CGEBRD (3lapack)

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 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) 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 bidiagonal 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 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 unitary 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/output) 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 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 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).