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cgebrd


 NAME
      CGEBRD - reduce a general complex M-by-N matrix A to upper
      or lower bidiagonal form B by a unitary transformation

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

          INTEGER        INFO, LDA, LWORK, M, N

          REAL           D( * ), E( * )

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

 PURPOSE
      CGEBRD reduces a general complex M-by-N matrix A to upper or
      lower bidiagonal form B by a unitary transformation: Q**H *
      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 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) 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) COMPLEX array dimension (min(M,N))
              The scalar factors of the elementary reflectors
              which represent the unitary matrix Q. See Further
              Details.  TAUP    (output) COMPLEX array, dimension
              (min(M,N)) The scalar factors of the elementary
              reflectors which represent the unitary matrix P. See
              Further Details.  WORK    (workspace) COMPLEX array,
              dimension (LWORK) On exit, if INFO = 0, WORK(1)
              returns the optimal LWORK.

      LWORK   (input) INTEGER
              The length of the array WORK.  LWORK >= max(1,M,N).
              For optimum performance LWORK >= (M+N)*NB, where NB
              is the optimal blocksize.

      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, and 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).