pracma (version 1.9.9)

horner: Horner's Rule

Description

Compute the value of a polynomial via Horner's Rule.

Usage

horner(p, x)
hornerdefl(p, x)

Arguments

p

Numeric vector representing a polynomial.

x

Numeric scalar, vector or matrix at which to evaluate the polynomial.

Value

horner returns a list with two elements, list(y=..., dy=...) where the first list elements returns the values of the polynomial, the second the values of its derivative at the point(s) x.hornerdefl returns a list list(y=..., dy=...) where q represents a polynomial, see above.

Details

horner utilizes the Horner scheme to evaluate the polynomial and its first derivative at the same time.

The polynomial p = p_1*x^n + p_2*x^{n-1} + ... + p_n*x + p_{n+1} is hereby represented by the vector p_1, p_2, ..., p_n, p_{n+1}, i.e. from highest to lowest coefficient.

hornerdefl uses a similar approach to return the value of p at x and a polynomial q that satisfies

p(t) = q(t) * (t - x) + r, r constant

which implies r=0 if x is a root of p. This will allow for a repeated root finding of polynomials.

References

Quarteroni, A., and Saleri, F. (2006) Scientific Computing with Matlab and Octave. Second Edition, Springer-Verlag, Berlin Heidelberg.

See Also

polyval

Examples

x <- c(-2, -1, 0, 1, 2)
p <- c(1, 0, 1)  # polynomial x^2 + x, derivative 2*x
horner(p, x)$y   #=>  5  2  1  2  5
horner(p, x)$dy  #=> -4 -2  0  2  4

p <- Poly(c(1, 2, 3))  # roots 1, 2, 3
hornerdefl(p, 3)          # q = x^2- 3 x + 2  with roots 1, 2