Computing the diagonal elements and inverse of a large numerator relationship matrix

TLDR: In this article, a modification of Henderson's procedure for finding the diagonal elements of an L (or A) matrix which does not require that L or A be stored in memory is described.

Abstract: A numerator relationship matrix for a group of animals is, by definition, the matrix with the ijth off-diagonal element equal to the numerator of Wright's [1922] coefficient of relationship between the ith and jth animals and with the ith diagonal element equal to 1 + fi where fi is Wright's [1922] coefficient of inbreeding for the ith animal. The numerator relationship matrix, say A, can be computed recursively (see Emik and Terrill [1949]), and for most situations, inbreeding and relationship coefficients can be calculated with a computer more rapidly in this manner than by path coefficient methods (Wright [1922]). The exception to this is when the dimension of A is too large for it to be stored in computer memory. Then computation of A is exceedingly time consuming. In addition to its usefulness for obtaining inbreeding and relationship coefficients, the inverse of A is required for best linear unbiased prediction of breeding values (Henderson [1973]) but, in general, A is too large to invert by conventional means. Recently, however, Henderson [1976] has described methods for computing a lower triangular matrix Z, defined such that LL' = A, with the object of computing A` = (L')1(LX1. He discovered that A-1 can be found directly from a list of sires and dams and the diagonal elements of L. Since the latter are functions of the diagonal elements of A, A1 for a noninbred population can be computed without having to compute either A or L. However, for an inbred population, the diagonal elements of L (or A) must first be found and when L is too large to store in computer memory, this can be very time consuming if Henderson's computing formulas are used. The purpose of this paper is to describe a modification of Henderson's procedure for finding the diagonal elements of an L (or A) matrix which does not require that L (or A) be stored in memory. It is therefore possible to compute rapidly inbreeding coefficients or the inverse of a numerator relationship matrix for very large numbers of animals. For example, less than three minutes were required by an IBM 370/135 to compute the diagonal elements and the inverse of a numerator relationship matrix for 1000 animals. Use of this procedure