Provided by: scalapack-doc_1.5-10_all 
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.