Abstract

Variances of errors of prediction and accuracies of solutions are functions of diagonal elements of the inverse of a coefficient matrix. Computing this inverse is expensive for large systems. Upper and lower bounds for diagonal elements of the inverse can be obtained from simple functions of elements of the original coefficient matrix. For a sire coefficient matrix of the form Z'MZ + lk, a particular sire's accuracy lies between two formulas that are functions of the appropriate diagonal element of Z'MZ and also the sum of squares of offdiagonal elements in that same row. Numerically, a sire whose true accuracy is .767 might be shown to have an accuracy between .762 and .774 with use of these bounds. Bounds are proved by partitioned matrices and positive definite quadratic forms. The method is demonstrated for only a simple model, but these procedures may provide a general approach for dealing with more complex models.