horner {freealg}

Horner's method

Description

Horner's method for multivariate polynomials

Usage

horner(P,v)

Arguments

P

Free algebra polynomial

v

Numeric vector of coefficients

Details

This function is (almost) the same as mvp::horner().

Given a polynomial

\[p(x) = a_0 +a_1+a_2x^2+\cdots + a_nx^n\]

it is possible to express p(x) in the algebraically equivalent form

\[p(x) = a_0 + x\left(a_1+x\left(a_2+\cdots + x\left(a_{n-1} +xa_n \right)\cdots\right)\right)\]

which is much more efficient for evaluation, as it requires only n multiplications and n additions, and this is optimal. Function horner() will take a freealg object for its first argument.

Author(s)

Robin K. S. Hankin

Examples

horner("x",  1:4)  # note constant term is 1.

horner("x+y",1:3) # note presence of xy and yx terms

horner("1+x+xyX",1:3)