Mathematica Eigensystem Calculation Taking Too Long

[thatboi]

Hey all,
I am currently trying to find the Eigensystem of the following 5x5 matrix in Mathematica. Unfortunately, doing so takes an insanely long time (I cannot even call Eigensystem[p][[1,1]] in a reasonable amount of time). Does anyone have any advice? Also I cannot simply redefine new variables for each of the differences x_{i}-x_{j} because I will need to solve for each x_{i} individually in an upcoming calculation so I do not think we can use some clever algebraic combination of the x_{i}-x_{j} to recover the x_{i} individually.

p = ({
{0, 1/(x1 - x2)^3, 1/(x1 - x3)^3, 1/(x1 - x4)^3, 1/(x1 - x5)^3},
{1/(x1 - x2)^3, 0, 1/(x2 - x3)^3, 1/(x2 - x4)^3, 1/(x2 - x5)^3},
{1/(x1 - x3)^3, 1/(x2 - x3)^3, 0, 1/(x3 - x4)^3, 1/(x3 - x5)^3},
{1/(x1 - x4)^3, 1/(x2 - x4)^3, 1/(x3 - x4)^3, 0, 1/(x4 - x5)^3},
{1/(x1 - x5)^3, 1/(x2 - x5)^3, 1/(x3 - x5)^3, 1/(x4 - x5)^3, 0}
});

Thanks!

 

[jim mcnamara]

Have you considered Parallelize[] ? I think it is often used for the issue you describe -

 

[pasmith]

Are you trying to compute the eigenvalues/vectors symbolically? That is (a) far more memory intensive than a numerical computation (according to https://reference.wolfram.com/language/ref/Eigensystem.html the memory required increases faster than exponentially with the size of the matrix), and (b) probably not possible for 5x5 or larger matrices in general, since it is not possible to solve a generic polynomial of degree 5 or higher by radicals. (I think it's significant that none of the examples on the page I linked attempt symbolic/exact calculations on anything larger than a 4x4 matrix.)

What is the actual problem you are trying to solve for the $x_i$? It may be possible to solve it numerically without trying to find symbolic expressions for the eigenvalues/vectors of a 5x5 matrix.