svd_truncated_test, a MATLAB code which calls MATLAB'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 Note that the transpose of V is used in the decomposition, and that the diagonal matrix S is typically stored as a vector.

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 MATLAB function

[ u, s, v ] = svd ( a )
returns the standard SVD. However, by including the switch 'econ', you can request the truncated SVD:
[ u, s, v ] = svd ( a, 'econ' )
This will automatically create the truncated V or truncated U decomposition, depending on whether M < N or N < M.


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.

Related Data and Programs:

fingerprints, a dataset directory which contains a few images of fingerprints.

svd_basis, a MATLAB code which computes a reduced basis for a collection of data vectors using the svd.

svd_demo, a MATLAB code which demonstrates the singular value decomposition (svd) for a simple example.

svd_fingerprint, a MATLAB code which reads a file containing a fingerprint image and uses the singular value decomposition (svd) to compute and display a series of low rank approximations to the image.

svd_gray, a MATLAB code which reads a gray scale image, computes the singular value decomposition (svd), and constructs a series of low rank approximations to the image.

svd_snowfall, a MATLAB code which reads a file containing historical snowfall data and analyzes the data with the singular value decomposition (svd).


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    Third Edition,
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    ISBN: 0898714478,
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    Johns Hopkins, 1996,
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    Prentice Hall, 1989,
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    Numerical Linear Algebra,
    SIAM, 1997,
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    LC: QA184.T74.

Source Code:

Last revised on 12 December 2018.