svd_test

svd_test, a Fortran90 code which demonstrates the computation of the singular value decomposition (SVD) and a few of its properties.

The singular value decomposition has uses in solving overdetermined or underdetermined linear systems, linear least squares problems, data compression, the pseudoinverse matrix, reduced order modeling, and the accurate computation of matrix rank and null space.

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;

Usage:

svd_test m n seed

• m is the number of rows in the random matrix;
• n is the number of columns in the random matrix;
• seed an optional seed for the random number generator;

Licensing:

The information on this web page is distributed under the MIT license.

Languages:

svd_test is available in a C version and a C++ version and a Fortran90 version and a MATLAB version and an Octave version and a Python 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_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.

SVD_TRUNCATED, a Fortran90 code which demonstrates the computation of the reduced or truncated Singular Value Decomposition (SVD) that is useful for cases when one dimension of the matrix is much smaller than the other.

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. Jack Dongarra, Jim Bunch, Cleve Moler, Pete Stewart,
   LINPACK User's Guide,
   SIAM, 1979,
   ISBN13: 978-0-898711-72-1,
   LC: QA214.L56.
3. Gene Golub, Charles VanLoan,
   Matrix Computations, Third Edition,
   Johns Hopkins, 1996,
   ISBN: 0-8018-4513-X,
   LC: QA188.G65.
4. David Kahaner, Cleve Moler, Steven Nash,
   Numerical Methods and Software,
   Prentice Hall, 1989,
   ISBN: 0-13-627258-4,
   LC: TA345.K34.
5. Lloyd Trefethen, David Bau,
   Numerical Linear Algebra,
   SIAM, 1997,
   ISBN: 0-89871-361-7,
   LC: QA184.T74.

Source Code:

• svd_test.f90, the source code.
• svd_test.sh, compiles the source code.

• svd_test_5_3.txt, the output from the command "svd_test 5 3 123456789".
• svd_test_5_5.txt, the output from the command "svd_test 5 5 123456789".
• svd_test_5_8.txt, the output from the command "svd_test 5 8 123456789".