SVD_TRUNCATED
The Truncated Singular Value Decomposition

SVD_TRUNCATED is a MATLAB program which demonstrates the computation of the reduced or truncated 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

U is an orthogonal matrix, whose columns are the left singular vectors;
S is a diagonal matrix, whose min(m,n) diagonal entries are the singular values;
V is an orthogonal matrix, whose columns are the right singular vectors;

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.

Licensing:

The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.

Languages:

SVD_TRUNCATED is available in 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 program which computes a reduced basis for a collection of data vectors using the SVD.

SVD_DEMO, a MATLAB program which demonstrates the singular value decomposition (SVD) for a simple example.

SVD_FINGERPRINT, a MATLAB program 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_SNOWFALL, a MATLAB program which reads a file containing historical snowfall data and analyzes the data with the Singular Value Decomposition (SVD).

Reference:

Edward Anderson, Zhaojun Bai, Christian Bischof, Susan Blackford, James Demmel, Jack Dongarra, Jeremy Du Croz, Anne Greenbaum, Sven Hammarling, Alan McKenney, Danny Sorensen,
LAPACK User's Guide,
Third Edition,
SIAM, 1999,
ISBN: 0898714478,
LC: QA76.73.F25L36
Gene Golub, Charles VanLoan,
Matrix Computations, Third Edition,
Johns Hopkins, 1996,
ISBN: 0-8018-4513-X,
LC: QA188.G65.
David Kahaner, Cleve Moler, Steven Nash,
Numerical Methods and Software,
Prentice Hall, 1989,
ISBN: 0-13-627258-4,
LC: TA345.K34.
Lloyd Trefethen, David Bau,
Numerical Linear Algebra,
SIAM, 1997,
ISBN: 0-89871-361-7,
LC: QA184.T74.

Source Code:

svd_truncated.m, the source code.
svd_truncated_output.txt, the output file.

You can go up one level to the MATLAB source codes.