HP HP-15C - Loop Control With Counters: ISG and DSE

Section 4: Using Matrix Operations 1 07
This result is correct to 10 digits, although you wouldn't be
expected to know this. This result is verifiably correct in the sense
that using the calculated inverse,
(LERr^LER) = (LERXLER)-1 = I (the identity matrix)
to 10 digits.
Then E-1 is calculated as
-2 X10"40 3 -1
3 -4 X 1040 2 X 1040
-1 2 X 1040 -1040
= R(LERPL =
which is correct to 10 digits.
If (LER)"1 is verifiably poor, you can repeat the scaling, using
LER in place of E and using new scaling matrices suggested by
LER and the calculated (LER)'1.
You can also apply scaling to solving a system of equations, for
example EX = B, where E is poorly scaled. When solving for X,
replace the system EX = B by a system (LER)Y = LB to be solved
for Y. The diagonal scaling matrices L and R are chosen as before
to make the matrix LER well-scaled. After you calculate Y from
the new system, calculate the desired solution as X = RY.
Preconditioning
Preconditioning is another method by which you can replace a
difficult system, EX = B, by an easier one, AX = D, with the same
solution X.
Suppose that E is ill-conditioned (nearly singular). You can detect
this by calculating the inverse E'1 and observing that I/HE"1!) is
very small compared to ||E|| (or equivalently by a large condition
number -K"(E)). Then almost every row vector UT will have the
property that ||ur|| / Hu^E"1!! is also very small compared with ||E||,
where E"1 is the calculated inverse. This is because most row
vectors u^will have Hu^E'11 not much smaller than Hu^lHlE'1!, and
HE"1! will be large. Choose such a row vector ur and calculate
vr= au^E'1. Choose the scalar a so that the row vector rr,
obtained by rounding every element of VT to an integer between
-100 and 100, does not differ much from VT. Then rris a row vector