The rate of convergence of a sequence describes how quickly its terms approach a limit. For a sequence 
whose errors 
are eventually nonzero, a common definition of
order 
is
 |
where 
.
This is called 
-order
to distinguish it from definitions based on bounds for the errors. Order 1 with 
gives linear convergence, while
orders 2 and 3 give quadratic and cubic convergence. Superlinear convergence means
,
even when no order 
exists in this definition.
For example, 
converges linearly to 0 with 
, while 
converges quadratically to 0 with 
. Near a simple root, Newton's
method applied to a twice continuously differentiable function
has an error bounded by a constant times the square
of the preceding error. This accounts for its characteristic quadratic convergence
when the leading error coefficient is nonzero.
