Sparse Matrix Multiplication Speed in Eigen - c++

I am using Sparse Matrices in Eigen and I observe the following behavior:
I have the following Sparse Matrices with Column Major storage
A [1,766,548 x 3,079,008] with 105,808,194 non-zero elements and
B [3,079,008 x 1,766,548] with 9,476,108 non-zero elements
When I compute the dot product AxB in takes almost 8 seconds.
When I want to compute transpose(Β) x transpose(A), the computation cost seems to increase a lot. In fact, this runs for about ~2,500 seconds.
Note that I load the transposed tables from files and I don't transpose them with Eigen.
I didn't expect the two approaches to have exactly the same computational cost but I don't really understand such a difference in execution time as in both approaches the two matrices have exactly the same number of non-zero elements.
I am using g++ 7.4 and Eigen 3.3.7

Related

Eigen: Modify Rows of Row-Major Sparse Matrix

I am using the Eigen library in C++ for solving sparse linear equations: Ax=b where, A is a square sparse matrix and b is a dense vector with ILU-preconditioned BiCGSTAB. I am initializing the matrix A using the setFromTriplets function. The linear system is generated from discretization of partial differential equations in space and time.
My application changes the matrix slightly at every time-step. I want to modify a small number of rows (around 1% rows) in the matrix in the beginning of each time-step. I am storing the matrix in the row-major format so that I can access the row directly. I don't want to re-assemble the entire matrix from triplets since the number of rows to be modified are around 1%. Moreover, the modification is such that the number of non-zeros in the row are exactly identical. I just want to change the column indices and values. Hence, I do not need to allocate extra memory for the row. After going through the Eigen documentation as well as the forum, I found the functions coeffRef and insert. Both of them will allocate extra memory if the element does not exist. I would like to avoid this since the number of non-zeros are not changing.
Any help is appreciated.

Right function for computing a limited number of eigenvectors of a complex symmetric matrix in Armadillo

I am using the Armadillo library to manually port a piece of Matlab code. The matlab code uses the eigs() function to find a small number (~3) of eigen vectors of a relative large(200x200) covariance matrix R. The code looks like this:
[E,D] = eigs(R,3,"lm");
In Armadillo there are two functions eigs_sym() and eigs_gen() however the former only support real symmetric matrix and the latter requires ARPACK (I'm building the code for Android). Is there a reason eigs_sym doesn't support complex matrices? Is there any other way to find the eigenvectors of a complex symmetric matrix?
The eigs_sym() and eigs_gen() functions (where the s in eigs stands for sparse) in Armadillo are for large sparse matrices. A "large" size in this context is roughly 5000x5000 or larger.
Your R matrix has a size of 200x200. This is very small by current standards. It would be much faster to simply use the dense eigendecomposition eig_sym() or eig_gen() functions to get all the eigenvalues / eigenvectors, followed by extracting a subset of them using submatrix operations like .tail_cols()
Have you tested constructing a 400x400 real symmetric matrix by replacing each complex value, a+bi, by a 2x2 matrix [a,b;-b,a] (alternatively using a block variant of this)?
This should construct a real symmetric matrix that in some way correspond to the complex one.
There will be a slow-down due to the larger size, and all eigenvalues will be duplicated (which may slow down the algorithm), but it seems straightforward to test.

Determinant Value For Very Large Matrix

I have a very large square matrix of order around 100000 and I want to know whether the determinant value is zero or not for that matrix.
What can be the fastest way to know that ?
I have to implement that in C++
Assuming you are trying to determine if the matrix is non-singular you may want to look here:
https://math.stackexchange.com/questions/595/what-is-the-most-efficient-way-to-determine-if-a-matrix-is-invertible
As mentioned in the comments its best to use some sort of BLAS library that will do this for you such as Boost::uBLAS.
Usually, matrices of that size are extremely sparse. Use row and column reordering algorithms to concentrate the entries near the diagonal and then use a QR decomposition or LU decomposition. The product of the diagonal entries of the second factor is - up to a sign in the QR case - the determinant. This may still be too ill-conditioned, the best result for rank is obtained by performing a singular value decomposition. However, SVD is more expensive.
There is a property that if any two rows are equal or one row is a constant multiple of another row we can say that determinant of that matrix is zero.It is applicable to columns as well.
From my knowledge your application doesnt need to calculate determinant but the rank of matrix is sufficient to check if system of equations have non-trivial solution : -
Rank of Matrix

Partitioned Matrix-Vector Multiplication

Given a very sparse nxn matrix A with nnz(A) non-zeros, and a dense nxn matrix B. I would like to compute the matrix product AxB. Since n is very large, if carried out naively, the dense matrix B cannot be put into the memory. I have the following two options, but not sure which one is better. Could you give some suggestions. Thanks.
Option1. I parition the matrix B into n column vectors [b1,b2,...,bn]. Then, I can put matrix A and any single vector bi into the memory, and calculate the A*b1, A*b2, ..., A*bn, respectively.
Option2. I partition the matrices A and B, respectively, into four n/2Xn/2 blocks, and then use the block matrix-matrix multiplications to calculate A*B.
Which of the above choice is better? Can I say that Option 1 has high performance in parallel calculation?
See a discussion of both approaches, though for two dense matrices, in this document from the Scalapack documentation. Scalapack is the one of the reference tools for distributed linear algebra.

Sparse Blas in Fortran 95

I want to use the Sparse Blas in Fortran95 just for the creation of the matrices and I am using the point entry construction. After creation of the matrix using the command
call duscr_begin(n,n,a,istat)
here a is the handle to the matrix n by n. After inserting value in it, how can I see the final matrix using its handles a ? As I want to use the matrix for some other operation, so I want to see the matrix in three vectors (sparse) form (row_index, Col_index, Value).
detail about this Sparse Blas is given in Chapter 3 and can be seen here
http://www.netlib.org/blas/blast-forum/
actually what i have asked is before 16 days and it is not just writing of a variable to thee screen. I was using some library known as Sparse Blas for creation of the Sparse matrices. Later on by digging in to the library i found the solution to my problem that using the handles how can we get the three vectors row, col and Val. The commands are something like
call accessdata_dsp(mat,a_handle,ierr)
call get_infoa(mat%INFOA,'n',nnz,ierr)
allocate(K0_row(nnz),K0_col(nnz),K0_A(nnz))
K0_row=mat%IA1; K0_col=mat%IA2; K0_A=mat%A
so here nnz are the non zeros entries in the sparse matrix while K0_row, K0_col and K0_A are our required three vectors, which can be used in further calculation.