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NAME

       PCGEBRD - reduce 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  B  by  an
       unitary transformation

SYNOPSIS

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

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

           INTEGER         DESCA( * )

           REAL            D( * ), E( * )

           COMPLEX         A( * ), TAUP( * ), TAUQ( * ), WORK( * )

PURPOSE

       PCGEBRD reduces 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  B by an
       unitary 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) COMPLEX 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  unitary  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  unitary
               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) COMPLEX 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  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.  WORK     (local  workspace/local  output)
               COMPLEX 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 complex  scalars,  and  v  and  u  are  complex
       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  complex  scalars,  and v and u are complex
       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 )