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 .
(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: !)