svd_truncated, a FORTRAN90 code 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 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 LAPACK routines CGESVD, DGESVD, SGESVD and ZGESVD compute the SVD for a rectangular matrix in single or double precision, real or complex arithmetic. These routines include some options that allow you to request a reduced or truncated SVD computation, but the exact details of how to do this may be a little obscure. This program demonstrates how it is done.

In order to compile and load this program, you need to have access to a copy of LAPACK.


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


svd_truncated is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version.

Related Data and Programs:

SVD_BASIS, a FORTRAN90 code which computes a reduced basis for a collection of data vectors using the SVD.

svd_test, a FORTRAN90 code which demonstrates the singular value decomposition (SVD) for a simple example.

SVD_SNOWFALL, a FORTRAN90 code which reads a file containing historical snowfall data and analyzes the data with the Singular Value Decomposition (SVD), and plots created by GNUPLOT.



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    ISBN: 0898714478,
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    Johns Hopkins, 1996,
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Source Code:

Last revised on 31 August 2020.