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NAME

        ZGEBRD  -  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( * )
 

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 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/output) COMPLEX*16
WORK(1)  returns
                the optimal LWORK.
 
        LWORK   (input) INTEGER
                The  length of the array WORK.  LWORK >= max(1,M,N).  For opti‐
                mum performance LWORK >= (M+N)*NB,  where  NB  is  the  optimal
                blocksize.
 
                If  LWORK  = -1, then a workspace query is assumed; the routine
                only calculates the optimal size of  the  WORK  array,  returns
                this  value  as the first entry of the WORK array, and no error
                message related to LWORK is issued by XERBLA.
 
        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, and v and u are  complex  vec‐
        tors;  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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