A purification method for computing the latent columns of numerical matrices and some integrals of differential equations

Abstract

The emphasis is on practical arithmetical methods suitable for application to matrices of so many rows and columns that their determinants are intractable. The arithmetic begins from an arbitrary, or guessed, column of numbers. This may be regarded as a raw material composed of the desired constituent plus an unknown assortment of impurities. Although the impurities are not known, their behaviour when subjected to certain computing operations can be foreseen sufficiently to allow them to be cleaned away gradually. These operations depend on all the latent roots. There is an initial stage of groping-about to find roughly where the latent roots are located. This leads on to a final stage of rapid routine in which roughly known latent roots are used to purify the column associated with the desired root. It is shown that the cleaning operation spreads over a considerable region of the Argand diagram; so that very imperfect information is, nevertheless, often effective. The proof of the numerical result is by comparison of independent estimates, and not by theorems about infinitely many operations. A closely similar purification method was applied in 1910 to a wide class of problems (so-called jury problems) involving differential equations, such as Laplace’s, or the equation of plane stress. In the present paper the choice of cleaning agents for such purposes is made more systematic by employing Legendre’s principle of least squares.