SVD_TRUNCATED, a C++ 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
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 LINPACK routine DSVDC can compute the "truncated U" version of the SVD. In order to compute the "truncated V" version with DSVDC, it's actually necessary to transpose the matrix, compute the truncated U version, and then transpose everything back...very carefully.
The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.
SVD_TRUNCATED is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version.
LINPACK_D, a C++ library which solves linear systems using double precision real arithmetic;
SVD_BASIS, a C++ program which computes a reduced basis for a collection of data vectors using the SVD.
SVD_SNOWFALL, a C++ library which reads a file containing historical snowfall data and analyzes the data with the Singular Value Decomposition (SVD), and plots created by GNUPLOT.