download searchable PDF of Circuit Design book - IEEE Global ...

-- -~---
Conjugate Gradient Search 133
the columns in matrix P; then (5.43) defines the substitutions
Using these in (5.22) produces
x,=y'+Y2'
x2= -y, +Y2'
Q(Y) = 72y, + 32Y2'
(5.45)
(5.46)
(5.47)
so that the cross terms are indeed removed, and the minimum could be found
in no more than two linear searches.
It is straightforward to show that changes in the gradient vectors of a
quadratic function are mapped by a constant linear transformation to the
corresponding changes in the variable vectors. As Figure 5.18 illustrates,
points A and B in the x space have gradient values (perpendicular to their
level curve), and these gradient vectors can be plotted in their own space.
There may be more than one x with the same g. Apply the gradient expression
(5.13) of a quadratic function to points x; and X;+1 and their corresponding
gradients ri and ri+'; the two equations may be subtracted to yield
(ri+1-g;)=A(x;+I-xl (5.48)
Using tJ. to indicate the differences and inverting (5.48), the mapping result is
tJ.x=A-'tJ.g. (5.49)
This result was anticipated by Newton's step in (5.34), which went to a
minimum where ri+ 1=0 was required. The importance of (5.49) is that it
shows the invariance of that mapping, independent of locations on any
quadratic surface.
Finally, it is shown that the altitude above the minimum value of a
quadratic surface at some point p is equal to a quadratic form composed of
the gradient at the point in question, g(p), and the inverse of its constant
lal
Ihl
Figure S.lS. A mapping of variable space to gradient space. (a) Constant objective function
curves in the variable space; (b) corresponding loci and points in the gradient space. [From
Davidon, 1959.] .