Evaluation of Polynomials

Evaluation Polynomials

PROBLEM

Compute the value of a given polynomial A.N*X**N+...+A.1*X+A.0 of degree N at a given point V.

ALGORITHM

A simple algorithm which avoids recomputation of the powers of X is known as Horner's rule. A degree N polynomial can be evaluated using only N-1 multiplications and N additions. This program assumes the array representation for polynomials.

IMPLEMENTATION

Unit: internal function

Global variables: vector A.0,A.1,...,A.N of coefficients from the set of real numbers

Parameters: a positive integer N - the degree of the polynomial, real number V - a given point

Returns: Evaluating the polynomial A. at the given point V

HORNER: procedure expose A.
parse arg N, V
P = A.N
do J = N - 1 to 0 by -1
P = P * V + A.J
end
return P

EXAMPLE

The value of a given polynomial 5*X**3+4*X-10 of degree N=3 at a given point V=8 is 2582. See the following fragment of program:

N = 3
A. = 0
A.3 = 5; A.1 = 4; A.0 = -10
V = 8
say HORNER(N, V)
exit
...

CONNECTIONS

Polynomial addition
Sparse polynomial addition
Evaluation of polynomial and its derivatives
Division of polynomials

Literature

Sedgewick R. Algorithms

Addison-Wesley, Reading, Massachusetts, 1984