Wynn's rho algorithm is an algorithm for convergence improvement by rational function extrapolation
(Wynn 1956, Weniger 1989). Let 
be the data at distinct increasing real nodes 
with 
. Starting from 
and 
, form, for 
, 1, ...,
 |
wherever the denominator is nonzero. For the standard choice 
, the numerator
is 
. This numerator distinguishes
the recurrence from Wynn's epsilon method,
whose corresponding numerator is 1. The even entries
are the extrapolated limits; the odd entries are auxiliary
values.
The table entries are reciprocal differences of the pairs 
. When the rational interpolation problem is
nonsingular, 
is the value at infinity of the rational
function whose numerator and denominator
have polynomial degrees at most 
interpolating the 
data values 
, ..., 
. For a positive
integer 
, the method is exact for data of the form
 |
with 
, provided the data and table entries required
by the recurrence are well defined (Weniger 1989). This interpretation relates the
algorithm to Thiele's interpolation formula.
It is especially suited to algebraically decaying errors, but does not accelerate
every convergent sequence.
