稀疏对角矩阵求解器

Question

I want to solve, in MatLab, a linear system (corresponding to a PDE system of two equations written in finite difference scheme). The action of the system matrix (corresponding to one of the diffusive terms of the PDE system) reads, symbolically ( u is one of the unknown fields, n is the time step, j is the grid point):

and fully:

The above matrix has to be intended as A , where A*U^n+1 = B is the system. U contains the 'u' and the 'v' (second unknown field of the PDE system) alternatively: U = [u_1,v_1,u_2,v_2,...,u_J,v_J]. So far I have been filling this matrix using spdiags and diag in the following expensive way:

    E=zeros(2*J,1);

    E(1:2:2*J) = 1;
    E(2:2:2*J) = 0;

    Dvec=zeros(2*J,1);

        for i=3:2:2*J-3
                 Dvec(i)=D_11((i+1)/2);    
        end

        for i=4:2:2*J-2
                 Dvec(i)=D_21(i/2);
        end

    A = diag(Dvec)*spdiags([-E,-E,2*E,2*E,-E,-E],[-3,-2,-1,0,1,2],2*J,2*J)/(dx^2);`

and for the solution

[L,U]=lu(A);
 y = L\B; 
 U(:) =U\y; 

where B is the right hand side vector.

This is obviously unreasonably expensive because it needs to build a JxJ matrix, do a JxJ matrix multiplication, etc.

Then comes my question: is there a way to solve the system without passing MatLab a matrix, eg, by passing the vector Dvec or alternatively directly D_11 and D_22 ? This would spare me a lot of memory and processing time!

Answer 1

Matlab doesn't store sparse matrices as JxJ arrays but as lists of size O(J). See http://au.mathworks.com/help/matlab/math/constructing-sparse-matrices.html Since you are using the spdiags function to construct A, Matlab should already recognize A as sparse and you should indeed see such a list if you display A in console view.

For a tridiagonal matrix like yours, the L and U matrices should already be sparse.

So you just need to ensure that the \\ operator uses the appropriate sparse algorithm according to the rules in http://au.mathworks.com/help/matlab/ref/mldivide.html . It's not clear whether the vector B will already be considered sparse, but you could recast it as a diagonal matrix which should certainly be considered sparse.