HP HP-15C

1 06 Section 4: Using Matrix Operations
is far from the true value
-2 X 1(T40 3 -1
3 -4 X 1040 2 X 1040
-1 2 X1040 -1040
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
105 0
0 105
1 2
l(r30 1(T30
ID'30 -1(T30
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 -4X1030 2X1030
-1 2 X 1030 -1030