HP -15C Advanced Functions Handbook page 110

108
Section 4: Using Matrix Operations
with integer elements with magnitudes less than 100. ||r
r
E|| will be
small compared with ||r
T
|| ||E||—the smaller the better.
Next, choose the kih element of r
T
having one of the largest
magnitudes. Replace the kth row of E by r
r
E and the kih row of B
by r
r
B. Provided that no roundoff has occurred during the
evaluation of these new rows, the new system matrix A should be
better conditioned (farther from singular) than E was, but the
system will still have the same solution X as before.
This process works best when E and A are both scaled so that
every row of E and of A have roughly the same norm as every
other. You can do this by multiplying the rows of the systems of
equations EX = B and AX — D by suitable powers of 10. If A is not
far enough from singular, though well scaled, repeat the
preconditioning process.
As an illustration of the preconditioning process, consider the
system EX = B, where
E =
X
y
y
y
y
y
X
y
y
y
y
y
X
y
y
y
y
y
X
y
y
y
y
y
X
,B =
1
0
0
0
0
and x = 8000.00002 and y = -1999.99998 . If you attempt to solve
this system directly, the HP-15C calculates the solution X and the
inverse E"
1
to be
X
2014.6
2014.6
2014.6
2014.6
2014.6
and E"
1
^ 2014.6
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1