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Ruffini-Horner Algorithm for Complex Arguments

​
n
2
3
4
5
6
z = u + v
u
-3
-2
-1
0
1
2
3
4
v
-3
-2
-1
1
2
3
4
new polynomial
steps
0
1
2
3
4
P(x)3
4
x
-
3
x
+2
2
x
-x+1
z
=
1+
Q(x)
2
x
-2x+2
p = 1
q = -2
3
-1
2
-1
1
-2
2
3
​
Suppose we need to calculate a value of the polynomial
P(x)=
a
n
n
x
+
a
n-1
n-1
x
+⋯+
a
1
x+
a
0
with real coefficients for the complex argument
z=u+iv
. We divide the polynomial by
(x-z)(x-
z
)=
2
x
-2uz+
2
u
+
2
v
=
2
x
-2px-q
, where
p=u
and
q=-
2
u
-
2
v
. The remainder is then a linear function and the value of the polynomial is the value of the remainder. In the table, that is the value at the bottom right.
The table is defined as follows, where the last row is the sum of the higher rows:
a
n
a
n-1
a
n-2
a
n-3
…
a
1
a
0
q
q
a
n
q
a
n
…
q
b
3
q
b
2
2p
2p
a
n
2p
b
n-1
2p
b
n-2
…
2p
b
2
b
n
=
a
n
b
b-1
b
n-2
b
n-3
…
b
1
b
0
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