Horner's
method is an efficient way
of evaluating polynomials and their derivatives at a given point. It is also
used for a compact presentation of the long division of a polynomial by a
linear polynomial.
A polynomial P_{n}(x)
can be written in the descending order of the powers of x:
P_{n}(x)
= a_{n}x^{n} + a_{n1}x^{n1} + ... + a_{1}x
+ a_{0}

or in the ascending
order of the exponents:
P_{n}(x)
= a_{0} + a_{1}x + ... + a_{n1}x^{n1} + a_{n}x^{n}.

The Horner scheme
computes the value P_{n}(z) of the polynomial P at x = z as the
sequence of steps starting with the leading coefficient a_{n} using:
b_{k} = z·b_{k+1} + a_{k}


c_{k} = z·c_{k+1} + b_{k}


The
next value of the x_{n} in the iterative step is found by,
x_{k+1
}= x_{k} – (b_{0}/c_{1}), where b_{0} = P_{n}(x_{k})
= b_{0} and c_{1} = P’_{n}(x_{k}) in shortcut.
Generally, the
iteration is continued until the required amount of precision is obtained.
Example: Solve
the equation x^{3}+9x^{2}18 using Horner’s Method where z= x_{0}
= 1.5.
Here, a_{0}=18,
a_{1}=0, a_{2}=9, and a_{3}=1.
Starting from a_{3},
b_{3}=c_{3}=a_{3}=1.
Hence,
b_{2} =
9+1x1.5 = 10.5 c_{2
}= 10.5+1x1.5 = 12
b_{1} =
0+10.5x1.5 = 15.75 c_{1}
= 15.75+12x1.5 = 33.75
b_{0} =
18+15.75x1.5 = 5.625_{ }
So, x_{1 }=
x_{0} – (b_{0}/c_{1}) = 1.5 – (5.625/33.75) = 1.333
Þ x_{1} = z = 1.333
Now, for the new
value of z,
b_{2} =
9 + 1x1.333 = 10.333 c_{1}
= 28.881
b_{1} =
0+10.333x1.333 = 13.774
b_{0} =
18 + 13.774x1.333 = 0.361
So, x_{2 }=
x_{1} – (b_{0}/c_{1}) = 1.333 – (0.361/28.881) = 1.320
Þ x_{2} = z = 1.320
Again, for new
value of z,
b_{0} =
0.018 c_{1} = 28.987
So, x_{3 }=
x_{2} – (b_{0}/c_{1}) = 1.320 – (0.018/28.987) = 1.320
Therefore,
Approximate Root = 1.320 correct to
3 decimal places.
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