How can I implement recurrence relations to find polynomials?

Hello everyone. I need to calculate some polynomials using a recurrence rule. I have the values of the polynomials of order zero and one. These are 1 and x. I also have the rule:

Where is the polynomial of order I and both, and are elements of two different vectors that I have stored in memory, so it won’t be difficult to access them. I also need to find the polynomials of order zero to four

Unfortunately, I don't know how to send this to Matlab to find the values of the different polynomials. I can try to do it by hand, but in the future if i need to calculate higher polynomials I will be in trouble.

I have tried to use a matrix to store the values as a matrix, e.g. the first row would be 0 0 0 1 and the second 0 0 x 0 and so on and a for loop to access said elements and then use the recurrence rule. However, I haven't managed to obtain any results.

Can someone please tell me how this could be done and how could I find said polynomials?

Regards.

Jaime.

Answer 1

[Alan Stevens]

Do you mean something along these lines:

order = 4;
a = [0, 0, 1.3, 1.4, 1.5];     % Obviously, replace with your own values.
b = [0, 3.5, 4.5, 5.5, 6.5];   % The initial zeros are simply placeholders
                               % they are never used here. For higher order 
                               % polynomials a and b will need to be 
                               % correspondingly bigger.

x = 0:0.1:5;
p = zeros(numel(x),1);

for i = 1:numel(x)
    p(i) = RPOLY(x(i), order, a, b);
end

plot(x,p),grid

function P = RPOLY(x, order, a, b)
         
         PR = zeros(order+1,1);
         PR(1) = 1;
         PR(2) = x;
         if order == 0
             P = PR(1);
         elseif order == 1
             P = PR(2);
         else
             for j = 3:order+1
                 PR(j) = ((x-a(j))*PR(j-1) - b(j-1)*PR(j-2))/b(j);
             end
             P = PR(order+1);
         end
         
end

[Jaime De La Mota Sanchis]

Hi. First of all, thanks for your answer, but I am afraid this isn't what I am looking for.

I have thwo polynomials, and and the recurrence rule which allows me to calculate the rest of the polynomials e.g.

I want to find the coefficients of each polynomial; in the case of , the coefficients are 0 for , , and x, but is 1 for the independent term. The same way, in the case of the coefficients are 0 for , , and the independent term, but it is 1 for x. My problem is that I don't know how to calculate the different polynomials of order 2 and higher using an automated method in matlab.

[Alan Stevens]

This expression

doesn't match the general expression

which, for P2(x), would include b1 in the numerator and b2 in the denominator, rather than b0 in both.

[Jaime De La Mota Sanchis]

You are right. I have fixed the error. Thanks.

[Alan Stevens]

Still doesn't match! The general expression, if correct, would give

Note the indices on the a and the b's.