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.