In numerical analysis, the speed at
which a convergent sequence approaches
its limit is called the rate of
convergence. Although strictly speaking,
a limit does not give information about
any finite first part of the sequence,
this concept is of practical importance
if we deal with a sequence of successive
approximations for an iterative method,
as then typically fewer iterations are
needed to yield a useful approximation
if the rate of convergence is higher.
This may even make the difference
between needing ten or a million
iterations insignificant.
Similar concepts are used for
discretization methods. The solution of
the discretized problem converges to the
solution of the continuous problem as
the grid size goes to zero, and the
speed of convergence is one of the
factors of the efficiency of the method.
However, the terminology in this case is
different from the terminology for
iterative methods.
Series acceleration is a collection of
techniques for improving the rate of
convergence of a series discretization.
Such acceleration is commonly
accomplished with sequence
transformations.
Convergence speed for iterative methods 
= Basic definition =
Suppose that the sequence {xk} converges
to the number L.
We say that this sequence converges
linearly to L, if there exists a number
μ ∈ such that
The number μ is called the rate of
convergence.
If the sequence converges, and
varies from step to step with  for ,
then the sequence is said to converge
superlinearly.
varies from step to step with  for ,
then the sequence is said to converge
sublinearly.
If the sequence converges sublinearly
and additionally
then it is said the sequence {xk}
converges logarithmically to L.
The next definition is used to
distinguish superlinear rates of
convergence. We say that the sequence
converges with order q to L for q>1 if
In particular, convergence with order
q = 2 is called quadratic convergence,
q = 3 is called cubic convergence,
etc.
This is sometimes called Q-linear
convergence, Q-quadratic convergence,
etc., to distinguish it from the
definition below. The Q stands for
"quotient," because the definition uses
the quotient between two successive
terms.
= Extended definition =
The drawback of the above definitions is
that these do not catch some sequences
which still converge reasonably fast,
but whose "speed" is variable, such as
the sequence {bk} below. Therefore, the
definition of rate of convergence is
sometimes extended as follows.
Under the new definition, the sequence
{xk} converges with at least order q if
there exists a sequence {εk} such that
and the sequence {εk} converges to zero
with order q according to the above
"simple" definition. To distinguish it
from that definition, this is sometimes
called R-linear convergence, R-quadratic
convergence, etc..
= Examples =
Consider the following sequences:
The sequence {ak} converges linearly to
0 with rate 1/2. More generally, the
sequence Cμk converges linearly with
rate μ if |μ| 
