Page 104 - Compact Numerical Methods For Computers

The Choleski decomposition 93
Using the right-hand side
5937938
2046485
T
B y = 3526413
1130177
5003
the forward- and back-substitution algorithm 8 computes a solution

-0·046192435
1·019386565
x = -0·159822924
-0·290376225
207·7826146
This is to be compared with solution (a) of table 3.2 or the first solution of
example 4.2 (which is on pp 62 and 63), which shows that the various methods all
give essentially the same solution under the assumption that none of the singular
values is zero. This is despite the fact that precautions such as subtracting means
have been ignored. This is one of the most annoying aspects of numerical
computation-the foolhardy often get the right answer! To underline, let us use
T T
the above data (that is B B and B y) in the Gauss elimination method, algorithms
5 and 6. If a Data General NOVA operating in 23-bit binary arithmetic is used,
the largest integer which can be represented exactly is
23
2 – 1 = 8388607
so that the original matrix of coefficients cannot be represented exactly. However,
T
the solution found by this method, which ignores the symmetry of B B, is
-4·62306E-2
1·01966
x =
-0·159942
-0·288716
207·426
While this is not as close as solution (a) of table 3.2 to the solutions computed in
comparatively double-length arithmetic on the Hewlett-Packard 9830, it retains
the character of these solutions and would probably be adequate for many
practitioners. The real advantage of caution in computation is not, in my opinion,
that one gets better answers but that the answers obtained are known not to be
unnecessarily in error.