Preconditioning 1 - HP -15C Advanced Functions Handbook

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
10
40
2 X
10
40
-1 2 X
10
40
-10
40
= 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 U
T
will have the
property that ||u
r
|| / 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'
1
1 not much smaller than Hu^lHlE'
1
!, and
HE"
1
! will be large. Choose such a row vector u
r
and calculate
v
r
= au^E'
1
. Choose the scalar a so that the row vector r
r
,
obtained by rounding every element of V
T
to an integer between
-100 and 100, does not differ much from V
T
. Then r
r
is a row vector