Session Date & Time**: 4D (Aug.24, 15:30-17:10)**Type**: Contributed Talk**Abstract**: Toeplitz and Toeplitz-like matrices arise in many fields; of special interest are the discretisations of differential equations. Matrix-less methods exploit the fact that we can view these Toeplitz(-like) matrices as part of a sequence of matrices and that the eigenvalues in the sequence behave as samplings of a function, called the symbol. We will discuss recent developments to handle the case of variable coefficient matrices, the approximation of eigenvectors, and Toeplitz(-like) matrices with non-monotone symbols.**Classification**:__65F15__,__15B05__,__15A18__**Author(s)**:**David Meadon**(Uppsala University)

## [00823] Weighted Trace-Penalty Minimization for Full Configuration Interaction

**Session Date & Time**: 4D (Aug.24, 15:30-17:10)**Type**: Contributed Talk**Abstract**: A novel unconstrained optimization model named weighted trace-penalty minimization (WTPM) is proposed to address the extreme eigenvalue problem arising from the Full Configuration Interaction (FCI) method. The coordinate descent method is adapted to WTPM and results in WTPM-CD method. With the sparse features of both FCI matrices and the global minimizers in mind, the reduction of computational and storage costs shows the efficiency of the algorithm.**Classification**:__65F15__,__Electron Structure Calculation__**Author(s)**:- Weiguo Gao (Fudan University)
- Yingzhou Li (Fudan University)
**Hanxiang Shen**(Fudan University)