I wish to solve exactly this formula involving sums and products ⟨ n l ⟩...

Eden Solomon

Eden Solomon

Answered question


I wish to solve exactly this formula involving sums and products
n l = [ 1 + k l ( e b N ( l k ) j l ( 1 e b ( l j ) ) j k ( 1 e b ( k j ) ) ) ] 1 ( N + h l 1 1 e b ( h l ) ) h l { [ 1 + k h ( e b N ( h k ) j h ( 1 e b ( h j ) ) j k ( 1 e b ( k j ) ) ) ] 1 ( 1 1 e b ( l h ) ) }
where b is a positive real number, N is a natural number, and all the sums and products are understood to run from 0 to M 1, where M is yet another natural number (all the l, h, j, k indices are therefore bound to this interval, with the appropriate restrictions written under the sums and the products). Eventually I would also like to take the limit as M goes to infinity, but I'm sure how well I can do that.
Do you think there is a hope to massage this formula to make it a little less ugly? For the moment, I've been trying to solve it numerically...

Answer & Explanation

Ryan Fitzgerald

Ryan Fitzgerald

Beginner2022-06-20Added 17 answers

Step 1
If I understand correctly your need, you are searching for a compact representation of the term n l .
A possible way to deal with the product terms is to use log. Consider the Toeplitz matrix B with elements B i j = 1 e b ( i j ) and compute the vector v = log ( B + I M ) 1 M .
Step 2
You will discover that p k = j k ( 1 e b ( k j ) ) can be computed as the k-th element of exp v.
Note here log and exp are computed elementwise and log returns complex values in case of negative values. So it holds p k = e k T log ( B + I M ) 1 M

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