SGEBD2 (3lapack)

reduce a real general m by n matrix A to upper or lower bidiagonal form B by an orthogonal transformation

SYNOPSIS

  SUBROUTINE SGEBD2( M, N, A, LDA, D, E, TAUQ, TAUP, WORK, INFO )

      INTEGER        INFO, LDA, M, N

      REAL           A( LDA, * ), D( * ), E( * ), TAUP( * ), TAUQ( * ), WORK(
                     * )

PURPOSE

  SGEBD2 reduces a real general m by n matrix A to upper or lower bidiagonal
  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) REAL 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 overwritten 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 orthogonal 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) REAL array, dimension (min(M,N))
          The diagonal elements of the bidiagonal matrix B: D(i) = A(i,i).

  E       (output) REAL 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) REAL array dimension (min(M,N))
          The scalar factors of the elementary reflectors which represent the
          orthogonal matrix Q. See Further Details.  TAUP    (output) REAL
          array, dimension (min(M,N)) The scalar factors of the elementary
          reflectors which represent the orthogonal matrix P. See Further
          Details.  WORK    (workspace) REAL 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 reflectors:

  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).