Eigenvalue for a big matrix

I have a matrix 24990001 x 24990001 and I want to calculate the smallest 10 eigenvalues of this matrix. But I get out of memory error. Is there a way to do that. (I have 16GB of Ram)

I used this code. There is no problem up to this code. It can form the matrix but cannot find the eigenvalues.

E=eigs(D,10,'smallestabs');

And my matrix is

N=5000;
a1=4;b1=-1; 
A =diag(a1*ones(1,N-1)) + diag(b1*ones(1,N-2),1) + diag(b1*ones(1,N-2),-1);
B=(-1)*eye(N-1); 
E0=speye(N);
E1=E0(2:end,1:end-1);
E0=E0(1:end-1,1:end-1);
D=kron(E0,A) + kron(E1,B)+kron(E1.',B);

[Matt J]

Not sure if this helps, but there appear to be some papers out there dealing with eigendecomposition of block tridiagonal matrices, e.g.,

https://arxiv.org/pdf/1306.0217.pdf

[Rena Berman]

(Answers Dev) Restored edit

Answer 1

[Maneet Kaur Bagga]

Hi,

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:

1. ARPACK: For more control or different configurations, consider using ARPACK directly from C++ or Fortran and interfacing with MATLAB via MEX files.
2. 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;
}

Hope this helps!