MATLAB: How to optimize the multiplication of large matrices in Matlab

matrix multiplicationsparse

I have 2 matrices A, B, and vector C that I need to multiply. They are fairly large.

Matrix A = 10000×10000

Matrix B = 10000×10000

Vector C = 10000×1

If I perform A*B*C, this takes a long time so I used sparse function which collapses the matrices/vectors by removing large number of zeros then I convert it back to a full matrix.

full(sparse(A)*sparse(B)*sparse(C))

It's faster but I was wondering if there are more efficient techniques for multiplying them together. Would it be better to write for loops?

Secondly, some of the elements in my matrix have values close to zero so I can replace these with zeroes before converting them to sparse matrices. What's the best way to do this?

Best Answer

Force the matrix-vector multipy to happen first. E.g.,

A*(B*C)

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MATLAB: How to calculate the first derivative of a vector with respect to a matrix

syms a [1 2]
syms B [2 4]
syms C [4 4]
x = a*B*C;
derivs = arrayfun(@(X) reshape(gradient(X, B(:)),size(B)), x, 'uniform', 0);

MATLAB: I am trying to convert a dense matrix into a sparse matrix to speed up the matrix multiplies, but it is running slowly. How to fix this

The computational bottleneck in this code comes from converting an existing dense matrix into a sparse matrix. If the matrix has only a few non-zero entries, sparse matrix multiplication will indeed be much more efficient. However, there may be a large overhead cost in constructing the sparse matrix from the dense matrix that exists. The cost of the conversion process may therefore negate the savings gained in the multiplication.The best option is to construct the matrix A to be sparse from the beginning, rather than building it as a dense matrix and converting it:

>> A = sparse(i, j, s, m, n);

This will provide the performance gain of the sparse multiplication without the overhead cost of converting a dense matrix into a sparse matrix. However, if it s not possible to do so, then it could prove more efficient to simply perform a dense matrix multiplication.

Best Answer

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MATLAB: How to calculate the first derivative of a vector with respect to a matrix

MATLAB: I am trying to convert a dense matrix into a sparse matrix to speed up the matrix multiplies, but it is running slowly. How to fix this

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