Page 288 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

second equation to solve for b . The value of b is then substituted in the first
2 2
equation to calculate b . This is the principle upon which the iterative procedures
1
are based. In the iterative procedure, the above process is continued until the solu-
tions for the b’s are more or less the same in each round of iteration and the equa-
tions are said to have converged. There are various iterative procedures that can
be used, and some are described below.

17.3.1 Jacobi iteration

One of the simplest methods is Jacobi iteration or total step iteration.
Consider the following set of simultaneous equations:
c ⎡ 11 c 12 c ⎤ ⎡ 1 b ⎤ ⎡ y y ⎤
13
1
⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎢
⎢ c 21 c 22 c 23 ⎥ ⎢ 2 b ⎥ = y 2⎥
⎢ c ⎣ 31 c 32 c ⎥ ⎢ 3 b ⎦ ⎥ ⎣ y ⎢ 3⎦ ⎥
33⎦ ⎣
These equations can also be written as:
11 1 cb + cb = y
cb + 12 2 13 3 1
cb + 22 2 23 3 2
21 1 cb + cb = y
31 1 cb + cb = y
cb + 32 2 33 3 3
+
or as:
Cb = y (17.1)

The system of equations is rearranged so that the first is solved for b , the second for
1
b and the third for b . Thus:
2 3
r+ 1 (y - r r
( /c )
1 b = 1 11 1 cb - 13 3 )
12 2 c b
r+ 1 (y - cb - r (17.2)
r
(
2 b = 1 /c ) 2 2 21 1 c b )
23 3
22
r+ 1 (y - r r
3 cb -
3 b = (1 /c ) 31 1 c b )
33
32 2
The superscript r refers to the number of the round of iteration. In the first round of
iteration, r equals 1 and b to b could be set to zero or an assumed set of values that
1 3
are used to solve the equations to obtain a new set of solutions (b terms). The process
is continued until two successive sets of solutions are within previously defined allow-
able deviations and the equations are said to converge. One commonly used conver-
gence criterion is the sum of squares of differences between the current and previous
solutions divided by the sum of squares of the current solution. Once this is lower
−9
than a predetermined value, for instance 10 , the equations are considered to have
converged.
From the set of equations above, the solution for b was obtained by divid-
i
ing the adjusted RHS by the diagonal (a ). It is therefore mandatory that the
ii
diagonal element, often called the pivot element, is not zero. If a zero pivot ele-
ment is encountered during the iterative process, the row containing the zero
should be exchanged with a row below it in which the element in that column
is not zero. To avoid the problem of encountering a zero pivot element and
272 Chapter 17