Horner's method
horner.RdHorner's method for Clifford objects
Details
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. The output
of horner() depends on the signature().
Note
Horner's method is not as cool for Clifford objects as it is for
(e.g.) multivariate polynomials or freealg objects. This is
because powers of Clifford objects don't get more complicated as the
power increases.
Examples
horner(1+e(1:3)+e(2:3) , 1:6)
#> Element of a Clifford algebra, equal to
#> + 511 + 490e_1 + 502e_23 + 502e_123
rcliff() |> horner(1:4)
#> Element of a Clifford algebra, equal to
#> + 12160 - 3156e_1 + 10800e_13 - 13968e_24 + 2400e_124 + 14202e_1234 + 8904e_5 -
#> 5490e_25 + 1500e_125 + 6912e_135 + 2880e_1235 - 4800e_245 - 720e_345 -
#> 576e_2345 + 5400e_12345 - 11046e_16 + 9660e_36 - 1800e_136 - 1714e_236 +
#> 144e_46 - 1350e_146 + 864e_246 + 8400e_1246 - 128e_1346 + 7200e_2346 -
#> 2304e_12346 - 448e_56 + 4200e_156 + 5250e_1256 - 3600e_356 - 80e_1356 +
#> 4500e_2356 - 1440e_12356 + 2160e_1456 - 5736e_12456 - 2264e_3456 - 4608e_23456