- #1
AiRAVATA
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I know that there is a theorem to calculate the coefficients for a multinomial expansion, but I'm having a hard time implementig the algorithm. What I need to know is if my procedure is correct:
[tex](A + B + C + D)^n = \sum_{i=0}^n \dbinom{n}{i} A^{n-i} (B + C + D)^i[/tex]
[tex]= \sum_{i=0}^n \dbinom{n}{i} A^{n-i} \sum_{j=0}^i \dbinom{i}{j} B^{i-j} (C + D)^j[/tex]
[tex]= \sum_{i=0}^n \dbinom{n}{i} A^{n-i} \sum_{j=0}^i \dbinom{i}{j} B^{i-j} \sum_{k=0}^j \dbinom{j}{k} C^{j-k} D^k,[/tex]
so
[tex](A + B + C + D)^n = \sum_{i=0}^n \sum_{j=0}^{i} \sum_{k=0}^{j}\dfrac{n!}{(n-i)!(i-j)!(j-k)!k!} A^{n-i} B^{i-j} C^{j-k} D^k [/tex]
Is that correct?
If so, I'm trying to compute the coefficients in Matlab in the following way
so the vector coef contains the coefficients of the polynomial. What do you think? Is my approach correct or am I doing something wrong?
[tex](A + B + C + D)^n = \sum_{i=0}^n \dbinom{n}{i} A^{n-i} (B + C + D)^i[/tex]
[tex]= \sum_{i=0}^n \dbinom{n}{i} A^{n-i} \sum_{j=0}^i \dbinom{i}{j} B^{i-j} (C + D)^j[/tex]
[tex]= \sum_{i=0}^n \dbinom{n}{i} A^{n-i} \sum_{j=0}^i \dbinom{i}{j} B^{i-j} \sum_{k=0}^j \dbinom{j}{k} C^{j-k} D^k,[/tex]
so
[tex](A + B + C + D)^n = \sum_{i=0}^n \sum_{j=0}^{i} \sum_{k=0}^{j}\dfrac{n!}{(n-i)!(i-j)!(j-k)!k!} A^{n-i} B^{i-j} C^{j-k} D^k [/tex]
Is that correct?
If so, I'm trying to compute the coefficients in Matlab in the following way
Code:
h = 1;
for i = 1:n
for j = 1:i
for k = 1:j
coef(h,1) = factorial(n)/(factorial(n-i)*factorial(i-j)*factorial(j-k)*factorial(k))
h = h+1;
end
end
end
so the vector coef contains the coefficients of the polynomial. What do you think? Is my approach correct or am I doing something wrong?