HP -15C Advanced Functions Handbook page 205

Appendix: Accuracy of Numerical Calculations
203
and
50,000 50,000 p q
0 0.00002 50,000.03 48,076.98077...
0 0 50,000 48,076.95192...
0 0 0 0.00001923076923...
Ideally, p = q = 0, but the HP-lSC's approximation to X
1
, namely
(T/x](X), has q = 9,643.269231 instead, a relative error
= 0.0964...,
llx-
1
!
nearly 10 percent. On the other hand, if X + AX differs from X only
in its second column where -50,000 and 50,000 are replaced
respectively by -50,000.000002 and 49,999.999998 (altered in the
llth significant digit), then (X + AX)"
1
differs significantly from
X'
1
only insofar as p — 0 and q = 0 must be replaced by p =
10,000.00600... and q = 9,615.396154.... Hence,
= 0.196...;
the relative error in (X + AX)
1
is nearly twice that in 11/x|(X). Do
not try to calculate (X + AX)"
1
directly, but use instead the formula
(X - cb
7
)-
1
= X-
1
+ X-WX-
1
/ (1 - b^X^c),
which is valid for any column vector c and row vector b
T
, and
specifically for
andb
r
= [0 0.000002 0 0] .
Despite that
it can be shown that no very small end-figure perturbation <5X
exists for which (X + 5X)'
1
matches 1 1/x|(X) to more than five
significant digits in norm.