NAME
PDTRRFS - provide error bounds and backward error estimates for the
solution to a system of linear equations with a triangular coefficient
matrix
SYNOPSIS
SUBROUTINE PDTRRFS( UPLO, TRANS, DIAG, N, NRHS, A, IA, JA, DESCA, B,
IB, JB, DESCB, X, IX, JX, DESCX, FERR, BERR, WORK,
LWORK, IWORK, LIWORK, INFO )
CHARACTER DIAG, TRANS, UPLO
INTEGER INFO, IA, IB, IX, JA, JB, JX, LIWORK, LWORK, N,
NRHS
INTEGER DESCA( * ), DESCB( * ), DESCX( * ), IWORK( * )
DOUBLE PRECISION A( * ), B( * ), BERR( * ), FERR( * ),
WORK( * ), X( * )
PURPOSE
PDTRRFS 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 PDTRTRS or some other means
before entering this routine. PDTRRFS 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 )**T * sub( X ) = sub( B ) (Conjugate transpose =
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) 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 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) 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 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) 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 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) DOUBLE PRECISION 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) 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.
IWORK (local workspace/local output) INTEGER array,
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.
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.