HP 15c - Manual (Page 168)

168 Appendix: Accuracy of Numerical Calculations
168
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 >7 or >8; we shall use the former norm, the row norm
 
j
ij
x
i
max X
in what follows. This norm has properties similar to those of the length of a vector and also
the multiplicative property
||XY|| ≤ ||X|| ||Y|| .
When the equation Ax = b is solved numerically with a given n × 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 + δA) c = b
with ||δA|| < 10
-9
n ||A||.
Consequently the residual b − Ac = (δA)c is always relatively small; quite often the residual
norm ||b − Ac|| smaller than x A b
 where x is obtained from the true solution x by
rounding each of its elements to 10 significant digits. Consequently, c can differ significantly
from x only if A is nearly singular, or equivalently only if ||A
-1
|| is relatively large compared
with 1/||A||;
||x − c|| = ||A
-1
(b − Ac)||
≤ ||A
-1
|| ||δA|| ||c||
≤ 10
-9
n ||c|| / σ(A)
where σ(A) = 1/(||A|| ||A -1
||) is the reciprocal of the condition number and measures how
relatively near to A is the nearest singular matrix S, since
.
σ
0 ) det(
min A (A) S A
S
 

These relations and some of their consequences are discussed extensively in section 4.
The calculation of A
-1
is more complicated. Each column of the calculated inverse ⁄(A) is
the corresponding column of some (A+δA)
-1
, but each column has its own small δA.
Consequently, no single small δA, with ||δA||≤10
-9
n ||A||, need exist satisfying
||(A+ δA)
-1 − ⁄ (A)|| ≤ 10 -9
||⁄ (A)||
roughly. Usually such a δA exists, but not always. This does not violate the prior assertion
that the matrix operations ⁄and ÷ lie in Level 2; they are covered by the second
assertion of the summary on page 162. The accuracy of ⁄ (A) can be described in terms of
the inverses of all matrices A + ΔA so near A that ||ΔA|| ≤ 10
-9
n||A||; the worst among those