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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 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 overwrit‐
                ten 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) 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.
        The matrices Q and P are represented as products of elementary  reflec‐
        tors:
 
        If m >= n,
 
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 vec‐
        tors; 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,
 
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).
 

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