Efficient Implementation of the New Restricted Maximum Likelihood Algorithms

Recently tridiagonalizafion and diagonalization have been proposed as methods to speed the EM algorithm for variance component estimation in restricted maximum likelihood. These methods require approximately the same computing resources, but only if the most efficient strategies are employed. When eigenvectors are explicitly calculated in diagonalization, computing requirements more than double. To avoid this problem, eigenvectors can be calculated implicitly, but this increases storage needs by a quadratic amount. A second solution is not to evaluate eigenvectors explicitly or implicitly but rather to embed appropriate orthogonal transformations in the diagonalization process. gested that this method can be applied to the broad class of models described by Smith and Graser (15). Smith and Graser (15) stated that diagonalization is an alternative to tridiagonalization. Both methods greatly reduce the computing cost per round of iteration and the amount of computer resources required by each is roughly equivalent. The same sequence of iterates is obtained whether tridiagonalization or diagonalization is employed (9). Unfortunately, specific REML algorithms are likely to be numerically inefficient, and the published literature offers little guidance to programmers. The purpose of this note is to describe efficient ways to implement the new REML algorithms. The differences between tridiagonalization and diagonalization will also be highlighted so as to help researchers choose between these valuable tools.