Provided by: scalapack-doc_1.5-11_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.