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NAME

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

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

           INTEGER         DESCA( * )

           REAL            A( * ), D( * ), E( * ), TAUP( * ), TAUQ( * ), WORK(
                           * )

PURPOSE

       PSGEBRD  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) REAL 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) REAL 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) REAL 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) REAL 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)  REAL 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)
               REAL  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 )