For a general function it may be difficult to find a
solution to
. However, if
is a linear
function, then we may solve for
as
.
Newton's method is a way to solve for by approximating
by a linear function.
If is differentiable, then for any point
we may
approximate
by the tangent line to the graph of
at
. That is,
So, if the tangent line is a good approximation to between
and a zero of
, then we may find an approximate zero
by solving for
giving
.
(The attribution of this method to
Raphson is
incorrect.)
Given: A differentiable function .
Goal: Find a solution (or an approximate solution) to .
Process:
Step 0: Guess an approximate zero and set
.
Step 1: Compute . If
is close enough to
zero for you, then set
and quit.
Step 2: Compute . If
, then
this method fails. Go back to step 0.
Step 3: Approximate
.
Set the approximation equal to zero and solve for
finding
. If this is the
iteration of this process, call this number
. Reset
and return to step 1.
(More on Newton's method.)
Find a solution to
.
We compute
.
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(Graphs)
Use Newton's method to approximate
.
Let
. (So that
.) We wish
to find
with
.
Start with . Then
. We compute
.
The next approximation is
while
.
Find a solution to .
Here
so that
. If we start with
, then
,
,
,
,
.
(Note: There is another zero: !)