svd_truncated_test, a Python code which calls Python's economy version of the Singular Value Decomposition (SVD) of an M by N rectangular matrix, in cases where M < N or N < M.
The singular value decomposition of an M by N rectangular matrix A has the form
A(mxn) = U(mxm) * S(mxn) * V'(nxn)
where
It is often the case that the matrix A has one dimension much bigger than the other. For instance, M = 3 and N = 10,000 might be such a case. For such examples, much of the computation and memory required for the standard SVD may not actually be needed. Instead, a truncated, or reduced version is appropriate. It will be computed faster, and require less memory to store the data.
If M < N, we have the "truncated V" SVD:
A(mxn) = U(mxm) * Sm(mxm) * Vm'(nxm)
Notice that, for our example, we will have to compute and store a Vm of size
30,000 instead of a V of size 1,000,000 entries.
If N < M, we have the "truncated U" SVD:
A(mxn) = Un(mxn) * Sn(nxn) * V'(nxn)
Similarly, in this case, the computation and storage of Un can be much reduced
from that of U.
The Python function
u, svec, v = np.linalg.svd ( a )returns the standard SVD. However, a variation of this call returns the truncated SVD:
u, svec, v = svd ( a, full_matrices = False )This will automatically create the truncated V or truncated U decomposition, depending on whether M < N or N < M.
Note that svec is a vector of singular values, and would have to be promoted to a matrix to represent the usual SVD decomposition. Also, the Python svd() function returns V, not V', unlike many other implementations of the SVD.
The computer code and data files described and made available on this web page are distributed under the MIT license
svd_truncated_test is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version and a Python version.
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