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NAME

       PDGEBRD  -  reduce  a real general M-by-N distributed matrix sub( A ) =
       A(IA:IA+M-1,JA:JA+N-1) to upper  or  lower  bidiagonal  form  B  by  an
       orthogonal transformation

SYNOPSIS

       SUBROUTINE PDGEBRD( M,  N,  A,  IA,  JA, DESCA, D, E, TAUQ, TAUP, WORK,
                           LWORK, INFO )

           INTEGER         IA, INFO, JA, LWORK, M, N

           INTEGER         DESCA( * )

           DOUBLE          PRECISION A( * ), D( * ), E( * ), TAUP( * ),  TAUQ(
                           * ), WORK( * )

PURPOSE

       PDGEBRD  reduces  a  real  general M-by-N distributed matrix sub( A ) =
       A(IA:IA+M-1,JA:JA+N-1) to upper  or  lower  bidiagonal  form  B  by  an
       orthogonal transformation: Q’ * sub( A ) * P = B.

       If M >= N, B is upper bidiagonal; if M < N, B is lower bidiagonal.

       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.

       A       (local input/local output) DOUBLE PRECISION 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 ). On exit, if M >= N,  the  diagonal
               and  the  first  superdiagonal of sub( A ) are overwritten with
               the upper bidiagonal matrix B; the elements below the diagonal,
               with  the  array  TAUQ,  represent the orthogonal matrix Q as a
               product of elementary reflectors, and the  elements  above  the
               first   superdiagonal,  with  the  array  TAUP,  represent  the
               orthogonal 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
               orthogonal matrix Q as a product of elementary reflectors,  and
               the elements above the diagonal, with the array TAUP, represent
               the orthogonal 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
               LOCc(JA+MIN(M,N)-1) if M >= N;  LOCr(IA+MIN(M,N)-1)  otherwise.
               The  distributed  diagonal elements of the bidiagonal matrix B:
               D(i) = A(i,i). 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(i,i+1)  for  i  =
               1,2,...,n-1;  if m < n, E(i) = A(i+1,i) for i = 1,2,...,m-1.  E
               is tied to the distributed matrix A.

       TAUQ    (local output) DOUBLE PRECISION array dimension
               LOCc(JA+MIN(M,N)-1).  The  scalar  factors  of  the  elementary
               reflectors  which  represent  the  orthogonal matrix Q. TAUQ is
               tied to the distributed matrix A. See  Further  Details.   TAUP
               (local    output)    DOUBLE    PRECISION    array,    dimension
               LOCr(IA+MIN(M,N)-1).  The  scalar  factors  of  the  elementary
               reflectors  which  represent  the  orthogonal matrix P. TAUP is
               tied to the distributed matrix A. See  Further  Details.   WORK
               (local   workspace/local   output)   DOUBLE   PRECISION  array,
               dimension (LWORK) On exit, WORK( 1 ) returns  the  minimal  and
               optimal LWORK.

       LWORK   (local or global input) INTEGER
               The dimension of the array WORK.  LWORK is local input and must
               be at least LWORK >= NB*( MpA0 + NqA0 + 1 ) + NqA0

               where NB = MB_A = NB_A, IROFFA = MOD( IA-1, NB ), ICOFFA = MOD(
               JA-1,  NB  ),  IAROW = INDXG2P( IA, NB, MYROW, RSRC_A, NPROW ),
               IACOL = INDXG2P( JA, NB, MYCOL, CSRC_A, NPCOL ), MpA0 = NUMROC(
               M+IROFFA,  NB,  MYROW, IAROW, NPROW ), NqA0 = NUMROC( N+ICOFFA,
               NB, MYCOL, IACOL, NPCOL ).

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

               If LWORK = -1, then LWORK is global input and a workspace query
               is assumed; the routine only calculates the minimum and optimal
               size for all work arrays. Each of these values is  returned  in
               the  first  entry of the corresponding work array, and no error
               message is issued by PXERBLA.

       INFO    (global output) INTEGER
               = 0:  successful exit
               < 0:  If the i-th argument is an array and the j-entry  had  an
               illegal  value, then INFO = -(i*100+j), if the i-th argument is
               a scalar and had an illegal value, then INFO = -i.

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 real scalars, and v and u are real vectors;
       v(1:i-1)  =  0,  v(i)  =  1,  and  v(i+1:m)  is  stored  on   exit   in
       A(ia+i:ia+m-1,ja+i-1);
       u(1:i)   =   0,  u(i+1)  =  1,  and  u(i+2:n)  is  stored  on  exit  in
       A(ia+i-1,ja+i+1:ja+n-1);
       tauq is stored in TAUQ(ja+i-1) and taup in TAUP(ia+i-1).

       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 real scalars, and v and  u  are  real  vectors;
       v(1:i)   =   0,  v(i+1)  =  1,  and  v(i+2: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+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).

       The  contents  of  sub(  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).

       Alignment requirements
       ======================

       The distributed submatrix sub( A ) must verify some  alignment  proper-
       ties, namely the following expressions should be true:
       ( MB_A.EQ.NB_A .AND. IROFFA.EQ.ICOFFA )