HP HP-15C

Appendix: Accuracy of Numerical Calculations 201
Consequently the residual b — Ac = (5A)c is always relatively
small; quite often the residual norm ||b — Ac|| is smaller than
|| b — Ax || 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 HA"1! is relatively large compared with 1/||A||;
where <r(A) = 1/(||A|| HA"1!!) is the reciprocal of the condition
number and measures how relatively near to A is the nearest
singular matrix S, since
min ||A-S|| = a(A)||A||.
det(S)=0
These relations and some of their consequences are discussed
extensively in section 4.
The calculation of A~l is more complicated. Each column of the
calculated inverse 1 1/x|(A) is the corresponding column of some
(A + 5A)"1, but each column has its own small <5A. Consequently,
no single small 5A, with ||<5A|| < l(T9n ||A||, need exist satisfying
roughly. Usually such a <5A exists, but not always. This does not
violate the prior assertion that the matrix operations 1 1/x| and Q
lie in Level 2; they are covered by the second assertion of the
summary on page 194. The accuracy of | l/x|(A) can be described in
terms of the inverses of all matrices A + AA so near A that
||AA||s£10~9n||A||; the worst among those (A + AA)'1 is at least
about as far from A"1 in norm as the calculated 1 1 /x | ( A). The figure
below illustrates the situation.
A + AA is in here
(A + AA) 1 is in here
1/x|(A) is in here