Provided by: scalapack-doc_1.5-10_all #### NAME

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

```

#### SYNOPSIS

```       SUBROUTINE PDGELS( 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( * )

DOUBLE         PRECISION A( * ), B( * ), WORK( * )

```

#### PURPOSE

```       PDGELS  solves  overdetermined  or underdetermined real linear systems involving an M-by-N
matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1), or its 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 = 'T' and m >= n:  find the minimum norm solution of
an undetermined system sub( A )**T * X = sub( B ).

4. If TRANS = 'T' and m < n:  find the least squares solution of
an overdetermined system, i.e., solve the least squares problem
minimize || sub( B ) - sub( A )**T * 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 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

```       TRANS   (global input) CHARACTER
= 'N': the linear system involves sub( A );
= 'T': the linear system involves sub( A )**T.

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) DOUBLE PRECISION 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  PDGEQRF;  if  M  <  N, sub( A ) is overwritten by
details of its LQ factorization as returned by PDGELQF.

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) DOUBLE PRECISION 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 = 'T' and M >= N, rows 1 to M of sub( B )  contain  the  minimum
norm  solution  vectors; if TRANS = 'T' 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) 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
>= 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.
```