Lecture 3
Newton’s Method and Loops
Solving equations numerically
For the next few lectures we will focus on the problem of solving an equation:
f
(
x
) = 0
.
(3.1)
As you learned in calculus, the final step in many optimization problems is to solve an equation of
this form where
f
is the derivative of a function,
F
, that you want to maximize or minimize. In real
engineering problems the function you wish to optimize can come from a large variety of sources,
including formulas, solutions of differential equations, experiments, or simulations.
Newton iterations
We will denote an actual solution of equation (3.1) by
x
∗
. There are three methods which you may
have discussed in Calculus: the bisection method, the secant method and Newton’s method.
All
three depend on beginning close (in some sense) to an actual solution
x
∗
.
Recall Newton’s method. You should know that the basis for Newton’s method is approximation of
a function by it linearization at a point, i.e.
f
(
x
)
≈
f
(
x
0
) +
f
′
(
x
0
)(
x

x
0
)
.
(3.2)
Since we wish to find
x
so that
f
(
x
) = 0, set the left hand side (
f
(
x
)) of this approximation equal
to 0 and solve for
x
to obtain:
x
≈
x
0

f
(
x
0
)
f
′
(
x
0
)
.
(3.3)
We begin the method with the initial guess
x
0
, which we hope is fairly close to
x
∗
. Then we define
a sequence of points
{
x
0
,x
1
,x
2
,x
3
,...
}
from the formula:
x
i
+1
=
x
i

f
(
x
i
)
f
′
(
x
i
)
,
(3.4)
which comes from (3.3). If
f
(
x
) is reasonably wellbehaved near
x
∗
and
x
0
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '08
 Young,T
 Calculus, Equations, Derivative, Newton’s method, Rootfinding algorithm

Click to edit the document details