PCTRRFS provides error bounds and backward error estimates for the
solution to a system of linear equations with a triangular
coefficient matrix.
The solution matrix X must be computed by PCTRTRS or some other
means before entering this routine. PCTRRFS does not do iterative
refinement because doing so cannot improve the backward error.
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
UPLO (global input) CHARACTER*1
= 'U': sub( A ) is upper triangular;
= 'L': sub( A ) is lower triangular.
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 )**H * sub( X ) = sub( B )
(Conjugate transpose)
DIAG (global input) CHARACTER*1
= 'N': sub( A ) is non-unit triangular;
= 'U': sub( A ) is unit triangular.
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) COMPLEX 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 original triangular
distributed matrix sub( A ).
If UPLO = 'U', the leading N-by-N upper triangular part of
sub( A ) contains the upper triangular part of the matrix,
and its strictly lower triangular part is not referenced.
If UPLO = 'L', the leading N-by-N lower triangular part of
sub( A ) contains the lower triangular part of the distribu-
ted matrix, and its strictly upper triangular part is not
referenced.
If DIAG = 'U', the diagonal elements of sub( A ) are also
not referenced and are assumed to be 1.
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) COMPLEX pointer into the local memory
to an array of local dimension (LLD_B, LOCc(JB+NRHS-1) ).
On entry, this array contains the the local pieces of the
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) COMPLEX pointer into the local memory
to an array of local dimension (LLD_X, LOCc(JX+NRHS-1) ).
On entry, this array contains the the local pieces of the
solution vectors sub( X ).
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) REAL array of local dimension
LOCc(JB+NRHS-1). The estimated forward error bounds for
each solution vector of sub( X ). If XTRUE is the true
solution, FERR bounds the magnitude of the largest entry
in (sub( X ) - XTRUE) divided by the magnitude of the
largest entry 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) REAL 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) COMPLEX 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 >= 2*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.
RWORK (local workspace/local output) REAL array,
dimension (LRWORK)
On exit, RWORK(1) returns the minimal and optimal LRWORK.
LRWORK (local or global input) INTEGER
The dimension of the array RWORK.
LRWORK is local input and must be at least
LRWORK >= LOCr( N + MOD( IB-1, MB_B ) ).
If LRWORK = -1, then LRWORK 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.
Notes
=====
This routine temporarily returns when N <= 1.
The distributed submatrices sub( X ) and sub( B ) should be
distributed the same way on the same processes. These conditions
ensure that sub( X ) and sub( B ) are "perfectly" aligned.
Moreover, this routine requires the distributed submatrices sub( A ),
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( IB, DESCB( MB_ ) ) = f( JB, DESCB( NB_ ) ) = 0, and
f( IX, DESCX( MB_ ) ) = f( JX, DESCX( NB_ ) ) = 0.