Newton-MVT - 0.1 Newton's method and the Mean Value Theorem...

0.1 Newton’s method and the Mean Value
Theorem
Newton’s method for computing the zeros of functions is a good example of
the practical application of the Mean Value Theorem. Let f ( x ) be a real-
valued function on the real line that has two continuous derivatives. We
are looking for a root of f , i.e., a point ˆ x such that f (ˆ x ) = 0. In Newton’s
method, which is geometrical, we consider the curve y = f ( x ). Then the
curve crosses the x -axis at the point (ˆ x,f (ˆ x )). Let x
0
be an initial guess
for the root. To improve on the guess we construct the tangent line to the
curve y = f ( x ) that passes through the point ( x
0
,f ( x
0
)) on the curve. This
tangent line satisfies the equation
y − f ( x
0
) = f
′
( x
0
)( x − x
0
) .
The tangent line crosses the x -axis at the point
x
1
= x
0
−
f ( x
0
)
f ′ ( x
0
)
,
and we take x
1
as our improved estimate of the root ˆ x . Now we repeat this
procedure with x
1
to get an improved estimate x
2
, and so on. Thus we have
a sequence { x
n
} such that
x
n +1
= x
n
−
f ( x
n
)
f ′ ( x
n
)
, n = 0 , 1 , · · · .
We need to give conditions that will guarantee that the sequence will converge
to a root of f ( x ), and will provide information about the rate of convergence.
To analyze this procedure we define an updating function T ( x ) by
T ( x ) = x −
f ( x )
f
′
( x )
.
We will not yet fix the domain D of this function, but it is clear that we
must require f ′ ( x ) negationslash = 0 for all x ∈ D . Then ˆ x will be a fixed point of T ,
( T (ˆ x ) = ˆ x ) if and only if f (ˆ x ) = 0. To get the growth rate for the iteration