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Quadratic Fonm and Ellipsoids 123
x, x,
Figure 5.9. A saddle point for a function of two variables. [From Murray, W. (1972). Numerical
Methods for Unconstrained Optimization. New York: Academic. Reprinted with kind permission
from the Institute of Mathematics and Its Applications.]
Figure 5.8c. This is the general situation when the quadratic form in (5.21)
may be positive or negative for all x. This produces a saddle point, as
illustrated in Figure 5.9. A saddle point occurs in function (5.1) at point
(2, 1.99759808), as readily determined using Program A5-1 with 0.1 % displacements.
5,1.4. Taylor Series. The reader should recall Taylor series of real variables.
An expansion of a function about the point x= a is
y(x)=y(a)+y'(a)(x-a)+ i,Y"(a)(x-a)2+ ...
It is important to define the difference,
Ax=x-a,
so that (5.26) reads:
y(Llx) = y(a) + y'(a) Llx + h"(a) Llx 2 + i, y"'(a)Llx' + ...
(5.26)
(5.27)
(5.28)
Figure 5.10 shows the situation for the Taylor series representation of a real
variable. Notice the slope and the "neighborhood" at x=a, in which a
truncated Taylor series might be valid, i.e., when all derivative terms greater
than a certain order in (5.28) may be ignored. On the other hand, if the
y(x)
y(a)
__ -
y'(al
~"igure 5.10. Taylor series representation in x or.6.x about the point x=a.
x