Page 287 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

17 Solving Linear Equations

17.1 Introduction

Different methods can be used to solve the MME covered in the previous chapters.
These various methods could broadly be divided into three main categories:
1. Direct inversion (Section 17.2).
2. Iteration on the MME (Section 17.3).
3. Iteration on the data (Section 17.4).
The manner in which the MME are set up depends on the method to be used in solving
these equations. As shown in Section 17.4, the third method, for instance, does not
involve setting up the MME directly.

17.2 Direct Inversion

The solutions to the MME in the various examples given so far in this book have
been based on this method. It involves setting up the MME and inverting the
coefficient matrix. Solutions are obtained by multiplying the right-hand side
(RHS) by the inverse of the coefficient matrix. Thus b, the vector of solution, is
calculated as:
ˆ
b = C y
−1
where C is the coefficient matrix and y is the RHS. Since the coefficient matrix is
symmetric, only the upper triangular portion is usually set up and inverted. The
major limitation of this approach is that it can only be applied to small data sets in
view of the memory requirements and computational difficulties of inverting large
matrices.

17.3 Iteration on the Mixed Model Equations

This involves setting up the MME and iterating on these equations until conver-
gence is achieved at a predetermined criterion. The iterative procedures are based
on the general theory for solving simultaneous equations. For instance, given two
simultaneous equations with unknown parameters, b and b , the first equation
1 2
can be solved for b in terms of b . This value of b can then be substituted in the
1 2 1

© R.A. Mrode 2014. Linear Models for the Prediction of Animal Breeding Values, 271
3rd Edition (R.A. Mrode)