Eigenvalue for a big matrix

From the above question it is my understanding that, you want to compute the smallest 10 eigen values of a large sparse matrix in MATLAB, following could be the possible workaround for the same:

To minimize the memory usage the matrix "D" should be a sparse matrix and specify options tailored for large scale problems:

D = kron(E0,A) + kron(E1,B) + kron(E1.',B);  % D should be sparse
% Assuming D is your large sparse matrix
opts.tol = 1e-3; % Adjust tolerance to manage computation time and memory
opts.maxit = 300; % Limit the maximum number of iterations
opts.isreal = true; % Set based on your matrix, can affect performance
opts.issym = true; % If D is symmetric, this can significantly improve efficiency
% Find the smallest 10 eigenvalues
[EigVec, EigVal] = eigs(D, 10, 'smallestabs', opts);

Assuming you have the "Parallel Computing Toolbox", you can utilize multiple cores to speed up the computation:

% Enable parallel computing
parpool; % Initializes a parallel pool of workers
% Use 'eigs' with parallel options if applicable
[EigVec, EigVal] = eigs(D, 10, 'smallestabs', opts); % The same as before, MATLAB will automatically use available workers

Another possible workaround for this is to integrate external libraries in the following way:

ARPACK: For more control or different configurations, consider using ARPACK directly from C++ or Fortran and interfacing with MATLAB via MEX files.
SLEPc: Using SLEPc (a scalable library for eigenvalue problem computations) involves more setup. You can write a C or C++ program that uses SLEPc for the eigenvalue computations, then call this program from MATLAB using the system command or compile it as a MEX file to be called directly from MATLAB.

Please refer to the following code below for reference:

// A very rough pseudo-code for using an external library like SLEPc
#include <slepc.h>
int main(int argc, char **argv) {
    // Initialize SLEPc
    SlepcInitialize(&argc,&argv,(char*)0,help);
    // Your matrix setup and eigenvalue computation goes here
    // Finalize SLEPc
    SlepcFinalize();
    return 0;
}