Biology Reference
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and so on. In general, for any guess x
n
we make, the ''improved guess''
can be calculated from the formula
x
n
Þ
f
0
ð
f
ð
x
n
þ
1
¼
x
n
Þ
:
(8-21)
x
n
The process terminates when two successive iterations of the same value
are produced.
2
Example 8-3
.......................
e
x
Use Newton's method to solve x
þ
¼
5.
S
OLUTION
:
e
x
and then want to solve the equation
We denote f
ð
x
Þ¼
5
x
e
2
< 0, a root must lie between
f (x)
¼
0. Since f (0)
¼
4 and f
ð
2
Þ¼
5
2
1. Since f
0
ð
e
x
x
¼
0 and x
¼
2. We make an initial guess x
0
¼
x
Þ¼
1
;
using Eq. (19), we calculate
e
x
0
5
x
0
5
1
e
x
1
¼
x
0
¼
1
¼
1
:
344707
:
1
e
x
0
1
e
With this value for x
1
and Eq. (8-20), we calculate
e
x
1
5
x
1
x
2
¼
x
1
¼
1
:
307128
;
1
e
x
1
and so on. Applying Eq. (8-21) in this case gives
e
x
n
5
x
n
x
n
þ
1
¼
x
n
:
1
e
x
n1
The process terminates when x
n
þ
1
¼
x
n
. Table 8-4 presents the values
of the consecutive iterations. Thus, we have found (to five decimal
places) that x
Iteration i
Guess x
i
e
x
0
1
þ
¼
5whenx
¼
1.306558.
1
1.344707
Choosing a good initial guess can be critical to the success of Newton's
method and is usually based on the experimental data and the model.
Recall the model P
2
1.307128
3
1.306558
5.3 e
rt
and that the least-squares estimate
for the parameter r is the solution of Eq. (8-11). In Chapter 1, we
described a way of estimating the value of the parameter r from the data
and found that the best value should be close to r*
¼
G(r; t)
¼
4
1.306558
TABLE 8-4.
Values of iterations for example 8-3.
0.3 (see Table 1-3 in
Chapter 1 and the preceding text). We can now use this approximation
as our initial guess r
0
for Newton's method to determine the least-
squares value of r*. The results of the iterative process described by
¼
2. In practice, the process is terminated when two successive iterations become
closer than a small tolerance value (e.g., 0.00000001) chosen in advance.
