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NAME
     zlabrd - reduce the first NB rows and columns of  a  complex
     general  m  by  n matrix A to upper or lower real bidiagonal
     form by a unitary transformation Q' * A * P, and returns the
     matrices  X  and Y which are needed to apply the transforma-
     tion to the unreduced part of A

SYNOPSIS
     SUBROUTINE ZLABRD( M, N, NB, A, LDA, D, E,  TAUQ,  TAUP,  X,
               LDX, Y, LDY )

     INTEGER LDA, LDX, LDY, M, N, NB

     DOUBLE PRECISION D( * ), E( * )

     COMPLEX*16 A( LDA, * ), TAUP( * ), TAUQ( * ), X( LDX,  *  ),
               Y( LDY, * )



     #include <sunperf.h>

     void zlabrd(int m, int n, int  nb,  doublecomplex  *za,  int
               lda,  double  *d,  double *e, doublecomplex *tauq,
               doublecomplex *taup, doublecomplex *zx,  int  ldx,
               doublecomplex *zy, int *ldy) ;

PURPOSE
     ZLABRD reduces the first NB rows and columns  of  a  complex
     general  m  by  n matrix A to upper or lower real bidiagonal
     form by a unitary transformation Q' * A * P, and returns the
     matrices  X  and Y which are needed to apply the transforma-
     tion to the unreduced part of A.

     If m >= n, A is reduced to upper bidiagonal form; if m <  n,
     to lower bidiagonal form.

     This is an auxiliary routine called by ZGEBRD


ARGUMENTS
     M         (input) INTEGER
               The number of rows in the matrix A.

     N         (input) INTEGER
               The number of columns in the matrix A.

     NB        (input) INTEGER
               The number of leading rows and columns of A to  be
               reduced.

     A         (input/output) COMPLEX*16 array, dimension (LDA,N)
               On entry, the m by n general matrix to be reduced.
               On  exit,  the  first  NB  rows and columns of the
               matrix are overwritten; the rest of the  array  is
               unchanged.   If  m >= n, elements on and below the
               diagonal in the first NB columns, with  the  array
               TAUQ,  represent the unitary matrix Q as a product
               of elementary reflectors; and elements  above  the
               diagonal  in  the  first  NB  rows, with the array
               TAUP, represent the unitary matrix P as a  product
               of  elementary  reflectors.   If  m  < n, elements
               below the diagonal in the first NB  columns,  with
               the  array TAUQ, represent the unitary matrix Q as
               a product of elementary reflectors,  and  elements
               on  and  above  the diagonal in the first NB rows,
               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 (NB)
               The diagonal elements of the  first  NB  rows  and
               columns of the reduced matrix.  D(i) = A(i,i).

     E         (output) DOUBLE PRECISION array, dimension (NB)
               The off-diagonal elements of the first NB rows and
               columns of the reduced matrix.

     TAUQ      (output) COMPLEX*16 array dimension (NB)
               The scalar factors of  the  elementary  reflectors
               which  represent the unitary matrix Q. See Further
               Details.   TAUP     (output)   COMPLEX*16   array,
               dimension  (NB)  The scalar factors of the elemen-
               tary reflectors which represent the unitary matrix
               P.  See  Further  Details.   X       (output) COM-
               PLEX*16  array,  dimension  (LDX,NB)  The  m-by-nb
               matrix  X required to update the unreduced part of
               A.

     LDX       (input) INTEGER
               The leading dimension  of  the  array  X.  LDX  >=
               max(1,M).

     Y         (output) COMPLEX*16 array, dimension (LDY,NB)
               The n-by-nb matrix Y required to update the  unre-
               duced part of A.

     LDY       (output) INTEGER
               The leading dimension  of  the  array  Y.  LDY  >=
               max(1,N).


FURTHER DETAILS
     The matrices Q and P are represented as products of  elemen-
     tary reflectors:

        Q = H(1) H(2) . . . H(nb)  and  P = G(1) G(2) . . . G(nb)

     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.

     If m >= n, v(1:i-1) = 0, v(i) = 1, and v(i:m) is  stored  on
     exit  in  A(i:m,i);  u(1:i) = 0, u(i+1) = 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).

     If m < n, v(1:i) = 0, v(i+1) = 1, and v(i+1:m) is stored  on
     exit  in  A(i+2:m,i);  u(1:i-1) = 0, u(i) = 1, and u(i:n) is
     stored on exit in A(i,i+1:n); tauq is stored in TAUQ(i)  and
     taup in TAUP(i).

     The elements of the vectors v and u together form the  m-by-
     nb matrix V and the nb-by-n matrix U' which are needed, with
     X and Y, to apply the transformation to the  unreduced  part
     of  the  matrix, using a block update of the form:  A := A -
     V*Y' - X*U'.

     The contents of A on exit are illustrated by the following examples with nb
     = 2:

     m = 6 and n = 5 (m > n):          m = 5 and n = 6 (m < n):

       (  1   1   u1  u1  u1 )           (  1   u1  u1  u1  u1  u1 )
       (  v1  1   1   u2  u2 )           (  1   1   u2  u2  u2  u2 )
       (  v1  v2  a   a   a  )           (  v1  1   a   a   a   a  )
       (  v1  v2  a   a   a  )           (  v1  v2  a   a   a   a  )
       (  v1  v2  a   a   a  )           (  v1  v2  a   a   a   a  )
       (  v1  v2  a   a   a  )

     where a denotes an element of the original matrix  which  is
     unchanged,  vi  denotes  an  element  of the vector defining
     H(i), and ui an element of the vector defining G(i).

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