PCGELS(3)     ScaLAPACK routine of NEC Numeric Library Collection    PCGELS(3)



NAME
       PCGELS - solve overdetermined or underdetermined complex linear systems
       involving an M-by-N matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1),

SYNOPSIS
       SUBROUTINE PCGELS( TRANS, M, N, NRHS, A, IA,  JA,  DESCA,  B,  IB,  JB,
                          DESCB, WORK, LWORK, INFO )

           CHARACTER      TRANS

           INTEGER        IA, IB, INFO, JA, JB, LWORK, M, N, NRHS

           INTEGER        DESCA( * ), DESCB( * )

           COMPLEX        A( * ), B( * ), WORK( * )

PURPOSE
       PCGELS  solves overdetermined or underdetermined complex linear systems
       involving an M-by-N matrix sub( A ) =  A(IA:IA+M-1,JA:JA+N-1),  or  its
       conjugate-transpose, using a QR or LQ factorization of sub( A ).  It is
       assumed that sub( A ) has full rank.

       The following options are provided:

       1. If TRANS = 'N' and m >= n:  find the least squares solution of
          an overdetermined system, i.e., solve the least squares problem
                       minimize || sub( B ) - sub( A )*X ||.

       2. If TRANS = 'N' and m < n:  find the minimum norm solution of
          an underdetermined system sub( A ) * X = sub( B ).

       3. If TRANS = 'C' and m >= n:  find the minimum norm solution of
          an undetermined system sub( A )**H * X = sub( B ).

       4. If TRANS = 'C' and m < n:  find the least squares solution of
          an overdetermined system, i.e., solve the least squares problem
                       minimize || sub( B ) - sub( A )**H * X ||.

       where sub( B ) denotes B( IB:IB+M-1, JB:JB+NRHS-1 ) when  TRANS  =  'N'
       and  B(  IB:IB+N-1,  JB:JB+NRHS-1  ) otherwise. Several right hand side
       vectors b and solution vectors x can be handled in a single call;  When
       TRANS  =  'N', the solution vectors are stored as the columns of the N-
       by-NRHS right hand side matrix sub( B ) and the  M-by-NRHS  right  hand
       side matrix sub( B ) otherwise.


       Notes
       =====

       Each  global data object is described by an associated description vec-
       tor.  This vector stores the information required to establish the map-
       ping 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  col-
       umn.
       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
       TRANS   (global input) CHARACTER
               = 'N': the linear system involves sub( A );
               = 'C': the linear system involves sub( A )**H.

       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.

       NRHS    (global input) INTEGER
               The  number  of right hand sides, i.e. the number of columns of
               the distributed submatrices sub( B ) and X.  NRHS >= 0.

       A       (local input/local output) COMPLEX pointer into the
               local  memory  to  an  array  of  local  dimension   (   LLD_A,
               LOCc(JA+N-1)  ).   On  entry,  the M-by-N matrix A.  if M >= N,
               sub( A ) is overwritten by details of its QR  factorization  as
               returned  by  PCGEQRF;  if  M  <  N, sub( A ) is overwritten by
               details of its LQ factorization as returned by PCGELQF.

       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.

       B       (local input/local output) COMPLEX pointer into the
               local  memory  to  an  array   of   local   dimension   (LLD_B,
               LOCc(JB+NRHS-1)).   On  entry,  this  array  contains the local
               pieces of the distributed matrix B of right hand side  vectors,
               stored  columnwise;  sub( B ) is M-by-NRHS if TRANS='N', and N-
               by-NRHS otherwise.  On exit, sub( B )  is  overwritten  by  the
               solution  vectors,  stored columnwise:  if TRANS = 'N' and M >=
               N, rows 1 to N of sub( B ) contain the least  squares  solution
               vectors;  the  residual sum of squares for the solution in each
               column is given by the sum of squares of elements N+1 to  M  in
               that  column; if TRANS = 'N' and M < N, rows 1 to N of sub( B )
               contain the minimum norm solution vectors; if TRANS = 'C' and M
               >= N, rows 1 to M of sub( B ) contain the minimum norm solution
               vectors; if TRANS = 'C' and M < N, rows 1 to M of sub( B ) con-
               tain  the  least  squares solution vectors; the residual sum of
               squares for the solution in each column is given by the sum  of
               squares of elements M+1 to N in that column.

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

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

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

       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 >= LTAU + MAX( LWF, LWS ) where If M >= N,
               then LTAU = NUMROC( JA+MIN(M,N)-1, NB_A, MYCOL,  CSRC_A,  NPCOL
               ),  LWF   =  NB_A  *  (  MpA0  +  NqA0  +  NB_A  )  LWS  = MAX(
               (NB_A*(NB_A-1))/2, (NRHSqB0 + MpB0)*NB_A ) + NB_A *  NB_A  Else
               LTAU = NUMROC( IA+MIN(M,N)-1, MB_A, MYROW, RSRC_A, NPROW ), LWF
               = MB_A * ( MpA0 + NqA0 + MB_A ) LWS  = MAX(  (MB_A*(MB_A-1))/2,
               (  NpB0  +  MAX(  NqA0  + NUMROC( NUMROC( N+IROFFB, MB_A, 0, 0,
               NPROW ), MB_A, 0, 0, LCMP ), NRHSqB0 ) )*MB_A ) + MB_A  *  MB_A
               End if

               where LCMP = LCM / NPROW with LCM = ILCM( NPROW, NPCOL ),

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

               IROFFB = MOD( IB-1, MB_B ), ICOFFB = MOD( JB-1, NB_B ), IBROW =
               INDXG2P( IB, MB_B, MYROW, RSRC_B, NPROW ), IBCOL = INDXG2P( JB,
               NB_B,  MYCOL,  CSRC_B,  NPCOL ), MpB0 = NUMROC( M+IROFFB, MB_B,
               MYROW, IBROW, NPROW ), NpB0 = NUMROC(  N+IROFFB,  MB_B,  MYROW,
               IBROW,  NPROW  ),  NRHSqB0  = NUMROC( NRHS+ICOFFB, NB_B, MYCOL,
               IBCOL, NPCOL ),

               ILCM, INDXG2P and NUMROC are ScaLAPACK tool  functions;  MYROW,
               MYCOL, NPROW and NPCOL can be determined by calling the subrou-
               tine 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.



ScaLAPACK routine               31 October 2017                      PCGELS(3)