Code Horner’s Method for Polynomial Evaluation - matlab

Question

I am trying to code Horner’s Method for Polynomial Evaluation but for some reason its not working for me and I'm not sure where I am getting it wrong.
These are the data I have:
nodes = [-2, -1, 1]
x = 2
c (coefficients) = [-3, 3, -1]
The code I have so far is:
function y = horner(x, nodes, c)
n = length(c);
y = c(1);
for i = 2:n
y = y * ((x - nodes(i - 1)) + c(i));
end
end
I am supposed to end up with a polynomial such as (−1)·(x+2)(x+1)+3·(x+2)−3·1 and if x =2 then I am supposed to get -3. But for some reason I don't know where I am going wrong.
Edit:
So I changed my code. I think it works but I am not sure:
function y = horner(x, nodes, c)
n = length(c);
y = c(n);
for k = n-1:-1:1
y = c(k) + y * (x - nodes((n - k) + 1));
end
end

Answer 1

This works:
function y = horner(x, nodes, c)
n = length(c);
y = 0;
for i = 1:n % We iterate over `c`
tmp = c(i);
for j = 1:i-1 % We iterate over the relevant elements of `nodes`
tmp *= x - nodes(j); % We multiply `c(i) * (x - nodes(1)) * (x -nodes(2)) * (x- nodes(3)) * ... * (x - nodes(i -1))
end
y += tmp; % We added each product to y
end
% Here `y` is as following:
% c(1) + c(2) * (x - nodes(1)) + c(3) * (x - nodes(1)) * (x - nodes(2)) + ... + c(n) * (x - nodes(1)) * ... * (x - nodes(n - 1))
end

Answer 2

(I'm sorry this isn't python but I don't know python)
In the case where we didn't have nodes, horner's method works like this:
p = c[n]
for i=n-1 .. 1
p = x*p + c[i]
for example for a quadratic (with coeffs a,b,c) this is
p = x*(x*a+b)+c
Note that if your language supports fma
fma(x,y,x) = x*y+z
then horner's method can be written
p = c[n]
for i=n-1 .. 1
p = fma( x, p, c[i])
When you do have nodes, the change is simple:
p = c[n]
for i=n-1 .. 1
p = (x-nodes[i])*p + c[i]
Or, using fma
p = c[n]
for i=n-1 .. 1
p = fma( (x-nodes[i]), p, c[i])
For the quadratic above this leads to
p = (x-nodes[1]*((x-nodes[2])*a+b)+c