Biomedical Engineering Reference
In-Depth Information
which I have shown as a full straight line on Figure 5.7. A little algebra shows that this line
crosses the
x
axis at
f
(1)
g
(1)
x
(2)
=
x
(1)
−
This crosses the
x
axis at
0.1883 and this value is taken as the next best estimate
x
(2)
of
the root. I then draw the tangent line at
x
(2)
, which is shown dotted in Figure 5.7, and so on.
Table 5.1 summarizes the iterations.
−
1.5
1
0.5
-1.5
-1
-0.5
0
0.5
1
1.5
-0.5
-1
-1.5
Figure 5.7
Newton-Raphson
Table 5.1
Newton-Raphson iterations
k
x
(
k
)
f
(
k
)
g
(
k
)
1
0.4
0.3409
0.5795
2
−
0.1882
−
0.1817
0.8968
3
1.4357
×
10
−2
1.4354
×
10
−2
0.9994
4
−
5.9204
×
10
−6
−
5.9204
×
10
−6
1.0000
5
0.0000
0.0000
1.0000
There are a few points worth noticing. First, the convergence is rapid. In fact, for a
quadratic function the Newton-Raphson method will locate the nearest root in just one step.
Second, the choice of starting point is crucial. If I start with
x
=±
0.5, the successive
estimates simply oscillate between
+
0.5 and
−
0.5. If I start with
|
x
|
< 0.5, then the method
converges to the root
x
> 0.5 then we find the infinite roots.
Themethod can be easilymodified to search for stationary points, which are characterized
by a zero gradient. The iteration formula
x
(
k
+
1)
=
0. If I start with
|
x
|
=
x
(
k
)
−
f
(
k
)
/
g
(
k
)
(5.5)
