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NAME

       PZLABRD  -  reduce  the  first NB rows and columns of a complex general M-by-N distributed
       matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) to upper or lower bidiagonal form by  an  unitary
       transformation  Q' * A * P, and returns the matrices X and Y which are needed to apply the
       transfor- mation to the unreduced part of sub( A )

SYNOPSIS

       SUBROUTINE PZLABRD( M, N, NB, A, IA, JA, DESCA, D, E, TAUQ, TAUP, X, IX, JX, DESCX, Y, IY,
                           JY, DESCY, WORK )

           INTEGER         IA, IX, IY, JA, JX, JY, M, N, NB

           INTEGER         DESCA( * ), DESCX( * ), DESCY( * )

           DOUBLE          PRECISION D( * ), E( * )

           COMPLEX*16      A( * ), TAUP( * ), TAUQ( * ), X( * ), Y( * ), WORK( * )

PURPOSE

       PZLABRD  reduces  the  first  NB  rows and columns of a complex general M-by-N distributed
       matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) to upper or lower bidiagonal form by  an  unitary
       transformation  Q' * A * P, and returns the matrices X and Y which are needed to apply the
       transfor- mation to the unreduced part of sub( A ).

       If M >= N, sub( A ) is reduced to upper bidiagonal form; if M <  N,  to  lower  bidiagonal
       form.

       This is an auxiliary routine called by PZGEBRD.

       Notes
       =====

       Each  global  data  object  is described by an associated description vector.  This vector
       stores the information required to establish the mapping between an object element and its
       corresponding process and memory location.

       Let  A be a generic term for any 2D block cyclicly distributed array.  Such a global array
       has an associated description vector DESCA.  In the following comments,  the  character  _
       should be read as "of the global array".

       NOTATION        STORED IN      EXPLANATION
       ---------------   --------------   --------------------------------------  DTYPE_A(global)
       DESCA( DTYPE_ )The descriptor type.  In this case,
                                      DTYPE_A = 1.
       CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating
                                      the BLACS process grid A is distribu-
                                      ted over. The context itself is glo-
                                      bal, but the handle (the integer
                                      value) may vary.
       M_A    (global) DESCA( M_ )    The number of rows in the global
                                      array A.
       N_A    (global) DESCA( N_ )    The number of columns in the global
                                      array A.
       MB_A   (global) DESCA( MB_ )   The blocking factor used to distribute
                                      the rows of the array.
       NB_A   (global) DESCA( NB_ )   The blocking factor used to distribute
                                      the columns of the array.
       RSRC_A (global) DESCA( RSRC_ ) The process row over which the first
                                      row of the array A is distributed.  CSRC_A (global)  DESCA(
       CSRC_ ) The process column over which the
                                      first column of the array A is
                                      distributed.
       LLD_A  (local)  DESCA( LLD_ )  The leading dimension of the local
                                      array.  LLD_A >= MAX(1,LOCr(M_A)).

       Let  K  be  the  number  of  rows  or columns of a distributed matrix, and assume that its
       process grid has dimension p x q.
       LOCr( K ) denotes the number of elements of K that a  process  would  receive  if  K  were
       distributed over the p processes of its process column.
       Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K
       were distributed over the q processes of its process row.
       The values of LOCr() and LOCc() may be  determined  via  a  call  to  the  ScaLAPACK  tool
       function, NUMROC:
               LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ),
               LOCc(  N  )  =  NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ).  An upper bound for these
       quantities may be computed by:
               LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A
               LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A

ARGUMENTS

       M       (global input) INTEGER
               The number of rows to be operated on, i.e. the number of rows of  the  distributed
               submatrix sub( A ). M >= 0.

       N       (global input) INTEGER
               The  number  of  columns  to  be  operated  on,  i.e. the number of columns of the
               distributed submatrix sub( A ). N >= 0.

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

       A       (local input/local output) COMPLEX*16 pointer into the
               local memory to an array of dimension (LLD_A,LOCc(JA+N-1)).  On entry, this  array
               contains  the  local  pieces  of  the  general  distributed  matrix sub( A ) to be
               reduced. On exit, the first NB rows and columns of the matrix are overwritten; the
               rest  of the distributed matrix sub( A ) 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.  IA      (global input) INTEGER The
               row index in the global array A indicating the first row of sub( A ).

       JA      (global input) INTEGER
               The column index in the global array A indicating the first column of sub( A ).

       DESCA   (global and local input) INTEGER array of dimension DLEN_.
               The array descriptor for the distributed matrix A.

       D       (local output) DOUBLE PRECISION array, dimension
               LOCr(IA+MIN(M,N)-1) if M >= N;  LOCc(JA+MIN(M,N)-1)  otherwise.   The  distributed
               diagonal  elements  of the bidiagonal matrix B: D(i) = A(ia+i-1,ja+i-1). D is tied
               to the distributed matrix A.

       E       (local output) DOUBLE PRECISION array, dimension
               LOCr(IA+MIN(M,N)-1) if M >= N;  LOCc(JA+MIN(M,N)-2)  otherwise.   The  distributed
               off-diagonal  elements  of  the bidiagonal distributed matrix B: if m >= n, E(i) =
               A(ia+i-1,ja+i) for i = 1,2,...,n-1; if m <  n,  E(i)  =  A(ia+i,ja+i-1)  for  i  =
               1,2,...,m-1.  E is tied to the distributed matrix A.

       TAUQ    (local output) COMPLEX*16 array dimension
               LOCc(JA+MIN(M,N)-1).  The  scalar  factors  of  the  elementary  reflectors  which
               represent the unitary matrix Q. TAUQ is tied to  the  distributed  matrix  A.  See
               Further   Details.    TAUP      (local   output)   COMPLEX*16   array,   dimension
               LOCr(IA+MIN(M,N)-1).  The  scalar  factors  of  the  elementary  reflectors  which
               represent  the  unitary  matrix  P.  TAUP is tied to the distributed matrix A. See
               Further Details.  X       (local output) COMPLEX*16 pointer into the local  memory
               to  an array of dimension (LLD_X,NB). On exit, the local pieces of the distributed
               M-by-NB matrix X(IX:IX+M-1,JX:JX+NB-1) required to update the  unreduced  part  of
               sub( A ).

       IX      (global input) INTEGER
               The row index in the global array X indicating the first row of sub( X ).

       JX      (global input) INTEGER
               The column index in the global array X indicating the first column of sub( X ).

       DESCX   (global and local input) INTEGER array of dimension DLEN_.
               The array descriptor for the distributed matrix X.

       Y       (local output) COMPLEX*16 pointer into the local memory
               to an array of dimension (LLD_Y,NB).  On exit, the local pieces of the distributed
               N-by-NB matrix Y(IY:IY+N-1,JY:JY+NB-1) required to update the  unreduced  part  of
               sub( A ).

       IY      (global input) INTEGER
               The row index in the global array Y indicating the first row of sub( Y ).

       JY      (global input) INTEGER
               The column index in the global array Y indicating the first column of sub( Y ).

       DESCY   (global and local input) INTEGER array of dimension DLEN_.
               The array descriptor for the distributed matrix Y.

       WORK    (local workspace) COMPLEX*16 array, dimension (LWORK)
               LWORK >= NB_A + NQ, with

               NQ = NUMROC( N+MOD( IA-1, NB_Y ), NB_Y, MYCOL, IACOL, NPCOL ) IACOL = INDXG2P( JA,
               NB_A, MYCOL, CSRC_A, NPCOL )

               INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can
               be determined by calling the subroutine BLACS_GRIDINFO.

FURTHER DETAILS

       The matrices Q and P are represented as products of elementary 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(ia+i-1:ia+m-1,ja+i-1); u(1:i) = 0, u(i+1) =  1,  and  u(i+1:n)  is  stored  on  exit  in
       A(ia+i-1,ja+i:ja+n-1); tauq is stored in TAUQ(ja+i-1) and taup in TAUP(ia+i-1).

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

       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:  sub( A ) := sub( A ) - V*Y' - X*U'.

       The contents of sub( 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).