Suppose integers
n
⩾
1
n \geqslant 1
and
σ
⩾
2
\sigma \geqslant 2
are given, together with n distinct points
z
2
,
…
,
z
n
{z_2}, \ldots ,{z_n}
, in the complex plane. Define
Φ
M
=
Φ
M
(
σ
;
z
1
,
…
,
z
n
)
{\Phi _M} = {\Phi _M}(\sigma ;{z_1}, \ldots ,{z_n})
to be the class of rational functions
ϕ
p
,
q
(
z
)
=
g
p
(
z
)
/
h
q
(
z
)
{\phi _{p,q}}(z) = {g_p}(z)/{h_q}(z)
(where g and h are polynomials of degree
p
⩾
1
p \geqslant 1
and
q
⩾
1
q \geqslant 1
, respectively) such that
(
1
)
p
+
q
+
1
=
M
,
(
2
)
ϕ
(1)\;p + q + 1 = M,(2)\;\phi
when iterated converges with order
σ
\sigma
at each
z
i
,
i
=
1
,
…
,
n
{z_i}, i = 1, \ldots ,n
. Then if
M
>
σ
n
,
Φ
M
M > \sigma n,{\Phi _M}
is null; if
M
=
σ
n
M = \sigma n
Φ
M
{\Phi _M}
contains exactly
σ
n
\sigma n
elements. For every
M
⩾
σ
n
M \geqslant \sigma n
, we show how to construct all the elements of
Φ
M
{\Phi _M}
by expressing, for each choice of p and q which satisfies
p
+
q
+
1
=
M
p + q + 1 = M
, the coefficients of
g
p
{g_p}
and
h
q
{h_q}
in terms of
M
−
σ
n
M - \sigma n
arbitrarily chosen values. In fact, these coefficients are expressed in terms of generalized Newton sums
S
n
j
,
k
=
S
n
j
,
k
(
z
1
,
…
,
z
n
)
S_n^{j,k} = S_n^{j,k}({z_1}, \ldots ,{z_n})
,
1
⩽
j
⩽
n
,
k
⩾
n
1 \leqslant j \leqslant n,k \geqslant n
, which we show may be calculated by recursion from the normal Newton sums
S
n
j
,
n
S_n^{j,n}
. Hence, given a polynomial
f
n
(
z
)
{f_n}(z)
with n distinct (unknown) zeros
z
1
,
…
,
z
n
{z_1}, \ldots ,{z_n}
, we may construct all
ϕ
p
,
q
(
z
)
{\phi _{p,q}}(z)
which converge to the
z
i
{z_i}
with order
σ
\sigma
in the case
σ
=
2
\sigma = 2
, the choice
p
=
n
p = n
,
q
=
n
−
1
q = n - 1
, yields the Newton-Raphson iteration
ϕ
n
,
n
−
1
∈
Φ
2
n
{\phi _{n,n - 1}} \in {\Phi _{2n}}
; the Schröder and König iterations are shown to be elements of
Φ
2
(
2
σ
−
3
)
(
n
−
1
)
+
2
{\Phi _{2(2\sigma - 3)(n - 1) + 2}}
and
Φ
2
(
σ
−
1
)
(
n
−
1
)
+
2
{\Phi _{2(\sigma - 1)(n - 1) + 2}}
, respectively. We show by example that there exist cases in which
ϕ
n
,
n
−
1
{\phi _{n,n - 1}}
has an undesirable property (attractive cycles) not shared by other iterating functions in the same class
Φ
2
n
{\Phi _{2n}}
.
