Page 244 - Compact Numerical Methods For Computers

Left-overs 231
required of the Hessian in §15.2. This behaviour is repeated in the early iterations
of case (c) above.
In conclusion, then, this problem presents several of the main difficulties which
may arise in function minimisation:

(i) it is highly nonlinear;
(ii) there are alternative optima; and
(iii) there is a possible scaling instability in that parameters 3 and 4 (v and w)
take values in the range 200-2000, whereas parameters 1 and 2 (t and p) are in
the range l-10.
These are problems which affect the choice and outcome of minimisation proce-
dures. The discussion leaves unanswered all questions concerning the reliability of
the model or the difficulty of incorporating other parameters, for instance to take
account of advertising or competition, which will undoubtedly cause the function
to be more difficult to minimise.

Example 18.2. Market equilibrium and the nonlinear equations that result
In example 12.3 the reconciliation of the market equations for supply
a
q=Kp
and demand

has given rise to a pair of nonlinear equations. It has been my experience that
such systems are less common than minimisation problems, unless the latter are
solved by zeroing the partial derivatives simultaneously, a practice which gener-
ally makes work and sometimes trouble. One’s clients have to be trained to
present a problem in its crude form. Therefore, I have not given any special
method in this monograph for simultaneous nonlinear equations, which can be
written
f (b)=0 (12.5)

preferring to solve them via the minimisation of
T
f f =S(b) (12.4)
which is a nonlinear least-squares problem. This does have a drawback, however,
in that the problem has in some sense been ‘squared’, and criticisms of the same
kind as have been made in chapter 5 against the formation of the sum-of-squares
and cross-products matrix for linear least-squares problems can be made against
solving nonlinear equations as nonlinear least-squares problems. Nonetheless, a
compact nonlinear-equation code will have to await the time and energy for its
development. For the present problem we can create the residuals

a
f = q- Kp
1
f =ln(q)-ln(Z)+bln(p).
2