Backward Error Analysis Of Matrix Inversion - HP -15C Advanced Functions Handbook

200 Appendix: Accuracy of Numerical Calculations
The procedure works impeccably on only certain machines like the
HP-15C, whose subtraction operation is free from avoidable error
and therefore enjoys the following property: Whenever y lies
between x/2 and 2x, the subtraction operation introduces no
roundoff error into the calculated value of x — y. Consequently,
whenever cancellation might leave relatively large errors contami-
nating a, b, or c, the pertinent difference (p — q) or (p — r) turns out
to be free from error, and then cancellation turns out to be
advantageous!
Cancellation remains troublesome on those other machines that
calculate (x + dx) — (y + 8y) instead of x — y even though neither
dx nor 8y amounts to as much as one unit in the last significant
digit carried in x or y respectively. Those machines deliver
Fc(P,Q,
r
) — f(P + 8p, q + dq, r + dr) with end-figure perturbations
8p, 5q, and 8r that always seem negligible from the viewpoint of
backward error analysis, but which can have disconcerting
consequences. For instance, only one of the triples (p,q,r) or
(p + 8p,q + dq,r + <5r), not both, might constitute the edge lengths
of a feasible triangle, so F
c
might produce an error message when
it shouldn't, or vice-versa, on those machines.
Backward Error Analysis of Matrix Inversion
The usual measure of the magnitude of a matrix X is a norm ||X||
such as is calculated by either | MATRIX 17 or | MATRIX| 8; we shall use
the former norm, the row norm
in what follows. This norm has properties similar to those of the
length of a vector and also the multiplicative property
When the equation Ax = b is solved numerically with a given n X n
matrix A and column vector b, the calculated solution is a column
vector c which satisfies nearly the same equation as does x,
namely
(A + <5A)c = b
with||5A||<Hr
9
n||A||.