ZGEBRD (3lapack)

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

SYNOPSIS

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

      INTEGER        INFO, LDA, LWORK, M, N

      DOUBLE         PRECISION D( * ), E( * )

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

PURPOSE

  ZGEBRD 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*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 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) 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, dimension (min(M,N)) The scalar factors of the elementary
          reflectors which represent the unitary matrix P. See Further
          Details.  WORK    (workspace/output) COMPLEX*16 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).