Multiple starting points for phase determination are created by the magic integer method which is described
in [39].
The values of 
phases, 
, may be expressed in the form of a set of equations by multiplying a
variable 
by successive magic integers, 
in each equation,
for 
. Any set of phases with values between 
and 
may be approximately
represented in this way as a set of values ranging from and 
with the appropriate choice of ,
where also takes values between 0 and 1.
There will in general be a residual error, 
, in the representation
of which depends on the choice of magic integers [79] [80].
The r.m.s. error in the phase representation is 
.
A starting set of phases is produced by choosing several values of at 
intervals which permute
the phases such that the mean change in , as x is incremented, is approximately equal
to . The advantage of magic integer permutations over quadrant permutations (where each
phase takes one of the four values, 
, 
, 
or 
) is the large reduction in the
number of permutations required to efficiently sample the -phase space.
Suitable magic integers have been
found to be those of the Fibonacci series defined as (
), with 
.