Page 78 - Compact Numerical Methods For Computers

Some comments on the formation of the cross-products matrix 67
where is the mean of the jth column of the m by n matrix A. Furthermore, the
right-hand side of the nth normal equation is
(5.9)

This permits x n to be eliminated by using the nth normal equation

(5.10)

or
(5.11)

When this expression is substituted into the normal equations, the kth equation
(note carefully the bars above the symbols) becomes

(5.12)

But since
(5.13)
and
(5.14)
equation (5.12) becomes

(5.15)

which defines a set of normal equations of order (n - 1)
T T
(A' ) A' x' = (A') b' (5.16)
where A' is formed from the (n – 1) non-constant columns of A each adjusted by
subtraction of the mean and where b' is formed from b by subtraction of . x' is
simply x without x .
n
Besides reducing the order of the problem, less information is lost in the
T T
formation of (A') A' than A A, since the possible addition of large numbers to the
matrix is avoided. These large numbers have subsequently to be subtracted from
each other in the solution process, and this subtraction leads to digit cancellation
which one should always seek to avoid.
As an example, consider the calculation of the variance of the column vector

(5.17)