Parallel Strategies for Computing the Orthogonal Factorizations Used in the Estimation of Econometric Models

Abstract

Parallel strategies based on compound disjoint Givens rotations are proposed for computing the main two factorizations that are used in the solution of seemingly unrelated regression and simultaneous equations models. The first factorization requires the triangularization of a set of upper-trapezoidals after deleting columns. The second factorization is equivalent to updating a lower-triangular matrix with a matrix having a block lower-triangular structure. Theoretical measures of complexity and examples are used for comparing and investigating the various parallel strategies. 1. Introduction. The estimation of simultaneous equations models (SEMs) is of great importance in econometric and financial modeling [13], [37]. The most commonly used estimation procedures are the Three Stage Least-Squares (3SLS) and the computationally expensive maximum likelihood [8], [12], [31], [33], [39]. Recently, new computational and numerical methods which substantially reduce the computational burden of deriving the 3SLS solution of SEMs have been proposed [2], [25]. These methods have, as a basic component, the QR decomposition (QRD) and its modification. The goal of this work is to provide theoretically-based efficient parallel strategies for computing these decompositions using as few steps as possible. These strategies will form the basis for implementing the new 3SLS and the simplified SURE model estimation algorithms on parallel systems [16], [20], [23].The SEM can be written in the form

Analysis of Radiative Radial Fin with Temperature-Dependent Thermal Conductivity Using Nonlinear Differential Transformation Methods

“…The Boubaker polynomials expansion scheme (BPES) [34][35][36][37][38][39][40][41][42][43][44][45][46][47][48][49][50][51][52] is a resolution protocol which has been successfully applied to several applied-physics and mathematics problems. The BPES protocol ensures the validity of the related boundary conditions regardless of main equation features.…”

“…Several solutions have been proposed through the BPES in many fields such as numerical analysis [34][35][36][37], theoretical physics [38][39][40][41], mathematical algorithms [42], heat transfer [43], homodynamic [44,45], material characterization [46], fuzzy systems modeling [47][48][49][50], and biology [51,52].…”

Algorithms for Computing the QR Decomposition of a Set of Matrices with Common Columns

“…whereQ i ∈ R n×n is orthogonal [9]- [11]. The orthogonal matrix Q i in (1) is defined by (5) Notice that the QRDs in (4) are equivalent to the re-triangularization of a set of uppertriangular matrices after deleting columns.…”

“…Sequential and parallel strategies which compute the QRD of RS i have been proposed [11], [12], [15]. These strategies use Givens rotations and exploit the non-full structure of RS i .…”

“…. , k i [11], [12], [15]. Then the number of Givens rotations needed to compute the QRD (4) is given by The complexity of computing the QRD (4) is given by…”