QR decomposition

About: QR decomposition is a research topic. Over the lifetime, 3504 publications have been published within this topic receiving 100599 citations. The topic is also known as: QR factorization.

Symmetric-Triangular Decomposition and its Applications Part I: Theorems and Algorithms

Abstract: A new decomposition of a nonsingular matrix, the Symmetric times Triangular (ST) decomposition, is proposed. By this decomposition, every nonsingular matrix can be represented as a product of a symmetric matrix S and a triangular matrix T. Furthermore, S can be made positive definite. Two numerical algorithms for computing the ST decomposition with positive definite S are presented.

Speeding Up PEEC Partial Inductance Computations Using a QR-Based Algorithm

Abstract: The partial element equivalent circuit (PEEC) approach has been used in different forms for the computation of equivalent circuit elements for quasi-static and full-wave electromagnetic models. In this paper, we focus on the topic of large scale inductance computations. For many problems as part of PEEC modeling, partial inductances need to be computed to model interactions between a large numbers of objects. These computations can be very time and memory consuming. To date, several techniques have been devised to reduce the memory and time required to compute the partial inductance entities, as well as the time required to use them in a circuit analysis compute step. Some of the existing methods use hierarchical compression while some others are based on issues like properties of the inverse of the partial inductance matrix. However, because of inherent limitations, most of these methods are less suitable for PEEC applications. In this paper, we present an approach which is based on the compression of the partial inductance matrix utilizing the QR decomposition of the far coefficients submatrices. The QR-decomposed form is represented as a compressed SPICE-compatible circuit. This yields an efficient and mathematically consistent approach for reducing the storage and time requirements

Interpolative Separable Density Fitting through Centroidal Voronoi Tessellation With Applications to Hybrid Functional Electronic Structure Calculations

Abstract: The recently developed interpolative separable density fitting (ISDF) decomposition is a powerful way for compressing the redundant information in the set of orbital pairs, and has been used to accelerate quantum chemistry calculations in a number of contexts. The key ingredient of the ISDF decomposition is to select a set of non-uniform grid points, so that the values of the orbital pairs evaluated at such grid points can be used to accurately interpolate those evaluated at all grid points. The set of non-uniform grid points, called the interpolation points, can be automatically selected by a QR factorization with column pivoting (QRCP) procedure. This is the computationally most expensive step in the construction of the ISDF decomposition. In this work, we propose a new approach to find the interpolation points based on the centroidal Voronoi tessellation (CVT) method, which offers a much less expensive alternative to the QRCP procedure when ISDF is used in the context of hybrid functional electronic structure calculations. The CVT method only uses information from the electron density, and can be efficiently implemented using a K-Means algorithm. We find that this new method achieves comparable accuracy to the ISDF-QRCP method, at a cost that is negligible in the overall hybrid functional calculations. For instance, for a system containing $1000$ silicon atoms simulated using the HSE06 hybrid functional on $2000$ computational cores, the cost of QRCP-based method for finding the interpolation points costs $434.2$ seconds, while the CVT procedure only takes $3.2$ seconds. We also find that the ISDF-CVT method also enhances the smoothness of the potential energy surface in the context of \emph{ab initio} molecular dynamics (AIMD) simulations with hybrid functionals.

A cmv-based eigensolver for companion matrices ∗

Abstract: In this paper we present a novel matrix method for polynomial rootfinding. We approximate the roots by computing the eigenvalues of a permuted version of the companion matrix associated with the polynomial. This form, referred to as a lower staircase form of the companion matrix in the literature, has a block upper Hessenberg shape with possibly nonsquare subdiagonal blocks. It is shown that this form is well suited to the application of the QR eigenvalue algorithm. In particular, each matrix generated under this iteration is block upper Hessenberg and, moreover, all its submatrices located in a specified upper triangular portion are of rank two at most, with entries represented by means of four given vectors. By exploiting these properties we design a fast and computationally simple structured QR iteration which computes the eigenvalues of a companion matrix of size $n$ in lower staircase form using $O(n^2)$ flops and $O(n)$ memory storage. So far, this iteration is theoretically faster than the fastest ...