Chemistry Reference
In-Depth Information
conjugate-gradient methods. In both cases we will only sketch the main con-
cepts. Resources that discuss the details of these methods are listed in the
Further Reading section at the end of the chapter.
The essence of quasi-Newton methods is to replace Eq. (3.15) by
A
1
x
iþ
1
¼ x
i
(
x
i
)
g
(
x
i
),
(3
:
16)
i
where
A
i
is a matrix that is defined to approximate the Jacobian matrix. This
matrix is also updated iteratively during the calculation and has the form
A
i
¼ A
i
1
þ F
[
x
i
,
x
i
1
,
G
(
x
i
),
g
(
x
i
1
)]
:
(3
:
17)
We have referred to quasi-Newton methods rather than the quasi-Newton
method because there are multiple definitions that can be used for the function
F
in this expression. The details of the function
F
are not central to our dis-
cussion, but you should note that this updating procedure now uses infor-
mation from the current and the previous iterations of the method. This is
different from all the methods we have introduced above, which only used
information from the current iteration to generate a new iterate. If you think
about this a little you will realize that the equations listed above only tell us
how to proceed once several iterations of the method have already been
made. Describing how to overcome this complication is beyond our scope
here, but it does mean than when using a quasi-Newton method, the conver-
gence of the method to a solution should really only be examined after
performing a minimum of four or five iterations.
The conjugate-gradient method to minimizing
E
(
x
) is based on an idea that
is quite different to the Newton-based methods. We will introduce these ideas
with a simple example: calculating the minima of a two-dimensional function
E
(
x
)
¼
3
x
2
þ y
2
, where
x ¼
(
x
,
y
). Hopefully, you can see right away what the
solution to this problem: the function has exactly one minimum and it lies at
x
(0,0). We will try to find this solution iteratively starting from
x
0
¼
(1,1). Since we are trying to minimize the function, it makes sense to
look in a direction that will cause the function to decrease. A basic result
from vector calculus is that the function
E
(
x
) decreases most rapidly along a
direction defined by the negative of the gradient of the function,
rE
(
x
).
This suggests we should generate a new estimate by defining
x
1
¼ x
0
a
0
rE
(
x
0
)
¼
(1
6
a
0
,1 2
a
0
)
:
(3
:
18)
This expression tells us to look along a particular line in space, but not where
on that line we should place our next iterate. Methods of this type are known as
