Biology Reference
In-Depth Information
We now outline the computational methods used to determine the least-
squares values of the parameters from the data.
III. A PRIMER FOR SOLVING NONLINEAR EQUATIONS
One way to use computers to solve nonlinear equations is by an
iterative process that assumes a function can be expanded as a Taylor series.
In one variable, this means that if we know the value of the function f (x)ata
point x
x
0
and we want to find the value of the function at another point x
that is close to x
0
,wecanusetheexpression
¼
f
00
ð
f
000
ð
x
0
Þ
2
x
0
Þ
2
3
f
0
ð
f
ð
x
Þ¼
f
ð
x
0
Þþ
x
0
Þð
x
x
0
Þþ
ð
x
x
0
Þ
þ
ð
x
x
0
Þ
þ:
3
!
!
What makes our technique work in most cases is that if our guess x
0
is
close to the value x we seek, then x
x
0
is small, and the sum of higher
order terms
f
00
ð
f
000
ð
x
0
Þ
x
0
Þ
2
3
ð
x
x
0
Þ
þ
ð
x
x
0
Þ
þ
2
3
!
!
will be negligible compared to f
ð
x
Þ¼
f
ð
x
0
Þþ
f
0
ð
x
0
Þð
x
x
0
Þ:
Thus:
f
0
ð
f
ð
x
Þ
f
ð
x
0
Þþ
x
0
Þð
x
x
0
Þ:
(8-16)
In two or more variables, the idea is similar. If we know the value
of f (x,y) at a point (x,y)
(x
0
, y
0
) and we want to find the value of the
function at another point (x,y) that is close to (x
0
,y
0
), we can use the
following expression:
¼
Þþ
@
f
ð
x
0
;
y
0
Þ
Þþ
@
f
ð
x
0
;
y
0
Þ
f
ð
x
;
y
Þ¼
f
ð
x
0
;
y
0
ð
x
x
0
ð
y
y
0
Þ
@
x
@
y
2
f
2
f
1
2
@
ð
x
0
;
y
0
Þ
1
2
@
ð
x
0
;
y
0
Þ
2
þ
ð
x
x
0
Þ
þ
ð
x
x
0
Þð
y
y
0
Þ
@
x
2
@
x
@
y
!
!
2
f
1
2
@
ð
x
0
;
y
0
Þ
2
þ
ð
y
y
0
Þ
þ
...
:
@
y
2
!
Like before, if x
x
0
and y
y
0
are both small, then the expression
Þþ
@
f
ð
x
0
;
y
0
Þ
Þþ
@
f
ð
x
0
;
y
0
Þ
f
ð
x
0
;
y
0
ð
x
x
0
ð
y
y
0
Þ
provides
@
x
@
y
a good approximation for f (x,y), and we write
y
0
Þþ
@
f
ð
x
0
;
y
0
Þ
x
0
Þþ
@
f
ð
x
0
;
y
0
Þ
f
ð
x
;
y
Þ
f
ð
x
0
;
ð
x
ð
y
y
0
Þ:
(8-17)
@
x
@
y
A. Newton's Method for One Variable
Suppose we have a function f (x) and want to find a point x* where
f (x*)
¼
0. We make an initial guess x
¼
x
0,
and then find the point where
