PZGELS(l)						   LAPACK routine (version 1.5) 						 PZGELS(l)

NAME

       PZGELS - 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 PZGELS( 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*16	  A( * ), B( * ), WORK( * )

PURPOSE

       PZGELS 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 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  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*16 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  over-
	       written	by details of its QR factorization as returned by PZGEQRF; if M <  N, sub( A ) is overwritten by details of its LQ factor-
	       ization as returned by PZGELQF.

       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*16 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  dis-
	       tributed  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 )  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 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 ) 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 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*16 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 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.

LAPACK version 1.5						    12 May 1997 							 PZGELS(l)