General Principles Of The Method Of Least Squares

by Dana P. Bartlett

• ISBN13: 9780486450797
• ISBN10: 0486450791
• Format: Hardcover
• Copyright: 2006-02-24
• Publisher: Dover Publications

Summary

This survey of the fundamental principles of the method of least squares includes everything essential to the solution of ordinary problems. Starting with an overview of general principles, the text proceeds to examine the adjustment of observations and the precision of observations, the computation of the precision measures, a survey of Gauss's method of substitution, more. 1915 edition.

Table of Contents

General Principles	1	(16)
Object of the Method of Least Squares
Errors
Constant Errors; Theoretical, Instrumental, Personal
Mistakes
Accidental Errors
Direct Observations; The Arithmetical Mean
Real Errors
Residuals
Weighted Observations
The General Mean
The Curve of Error
Laws of Errors of Observation
Derivation of the Equation of the Curve of Error
The Method of Least Squares
The Adjustment Of Observations	17	(20)
Indirect Observations
Rules for Forming the Normal Equations
Reduction of Equations to Weight Unity
Relation Between the Weight of an Observation and its Measure of Precision
Computation of Corrections
Significant Figures
Conditioned Observations
Special Cases
Empirical Formulas and Constants
Periodic Phenomena
The Logarithmic Solution
Reduction of Equations to the Linear Form
The Precision Of Observations	37	(10)
The Constant k
The Value of k in Terms of h
The Mean of the Errors, or Average Deviation
The Mean Error
The Probable Error
The Relations between μ, r, a.d., h, ρ, and p
Representation of μ, a.d., and r on the Curve of Error
Computation Of The Precision Measures	47	(36)
Direct Observations all of the Same Weight
Direct Observations, the Weights Not Being All Alike
Functions of Independent Observed Quantities
The Precision of Measurements
Functions of the Same Variables
Indirect Observations
First Method of Computing the Weights
Rule I
Second Method of Computing the Weights
Rule II
Third Method of Computing the Weights
Rule III
The Mean Error of an Observation
Observations of Unequal Weights
Conditioned Observations
Miscellaneous Theorems	83	(14)
The Distribution of Errors
The Rejection of Observations
Criterion for the Rejection of a Single Doubtful Observation
The Huge Error
Constant Errors
Combination of Determinations having Different Constant Errors
The Weighting of Observations
Special Laws of Error
Contradictory Observations
Gauss's Method Of Substitution	97
Checks on the Formation of the Normal Equations
The Reduced Normal Equations and the Elimination Equations
Checks on the Solution of the Normal Equations
Most Convenient Arrangement of the Computations
Application of the Checks
Solution of the Elimination Equations
The Weights of the Unknown Quantities
Gauss's Method Of Correlatives	111	(6)
Examples	117
Appendix. The Theory Of Probability
Definition; Simple Events
Compound Events
Dependent Events
Bibliography
Tables