Ruffini-Horner Algorithm for Complex Arguments
Ruffini-Horner Algorithm for Complex Arguments
Suppose we need to calculate a value of the polynomial with real coefficients for the complex argument . We divide the polynomial by , where and . 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.
P(x)=++⋯+x+
a
n
n
x
a
n-1
n-1
x
a
1
a
0
z=u+iv
(x-z)(x-)=-2uz++=-2px-q
z
2
x
2
u
2
v
2
x
p=u
q=--
2
u
2
v
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 |