PZPORFS - improve the computed solution to a system of linear equations when the
coefficient matrix is Hermitian positive definite and provides error bounds
and backward error estimates for the solutions
- SUBROUTINE PZPORFS(
- UPLO, N, NRHS, A, IA, JA, DESCA, AF, IAF, JAF, DESCAF, B, IB, JB, DESCB,
X, IX, JX, DESCX, FERR, BERR, WORK, LWORK, RWORK, LRWORK, INFO )
CHARACTER UPLO INTEGER IA, IAF, IB, INFO, IX, JA, JAF, JB, JX, LRWORK, LWORK, N,
NRHS INTEGER DESCA( * ), DESCAF( * ), DESCB( * ), DESCX( * ) COMPLEX*16 A( *
), AF( * ), B( * ), WORK( * ), X( * ) DOUBLE PRECISION BERR( * ), FERR( * ),
RWORK( * )
PZPORFS improves the computed solution to a system of linear equations when the
coefficient matrix is Hermitian positive definite 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).
- UPLO (global input) CHARACTER*1
- Specifies whether the upper or lower triangular part of the Hermitian
matrix sub( A ) is stored. = 'U': Upper triangular
= 'L': Lower 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*16 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 N-by-N Hermitian distributed matrix sub(
A ) to be factored. 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.
- 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) COMPLEX*16 pointer into the local memory
- to an array of local dimension (LLD_AF,LOCc(JA+N-1)). On entry, this array
contains the factors L or U from the Cholesky factorization sub( A ) =
L*L**H or U**H*U, as computed by PZPOTRF.
- 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.
- B (local input) COMPLEX*16 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*16 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 ). On
exit, it contains 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) COMPLEX*16 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) DOUBLE PRECISION 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.
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.