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sgebd2


 NAME
      SGEBD2 - 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 bidi-
      agonal.

 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 bidiag-
              onal 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 lead-
              ing 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 elemen-
      tary 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).