Biomedical Engineering Reference
In-Depth Information
surface. For example, if we consider the function
f
(
x
,
y
)
=
x
2
+
y
2
which has a minimum
at
x
=
0,
y
=
0 we have
2
x
2
y
H
20
02
g
=
=
⎛
⎝
⎞
⎠
1
2
0
H
−
1
=
1
2
0
If we start from the point (2, 2) we calculate
⎛
⎞
1
2
2
2
4
4
0
⎝
⎠
X
(2)
=
−
1
2
0
which is a null vector.
In practice molecular potential energy surfaces are rarely quadratic and so an infinite
number of steps will be required. We simply stop the calculation once a given number of
decimal places have been achieved.
5.8.2 Block Diagonal Newton-Raphson
There are a number of variations on the Newton-Raphson method, many of which aim
to eliminate the need to calculate the full Hessian. A widely used algorithm is the
block
diagonal Newton-Raphson
method, where just one atom is moved at each iteration. This
means that all the elements of the Hessian are zero except for a block along the diagonal
describing the atom in question.
5.8.3 Quasi-Newton-Raphson
In addition, there are a number of so-called
quasi-Newton-Raphson
methods that gradually
build up the inverse Hessian in successive iterations. At each iteration, the vector
X
(
k
)
is
updated to
X
(
k
+
1)
−
H
(
k
)
−
1
g
(
k
)
using the gradient and the current estimate of the inverse Hessian. Having made the move to
the new position,
H
−
1
is updated from its value at the previous step by an approximate pro-
cedure depending on the algorithm employed. The methods of Davidon-Fletcher-Powell,
Broyden-Fletcher-Goldfarb-Shanno and Murtaugh-Sargent are often encountered, but
there are many others.
X
(
k
+
1)
=
X
(
k
)
5.8.4 Fletcher-Powell Algorithm
The Fletcher-Powell algorithm (Fletcher and Powell 1963) is a derivative method where
elements of the gradient and the Hessian are estimated numerically each cycle. Essentially,
it assumes that the surface is quadratic around each chosen point, and finds the minimum
for the approximate quadratic surface. Steps in the algorithm are as follows.
