Eigenvalues of an integral operators
 
  
 
  
 
 
 
 
 
 
    
        
        asked 2 years ago  
 
 Olivier R. gravatar image  
 
 
 
 
    
        
        updated 2 years ago  
 
 Max Alekseyev gravatar image  
 
 
 
 
 
Dear all, 
  I want to find the first eigenvalues (and even the eigenvectors if possible) of a difference operator
 
G(y)=∫10V(t−y)F(t)dt
 
The function V is given by some infinite series and is easily computable to any decent degree of accuracy.
This is a very classical problem, but I wouldn't want to rediscover the wheel if there is a shop that sells some :)
  Many thanks in advance, Olivier
 
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Do you mean G(F)(y)? Or what is F?
Max Alekseyev gravatar imageMax Alekseyev (
    
        2 years ago
    )
Yes, G(F)(y).
A basic answer consists in approximating the integral. Wiith Simpson's formula, this gives
def GetEigenvalues (bigK):
    ak = [1, 4] + [2 for k in range(3-1, bigK-1)] + [4, 1]
    coefs = [0] + [ Wstar(k/bigK) / 3/bigK for k in range(1, bigK + 1)]
    myMat = matrix([[ ak[k]*coefs[abs(k-ell) + 1]
                                  for ell in range(1, bigK + 1)]
                                for k in range(1, bigK + 1)])
   myEigen = myMat.eigenvalues()
   myOrderedEigen = sorted(myEigen, key = lambda x: -abs(x))
   return(myOrderedEigen)
where Wstar(y) is my kernel function (which is even and vanishes at 0). This gives an idea, but the above has no control of approximation.
I would like to find something more refined.
Best, Olivier
Olivier R. gravatar imageOlivier R. (
    
        2 years ago
    )
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