# SVD_DEMO Demonstration of the Singular Value Decomposition

SVD_DEMO is a FORTRAN77 program which demonstrates the computation of the singular value decomposition 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)

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.

### Usage:

svd_demo m n seed
where
• 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;

### Languages:

SVD_DEMO 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, a FORTRAN77 library which includes routines to carry out the singular value decomposition.

QR_SOLVE, a FORTRAN77 library which computes the least squares solution of a linear system A*x=b.

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

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.

SVD_TRUNCATED, a FORTRAN90 program 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.

TOMS581, a FORTRAN77 library which implements an improved algorithm for computing the singular value decomposition (SVD) of a rectangular matrix; this is ACM TOMS algorithm 571, by Tony Chan.

### 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.

### List of Routines:

• MAIN is the main program for SVD_DEMO.
• SVD_SUB carries out the demonstration, after dimensions are known.
• COMPARE_LINPACK_LAPACK compares the SVD's from LINPACK and LAPACK.
• GET_SEED returns a seed for the random number generator.
• GET_SVD_LAPACK gets the SVD of a matrix using a call to LAPACK.
• GET_SVD_LINPACK gets the SVD of a matrix using a call to LINPACK.
• PSEUDO_INVERSE computes the pseudoinverse.
• PSEUDO_LINEAR_SOLVE_TEST uses the pseudoinverse for linear systems.
• PSEUDO_PRODUCT_TEST examines pseudoinverse products.
• R8MAT_DIF_FRO returns the Frobenius norm of the difference of two R8MAT's.
• R8MAT_MM multiplies two R8MAT's.
• R8MAT_MV multiplies a matrix times a vector.
• R8MAT_NORM_FRO returns the Frobenius norm of an R8MAT.
• R8MAT_PRINT prints an R8MAT.
• R8MAT_PRINT_SOME prints some of an R8MAT.
• R8MAT_TRANSPOSE makes a transposed copy of a matrix.
• R8MAT_UNIFORM_01 returns a unit pseudorandom R8MAT.
• R8VEC_NORM_L2 returns the L2 norm of an R8VEC.
• R8VEC_UNIFORM_01 returns a unit pseudorandom R8VEC.
• RANK_ONE_PRINT_TEST prints the sums of rank one matrices.
• RANK_ONE_TEST compares A to the sum of rank one matrices.
• S_LEN_TRIM returns the length of a string to the last nonblank.
• S_TO_I4 reads an I4 from a string.
• SVD_PRODUCT_TEST tests that A = U * S * V'.
• TIMESTAMP prints out the current YMDHMS date as a timestamp.

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

Last revised on 18 June 2012.