Horner's Rule
Compute the value of a polynomial via Horner's Rule.
horner(p, x) hornerdefl(p, x)
p |
Numeric vector representing a polynomial. |
x |
Numeric scalar, vector or matrix at which to evaluate the polynomial. |
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.
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.
For fast evaluation, there is no error checking for p and x,
which both must be numerical vectors
(x can be a matrix in horner).
Quarteroni, A., and Saleri, F. (2006) Scientific Computing with Matlab and Octave. Second Edition, Springer-Verlag, Berlin Heidelberg.
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
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