Horner's Method
Horner's Method
Horner's method for computing a polynomial both reduces the number of multiplications and results in greater numerical stability by potentially avoiding the subtraction of large numbers. It is based on successive factorization to eliminate powers of greater than 1.
x
Suppose ; then the method rewrites .
P(x)=++⋯+
a
n
n
x
a
n-
1
n-1
x
a
0
P(x)=((x+)x+⋯)x+
a
n
a
n-1
a
0
To compute , we find
P(u)
b
n-1
a
n
b
n-2
a
n-1
b
n-1
…,
b
0
a
1
b
1
b
-1
a
0
b
0
P(u)=
b
-1
The factor polynomial is given by .
Q(x)=P(x)/(x-u)=++...++=Q(x)+
b
n-1
n-1
x
b
n-2
n-2
x
b
0
b
-1
x-u
b
-1
x-u
You can select , the degree of the polynomial; , the desired value at ; and , the number of steps in which is replaced by . The table shows the new coefficients after each step.
n
u
u
k
x
u
External Links
External Links
Permanent Citation
Permanent Citation
Izidor Hafner
"Horner's Method"
http://demonstrations.wolfram.com/HornersMethod/
Wolfram Demonstrations Project
Published: January 8, 2016