A fast scheme for evaluating a polynomial such as:

 



−
19
+
7
x
−
4

x

2


+
6

x

3





{\displaystyle -19+7x-4x^{2}+6x^{3}\,}


when

 



x
=
3



{\displaystyle x=3\;}

.
is to arrange the computation as follows:

 



(
(
(
(
0
)
x
+
6
)
x
+
(
−
4
)
)
x
+
7
)
x
+
(
−
19
)



{\displaystyle ((((0)x+6)x+(-4))x+7)x+(-19)\;}


And compute the result from the innermost brackets outwards as in this pseudocode:

coefficients := [-19, 7, -4, 6] # list coefficients of all x^0..x^n in order
x := 3
accumulator := 0
for i in length(coefficients) downto 1 do
    # Assumes 1-based indexing for arrays
    accumulator := ( accumulator * x ) + coefficients[i]
done
# accumulator now has the answer

Task Description

Create a routine that takes a list of coefficients of a polynomial in order of increasing powers of x; together with a value of x to compute its value at, and return the value of the polynomial at that value using Horner's rule.
Cf. Formal power series

 
>function horner (x,v) ...
$  n=cols(v); res=v{n};
$  loop 1 to n-1; res=res*x+v{n-#}; end;
$  return res
$endfunction
>v=[-19,7,-4,6]
 [ -19  7  -4  6 ]
>horner(2,v) // test Horner
 27
>evalpoly(2,v) // built-in Horner
 27
>horner(I,v) // complex values
 -15+1i
>horner(1±0.05,v) // interval values
 ~-10.9,-9.11~
>function p(x) &= sum(@v[k]*x^(k-1),k,1,4) // Symbolic Polynomial
                            3      2
                         6 x  - 4 x  + 7 x - 19
 