HP -15C Advanced Functions Handbook page 108

1 06 Section 4: Using Matrix Operations
is far from the true value
-2 X 1(T
40
3 -1
3 -4 X 10
40
2 X 10
40
-1 2 X10
40
-10
40
Multiplying the calculated inverse and the original matrix verifies
that the calculated inverse is poor.
The trouble is that E is badly scaled. A well-scaled matrix, like A,
has all its rows and columns comparable in norm and the same
must hold true for its inverse. The rows and columns of E are about
as comparable in norm as those of A, but the first row and column
of E"
1
are small in norm compared with the others. Therefore, to
achieve better numerical results, the rows and columns of E should
be scaled before the matrix is inverted. This means that the
diagonal matrices L and R discussed earlier should be chosen to
make LER and (LER)'
1
= R^E'
1
!/
1
not so badly scaled.
In general, you can't know the true inverse of matrix E in advance.
So the detection of bad scaling in E and the choice of scaling
matrices L and R must be based on E and the calculated E"
1
. The
calculated E"
1
shows poor scaling and might suggest trying
10"
5
0
0
Using these scaling matrices,
3X10"
10
LER-
0 0
10
5
0
0 10
5
1 2
l(r
30
1(T
30
ID'
30
-1(T
30
which is still poorly scaled, but not so poorly that the HP-15C can't
cope. The calculated inverse is
(LER)
-2 X10"
30
3 -1
3 -4X10
30
2X10
3 0
-1 2 X 10
30
-10
30