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dgebd2 examples
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Man Page Details
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NAME
       DGEBD2  -  a  real general m by n matrix A to upper or lower bidiagonal
       form B by an orthogonal transformation

SYNOPSIS
       SUBROUTINE DGEBD2( M, N, A, LDA, D, E, TAUQ, TAUP, WORK, INFO )

           INTEGER        INFO, LDA, M, N

           DOUBLE         PRECISION A( LDA, * ), D( * ), E( * ),  TAUP(  *  ),
                          TAUQ( * ), WORK( * )

PURPOSE
       DGEBD2 reduces a real general m by n matrix A to upper or lower bidiag-
       onal form B by an orthogonal transformation: Q' * A * P = B.

       If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal.


ARGUMENTS
       M       (input) INTEGER
               The number of rows in the matrix A.  M >= 0.

       N       (input) INTEGER
               The number of columns in the matrix A.  N >= 0.

       A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
               On entry, the m by n general matrix to be reduced.  On exit, if
               m  >= n, the diagonal and the first superdiagonal are overwrit-
               ten with the upper bidiagonal matrix B; the elements below  the
               diagonal,  with the array TAUQ, represent the orthogonal matrix
               Q as a product of elementary reflectors, and the elements above
               the  first  superdiagonal,  with  the array TAUP, represent the
               orthogonal matrix P as a product of elementary reflectors; if m
               <  n,  the  diagonal  and the first subdiagonal are overwritten
               with the lower bidiagonal matrix  B;  the  elements  below  the
               first  subdiagonal, with the array TAUQ, represent the orthogo-
               nal matrix Q as a product of  elementary  reflectors,  and  the
               elements above the diagonal, with the array TAUP, represent the
               orthogonal matrix P as a product of elementary reflectors.  See
               Further Details.  LDA     (input) INTEGER The leading dimension
               of the array A.  LDA >= max(1,M).

       D       (output) DOUBLE PRECISION array, dimension (min(M,N))
               The diagonal elements  of  the  bidiagonal  matrix  B:  D(i)  =
               A(i,i).

       E       (output) DOUBLE PRECISION array, dimension (min(M,N)-1)
               The  off-diagonal  elements of the bidiagonal matrix B: if m >=
               n, E(i) = A(i,i+1) for i =  1,2,...,n-1;  if  m  <  n,  E(i)  =
               A(i+1,i) for i = 1,2,...,m-1.

       TAUQ    (output) DOUBLE PRECISION array dimension (min(M,N))
               The scalar factors of the elementary reflectors which represent
               the orthogonal matrix Q. See Further Details.  TAUP    (output)
               DOUBLE PRECISION array, dimension (min(M,N)) The scalar factors
               of the elementary reflectors  which  represent  the  orthogonal
               matrix P. See Further Details.  WORK    (workspace) DOUBLE PRE-
               CISION array, dimension (max(M,N))

       INFO    (output) INTEGER
               = 0: successful exit.
               < 0: if INFO = -i, the i-th argument had an illegal value.

FURTHER DETAILS
       The matrices Q and P are represented as products of elementary  reflec-
       tors:

       If m >= n,

          Q = H(1) H(2) . . . H(n)  and  P = G(1) G(2) . . . G(n-1)

       Each H(i) and G(i) has the form:

          H(i) = I - tauq * v * v'  and G(i) = I - taup * u * u'

       where  tauq  and  taup  are real scalars, and v and u are real vectors;
       v(1:i-1) = 0, v(i) = 1, and v(i+1:m) is stored on exit  in  A(i+1:m,i);
       u(1:i)  =  0, u(i+1) = 1, and u(i+2:n) is stored on exit in A(i,i+2:n);
       tauq is stored in TAUQ(i) and taup in TAUP(i).

       If m < n,

          Q = H(1) H(2) . . . H(m-1)  and  P = G(1) G(2) . . . G(m)

       Each H(i) and G(i) has the form:

          H(i) = I - tauq * v * v'  and G(i) = I - taup * u * u'

       where tauq and taup are real scalars, and v and  u  are  real  vectors;
       v(1:i)  =  0, v(i+1) = 1, and v(i+2:m) is stored on exit in A(i+2:m,i);
       u(1:i-1) = 0, u(i) = 1, and u(i+1:n) is stored on exit  in  A(i,i+1:n);
       tauq is stored in TAUQ(i) and taup in TAUP(i).

       The contents of A on exit are illustrated by the following examples:

       m = 6 and n = 5 (m > n):          m = 5 and n = 6 (m < n):

         (  d   e   u1  u1  u1 )           (  d   u1  u1  u1  u1  u1 )
         (  v1  d   e   u2  u2 )           (  e   d   u2  u2  u2  u2 )
         (  v1  v2  d   e   u3 )           (  v1  e   d   u3  u3  u3 )
         (  v1  v2  v3  d   e  )           (  v1  v2  e   d   u4  u4 )
         (  v1  v2  v3  v4  d  )           (  v1  v2  v3  e   d   u5 )
         (  v1  v2  v3  v4  v5 )

       where  d  and  e  denote  diagonal  and  off-diagonal elements of B, vi
       denotes an element of the vector defining H(i), and ui  an  element  of
       the vector defining G(i).

