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zgebd2


 NAME
      ZGEBD2 - reduce a complex general m by n matrix A to upper
      or lower real bidiagonal form B by a unitary transformation

 SYNOPSIS
      SUBROUTINE ZGEBD2( M, N, A, LDA, D, E, TAUQ, TAUP, WORK,
                         INFO )

          INTEGER        INFO, LDA, M, N

          DOUBLE         PRECISION D( * ), E( * )

          COMPLEX*16     A( LDA, * ), TAUP( * ), TAUQ( * ), WORK(
                         * )

 PURPOSE
      ZGEBD2 reduces a complex general m by n matrix A to upper or
      lower real bidiagonal form B by a unitary 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) COMPLEX*16 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 unitary matrix Q as a
              product of elementary reflectors, and the elements
              above the first superdiagonal, with the array TAUP,
              represent the unitary matrix P as a product of ele-
              mentary 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 uni-
              tary matrix Q as a product of elementary reflectors,
              and the elements above the diagonal, with the array
              TAUP, represent the unitary 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) COMPLEX*16 array dimension (min(M,N))
              The scalar factors of the elementary reflectors
              which represent the unitary matrix Q. See Further
              Details.  TAUP    (output) COMPLEX*16 array, dimen-
              sion (min(M,N)) The scalar factors of the elementary
              reflectors which represent the unitary matrix P. See
              Further Details.  WORK    (workspace) COMPLEX*16
              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 complex scalars, and v and u are
      complex 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 complex scalars, v and u are complex
      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).