Newton's iteration is an algorithm for computing the square root 
of a positive real number 
via the recurrence
equation
 |
(1)
|
where 
.
For 
,
the recurrence equation converges to 
with quadratic convergence
order. For 
,
every iterate equals 1.
Newton's iteration is simply an application of Newton's method for solving the equation
 |
(2)
|
For example, when applied numerically, the first few iterations to Pythagoras's constant 
are 1, 1.5, 1.41667, 1.41422, 1.41421, ....
The first few approximants 
, 
, ... to 
are given by
 |
(3)
|
For nonnegative integer 
, these are given by the analytic formula
 |  |  |
(4)
|
For 
,
this also equals 
,
using the principal value of the inverse
hyperbolic tangent. The formula can be derived by noting that, for 
, the recurrence equation
can be written as
 |
(5)
|
which has the closed-form solution
 |
(6)
|
Solving for 
then gives the solution derived above.
The following table summarizes the first few iterates for small positive integer 
, 
, ...