SVD_TRUNCATED
The Truncated Singular Value Decomposition

SVD_TRUNCATED is a FORTRAN77 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)
      

• 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;

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)
      

If N < M, we have the "truncated U" SVD:

        A(mxn) = Un(mxn) * Sn(nxn) * V'(nxn)
      

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.

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 C++ version and a FORTRAN77 version and a FORTRAN90 version and a MATLAB version.

Related Data and Programs:

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

LAPACK_EXAMPLES, a FORTRAN77 program which demonstrates the use of the LAPACK linear algebra library.

LINPACK_D, a FORTRAN77 library which solves linear systems using double precision real arithmetic;

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

SVD_DEMO, a FORTRAN77 program which demonstrates the singular value decomposition (SVD) for a simple example.

SVD_SNOWFALL, a FORTRAN77 program which reads a file containing historical snowfall data and analyzes the data with the Singular Value Decomposition (SVD), and plots created by GNUPLOT.

TOMS358, a FORTRAN77 routine which computes the singular value decomposition for a complex matrix.

Reference:

1. 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
2. Gene Golub, Charles VanLoan,
   Matrix Computations, Third Edition,
   Johns Hopkins, 1996,
   ISBN: 0-8018-4513-X,
   LC: QA188.G65.
3. David Kahaner, Cleve Moler, Steven Nash,
   Numerical Methods and Software,
   Prentice Hall, 1989,
   ISBN: 0-13-627258-4,
   LC: TA345.K34.
4. Lloyd Trefethen, David Bau,
   Numerical Linear Algebra,
   SIAM, 1997,
   ISBN: 0-89871-361-7,
   LC: QA184.T74.

Source Code:

• svd_truncated.f, the source code.
• svd_truncated.sh, commands to compile and load the source code.
• svd_truncated_output.txt, the output file.

List of Routines:

• MAIN is the main program for SVD_TRUNCATED.
• SVD_TRUNCATED_U_TEST tests SVD_TRUNCATED_U.
• SVD_TRUNCATED_U computes the SVD when N <= M.
• SVD_TRUNCATED_V_TEST tests SVD_TRUNCATED_V.
• SVD_TRUNCATED_V computes the SVD when M <= N.
• R8MAT_PRINT prints an R8MAT.
• R8MAT_PRINT_SOME prints some of an R8MAT.
• TIMESTAMP prints the current YMDHMS date as a time stamp.

You can go up one level to the FORTRAN77 source codes.