cgebd2(3F)

NAME
     CGEBD2 - reduce a complex general m by n matrix A to upper or lower real
     bidiagonal form B by a unitary transformation

SYNOPSIS
     SUBROUTINE CGEBD2( M, N, A, LDA, D, E, TAUQ, TAUP, WORK, INFO )

         INTEGER        INFO, LDA, M, N

         REAL           D( * ), E( * )

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

PURPOSE
     CGEBD2 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 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) COMPLEX 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 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, 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).