PDGERFS improves the computed solution to a system of linear
equations and provides error bounds and backward error estimates for
the solutions.
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
In the following comments, sub( A ), sub( X ) and sub( B ) denote
respectively A(IA:IA+N-1,JA:JA+N-1), X(IX:IX+N-1,JX:JX+NRHS-1) and
B(IB:IB+N-1,JB:JB+NRHS-1).
ARGUMENTS
TRANS (global input) CHARACTER*1
Specifies the form of the system of equations.
= 'N': sub( A ) * sub( X ) = sub( B ) (No transpose)
= 'T': sub( A )**T * sub( X ) = sub( B ) (Transpose)
= 'C': sub( A )**T * sub( X ) = sub( B )
(Conjugate transpose = Transpose)
N (global input) INTEGER
The order of the matrix sub( A ). N >= 0.
NRHS (global input) INTEGER
The number of right hand sides, i.e., the number of columns
of the matrices sub( B ) and sub( X ). NRHS >= 0.
A (local input) DOUBLE PRECISION pointer into the local
memory to an array of local dimension (LLD_A,LOCc(JA+N-1)).
This array contains the local pieces of the distributed
matrix sub( A ).
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.
AF (local input) DOUBLE PRECISION pointer into the local
memory to an array of local dimension (LLD_AF,LOCc(JA+N-1)).
This array contains the local pieces of the distributed
factors of the matrix sub( A ) = P * L * U as computed by
PDGETRF.
IAF (global input) INTEGER
The row index in the global array AF indicating the first
row of sub( AF ).
JAF (global input) INTEGER
The column index in the global array AF indicating the
first column of sub( AF ).
DESCAF (global and local input) INTEGER array of dimension DLEN_.
The array descriptor for the distributed matrix AF.
IPIV (local input) INTEGER array of dimension LOCr(M_AF)+MB_AF.
This array contains the pivoting information as computed
by PDGETRF. IPIV(i) -> The global row local row i
was swapped with. This array is tied to the distributed
matrix A.
B (local input) DOUBLE PRECISION pointer into the local
memory to an array of local dimension
(LLD_B,LOCc(JB+NRHS-1)). This array contains the local
pieces of the distributed matrix of right hand sides
sub( B ).
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.
X (local input and output) DOUBLE PRECISION pointer into the
local memory to an array of local dimension
(LLD_X,LOCc(JX+NRHS-1)). On entry, this array contains
the local pieces of the distributed matrix solution
sub( X ). On exit, the improved solution vectors.
IX (global input) INTEGER
The row index in the global array X indicating the first
row of sub( X ).
JX (global input) INTEGER
The column index in the global array X indicating the
first column of sub( X ).
DESCX (global and local input) INTEGER array of dimension DLEN_.
The array descriptor for the distributed matrix X.
FERR (local output) DOUBLE PRECISION array of local dimension
LOCc(JB+NRHS-1).
The estimated forward error bound for each solution vector
of sub( X ). If XTRUE is the true solution corresponding
to sub( X ), FERR is an estimated upper bound for the
magnitude of the largest element in (sub( X ) - XTRUE)
divided by the magnitude of the largest element in sub( X ).
The estimate is as reliable as the estimate for RCOND, and
is almost always a slight overestimate of the true error.
This array is tied to the distributed matrix X.
BERR (local output) DOUBLE PRECISION array of local dimension
LOCc(JB+NRHS-1). The componentwise relative backward
error of each solution vector (i.e., the smallest re-
lative change in any entry of sub( A ) or sub( B )
that makes sub( X ) an exact solution).
This array is tied to the distributed matrix X.
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 >= 3*LOCr( N + MOD(IA-1,MB_A) )
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.
dimension (LIWORK)
On exit, IWORK(1) returns the minimal and optimal LIWORK.
LIWORK (local or global input) INTEGER
The dimension of the array IWORK.
LIWORK is local input and must be at least
LIWORK >= LOCr( N + MOD(IB-1,MB_B) ).
If LIWORK = -1, then LIWORK 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.
PARAMETERS
ITMAX is the maximum number of steps of iterative refinement.
Notes
=====
This routine temporarily returns when N <= 1.
The distributed submatrices op( A ) and op( AF ) (respectively
sub( X ) and sub( B ) ) should be distributed the same way on the
same processes. These conditions ensure that sub( A ) and sub( AF )
(resp. sub( X ) and sub( B ) ) are "perfectly" aligned.
Moreover, this routine requires the distributed submatrices sub( A ),
sub( AF ), sub( X ), and sub( B ) to be aligned on a block boundary,
i.e., if f(x,y) = MOD( x-1, y ):
f( IA, DESCA( MB_ ) ) = f( JA, DESCA( NB_ ) ) = 0,
f( IAF, DESCAF( MB_ ) ) = f( JAF, DESCAF( NB_ ) ) = 0,
f( IB, DESCB( MB_ ) ) = f( JB, DESCB( NB_ ) ) = 0, and
f( IX, DESCX( MB_ ) ) = f( JX, DESCX( NB_ ) ) = 0.