The Solution of ‘Ill‐Conditioned’ Linear Simultaneous Equations: Methods for Reducing the Influence of Experimental Errors

When linear simultaneous equations are ‘ill‐conditioned’, small variations in the (usually measured) quantities on the right‐hand sides of the equations are associated with large variations in the values of some of the unknowns. The basic reason for this is that the matrix of the coefficients has at least one small latent root. Associated with each small latent root λr is a quantity Ur, a linear combination of the quantities on the right‐hand sides of the original equations, and the uncertainty associated with the ‘ill‐condition’ is essentially uncertainty as to the values of the quantities Ur; the original equations can be replaced by another set of equations which is not ill‐conditioned but involves the Ur. General methods are used, but applied to two specific cases; in one of these there are two equal small latent roots of the matrix of coefficients and the necessary adjustments for this case are explained. Two alternative methods for determining the small latent roots of a matrix are discussed.