I have a matrix problem in Matlab.
I have a 1million x 1million sparse matrix, and I keep using null. Usually, the problem is that I run out of memory. I tried svds (which is used for svd for sparse matrix), but my issue is I run out of memory as well. Is there a possible work around for large sparse matrices for the null() function in Matlab?
In general, the null space of a matrix, or the unitary matrices (U and V) of the singular values decomposition are NOT sparse even if the input matrix is sparse. Therefore, if you try to work with a 1M-by-1M matrix, even though it is sparse, the outputs of your operations are NOT and therefore you run out of memory.
What can you do?
If your input matrix has a certain structure to it (in addition to its sparseness) you might find some algebraic method to take advantage of this structure.
Another path you should consider, is why you need to compute the null space of the matrix? Can you achieve the same goal without explicitly estimating the null space?
Related
I have a 40x43 matrix and I would like to use this matrix a building block to generate larger matrix.
I want to generate a structure like the image attached and the building block is the 40x43 matrix. I tried using [A zeros(20,43); zeros(20,43) A] but as I had guessed, the horzcat did't work. I would ideally like to use this block 1000 times to extend the structure of matrix. Could anyone tell me an efficient way to concatenate the small matrix?
Try using kron. This performs what is known as the Kronecker product such that for two matrices A and B, the result is:
In this case, we can replicate what you want exactly by setting A to be the identity matrix of size 1000 x 1000 and B to be the matrix you want to replicate. However, to promote computational savings and memory usage, make sure you use the sparse version of the identity matrix. This will convert the output matrix to sparse form. If you want to replicate this 1000 times, you are creating a 40000 x 43000 matrix and this requires 13.76 GB of memory and you probably don't have enough memory available for this matrix. Since most of the elements are zero, use the sparse version instead:
N = 1000;
B = kron(speye(N), A);
I have to perform this operation:
N = A'*P*A
The structure of the P matrix is block diagonal while the A matrix is largely sparse (also in a banded structure). The multiplication is performed in blocks. But the problem is storage.
The N matrix is too huge to store in full (out of memory when trying to allocate). So, I want to store in a sparse fashion. While the sparse command generates only the values in row,column format, can it be applied to store banded matrices with the row column as the index of the block?
I have tried spalloc given in the this question but it hasnt helped storing the row and index of the block.
Thank you.
Image for A P A' formation
The problem lies in the blocks. The blocks are themselves sparse. So is it possible to make blocks as sparse matrices themselves while saving.
So, if a block has a row = 1 and col = 1, then can this be done?
N(row,col) = sparse(A'*P*A)
There may be some additional tricks to play but the first thing to try is to make sure the full matrix N is never created in memory. The immediate problem is that if you call sparse(A'*P*A) then you multiple A'*P then (A'*P)*A and only then do you make it sparse and take out the zeros. Right before making it sparse, the entire non-sparse matrix representation of N is in memory. To force MATLAB to be smarter do the following:
SA = sparse(A);
N = SA'*sparse(P)*SA;
whos N
You should see that N is sparse but, more importantly, each multiplication result is sparse as well because you are multiplying a sparse matrix times a sparse matrix.
Over the course of finite difference discretization of an elliptic equation
and application of Neumann BCs on all sides of the 2D domain I have a large
sparse matrix. I need to find null space of its transpose to enforce
consistency condition on either side. For a computational domain of 50X50
and 100X100 I am able to do this with the available RAM of 32 GB in
Mathematica and MATLAB with the available full matrix commands of NullSpace and null respectively.
If the computational domain is 500X250 (which is of
the order of what I would have in general) the RAM required for the storage of the
matrix of size (500X250)X(500X250) is 125 GB and is highly prohibitive. I use sparse matrices to store this super-matrix and I don't have have the space constraint anymore. But I can't use the 'null" command on this as it is for only full matrices. MATLAB suggests to use "SVDS" command on sparse matrices. SVDS(A) gives only the first 6 singular values and singular vectors. There is another command SVDS(A,k,sigma) which gives "k" singular values and vectors around the scalar singular value of "sigma". When I use sigma=0 so as to find the singular vector corresponding to the the "zero" value, which would be the vector from null space basis I get an error that the "sigma" is too close to the eigen value of the matrix.
My matrix is singular and hence one of its Eigen values is "zero". How do I circumvent this error? Or Is there a better way to do this with the tools at hand. I have both MATLAB and Mathematica at hand.
Thanks in advance for your help and suggestions.
Best
Trinath
I guess you can try with some sort of decomposition.
http://www.mathworks.co.uk/matlabcentral/fileexchange/11120-null-space-of-a-sparse-matrix
Have you tried this?
Or maybe this?
http://www.mathworks.co.uk/matlabcentral/newsreader/view_thread/249467
I am confident they should work, but I have not tried them myself. Another way of proceeding would be to get into QR decomposition (which would give you the permutation of the first k independent columns, if k is the rank of your matrix. Then the vectors from k+1 to n would provide a basis for your null space).
Hope this helps.
Cheers,
GL
I have a sparse matrix in Matlab 43916x43916, which is calculated by this equation:
B=(speye(nV,nV)-alpha*NeMatrix+beta*NeMatrix*NeMatrix);
being nVa int, alphaa int, NeMatrix a sparse matrix and beta a int.
I can't do inv(B) because it increases the use of RAM till it crashes. I've tried LU already with no success.
How can I alternatively calculate this inverse matrix?
The inverse will be a dense matrix. Thus you should check, whether you can handle a matrix of this size. Try, e.g., to set up ones(nV,nV) ...
If you have enough storage, you may consider to compute the inverse column wise. The i-th column would be B\ei, where ei is the i-th unit vector.
HOWEVER, in numerical computations you hardly ever need the inverse of a matrix B. Most times B\v is enough, where v is a vector. So you better check, whether you really need the full inverse...
I have a big (400K*400K) sparse matrix and I need to calculate the largest eigenvalue of A'*A.
The problem is that Matlab can't even calculate A' due to memory problems.
I also tried [a,b,c] = find(A) and then transpose by creating a transpose sparse matrix, but although the find() works, the sprase creation doesn't.
Is there a nice solution for this? it can be either in a matlab function or in another technique to calculate the largest eigenvalue for this kind of multiplication.
Thanks.
If A is sparse, see this thread and some discussion in this documentation (basically do it part by part) for a way to transpose it etc.
But now you need to calculate B=A'*A. The question is, is it still sparse? assuming it is, there shouldn't be a problem to proceed using the previous technique mentioned in the link.
Then after you've obtained B=A'*A, use eigs
eigs(B,1)
to obtain the largest magnitude eigenvalue.