We first note how the polynomial in the previous exercise
can be written in the nested multiplication method:
8 + x(−10 + x(7 + x(−2 + x(3 + 12x)))).
Using this representation, the following pseudocode procedure
(implementing Horner’s method) can be used to evaluate the
given polynomial.
procedure PolynomialEvaluation2
(n: nonnegative integer;
r,a0,a1,a2,. . .,an: real)
begin
value := an
for j := n - 1 down to 0 do
value := aj + r * value
end
The given pseudocode procedure implements Horner's method to evaluate a polynomial. It takes the following parameters:
- n: a nonnegative integer representing the degree of the polynomial
- r: a real number representing the value at which the polynomial needs to be evaluated
- a0, a1, a2, ..., an: real numbers representing the coefficients of the polynomial in decreasing order of degree
The procedure follows the steps of Horner's method, which is an efficient algorithm to evaluate a polynomial without repeatedly performing exponentiations. Here's a step-by-step breakdown of the procedure:
1. Set the variable "value" to the coefficient of the highest degree term (an).
2. Start a loop that iterates from n - 1 down to 0.
3. Inside the loop, update the value of "value" by multiplying it with r and adding the coefficient of the current term (aj).
4. Repeat steps 3 and 4 until the loop completes.
5. The final value of "value" will be the evaluation of the polynomial at the given value of r.
It is important to note that Horner's method is based on the factorization of the polynomial. The nested multiplication representation of the polynomial allows us to perform a series of multiplications and additions, which are more efficient computationally compared to multiple exponentiations.
By following this pseudocode procedure, you can evaluate the given polynomial using Horner's method. Simply provide the appropriate values for the parameters n, r, and the coefficients a0, a1, a2, ..., an.