Horner's method — horner • clifford

Horner's method for Clifford objects

Usage

horner(P,v)

Arguments

P: A Clifford object
v: Numeric vector of coefficients

Details

Given a polynomial

$p (x) = a_{0} + a_{1} + a_{2} x^{2} + \dots + a_{n} x^{n}$

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

$p (x) = a_{0} + x (a_{1} + x (a_{2} + \dots + x (a_{n - 1} + x a_{n}) \dots))$

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().

Author

Robin K. S. Hankin

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