Linear Least Squares Computations

R.W. Farebrother

• Publisher: Routledge
• Year: 2018
• ISBN:
  9781351435253
• Edition: 1
• Topic: Mathematics
• Subtopic: Mathematics General
• Index: Mathematics

Presenting numerous algorithms in a simple algebraic form so that the reader can easilytranslate them into any computer language, this volume gives details of several methodsfor obtaining accurate least squares estimates. It explains how these estimates may beupdated as new information becomes available and how to test linear hypotheses.Linear Least Squares Computations features many structured exercises that guidethe reader through the available algorithms, plus a glossary of commonly used terms anda bibliography of supplementary reading... collects "ancient" and modem results onlinear least squares computations in a convenient single source... develops the necessarymatrix algebra in the context of multivariate statistics... only makes peripheral use ofconcepts such as eigenvalues and partial differentiation... interprets canonical formsemployed in computation... discusses many variants of the Gauss, Laplace-Schmidt, Givens, and Householder algorithms... and uses an empirical approach for the appraisalof algorithms.Linear Least Squares Computations serves as an outstanding reference forindustrial and applied mathematicians, statisticians, and econometricians, as well as atext for advanced undergraduate and graduate statistics, mathematics, and econometricscourses in computer programming, linear regression analysis, and applied statistics.

1

The Gauss and Gauss-Jordan Methods

Aspects of Computer Programming

1.1 INTRODUCTION

This book is concerned with the solution of systems of n ≣ p linear equations in p unknowns. For example, if n = 6 and p = 3, then we may be interested in solving the six equations in three unknowns

$\begin{array}{rrr}\hfill x+y& \hfill & \hfill =1\\ \hfill x& \hfill +z& \hfill =1\\ \hfill & \hfill y+z& \hfill =1\\ \hfill & \hfill y& \hfill =1\\ \hfill & \hfill & \hfill z=1\\ \hfill & \hfill & \hfill z=1\end{array}$

Such systems of equations are usually inconsistent if the number of equations exceeds the number of unknowns. This feature is revealed in our example by the fact that the fourth equation sets y equal to one, the fifth sets z equal to one, and the third requires that they should also sum to one. These are clearly contradictory requirements which cannot be satisfied simultaneously.

Nevertheless, this impasse may be resolved if we are willing to assume that the equations are not exact but inexact equalities whose right sides may be adjusted in such a way as to yield a system of equations which is consistent. This assumption and several others are embodied in the algebraic and statistical models presented in Chapters 3 and 9, which have been accepted as the basis of much quantitative analysis in a wide range of disciplines, including astronomy, physics, chemistry, medicine, psychology, geodesy, engineering, sociology, economics, and education.

The adjustments to the system of n equations in p unknowns are usually made indirectly by solving a system of p equations in p unknowns which is closely related to the original system. For example, if w...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Table of Contents
• PREFACE
• 1 THE GAUSS AND GAUSS-JORDAN METHODS: ASPECTS OF COMPUTER PROGRAMMING
• 2 MATRIX ANALYSIS OF GAUSS’S METHOD: THE CHOLESKY AND DOOLITTLE DECOMPOSITIONS
• 3 THE LINEAR ALGEBRAIC MODEL: THE METHOD OF AVERAGES AND THE METHOD OF LEAST SQUARES
• 4 THE CAUCHY-BIENAYMÉ, LAPLACE, AND SCHMIDT PROCEDURES “
• 5 HOUSEHOLDER’S PROCEDURE
• 6 GIVENS’S PROCEDURE
• 7 UPDATING THE QU DECOMPOSITION
• 8 PSEUDO-RANDOM NUMBERS
• 9 THE STANDARD LINEAR MODEL
• 10 CONDITION NUMBERS
• 11 INSTRUMENTAL VARIABLE ESTIMATORS
• 12 GENERALIZED LEAST SQUARES ESTIMATION
• 13 ITERATIVE SOLUTIONS OF LINEAR AND NONLINEAR LEAST SQUARES PROBLEMS
• 14 CANONICAL EXPRESSIONS FOR THE LEAST SQUARES ESTIMATORS AND TEST STATISTICS
• 15 TRADITIONAL EXPRESSIONS FOR THE LEAST SQUARES UPDATING FORMULAS AND TEST STATISTICS
• 16 LEAST SQUARES ESTIMATION SUBJECT TO LINEAR EQUALITY CONSTRAINTS
• SUPPLEMENTARY READING
• GLOSSARY OF MATRIX ALGEBRA
• AUTHOR INDEX
• SUBJECT INDEX