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PDPORFS

improve the computed solution to a system of linear equations when the coefficient matrix is symmetric positive definite and provides error bounds and backward error estimates for the solutions

UPLO, N, NRHS, A, IA, JA, DESCA, AF, IAF, JAF, DESCAF, B, IB, JB, DESCB, X, IX, JX, DESCX, FERR, BERR, WORK, LWORK, IWORK, LIWORK, INFO )

CHARACTER UPLO INTEGER IA, IAF, IB, INFO, IX, JA, JAF, JB, JX, LIWORK, LWORK, N, NRHS INTEGER DESCA( * ), DESCAF( * ), DESCB( * ), DESCX( * ), IWORK( * ) DOUBLE PRECISION A( * ), AF( * ), B( * ), BERR( * ), FERR( * ), WORK( * ), X( * )

PDPORFS improves the computed solution to a system of linear equations when the coefficient matrix is symmetric 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).

Specifies whether the upper or lower triangular part of the symmetric matrix sub( A ) is stored. = 'U': Upper triangular
= 'L': Lower triangular
The order of the matrix sub( A ). N >= 0.
The number of right hand sides, i.e., the number of columns of the matrices sub( B ) and sub( X ). NRHS >= 0.
memory to an array of local dimension (LLD_A,LOCc(JA+N-1) ). This array contains the local pieces of the N-by-N symmetric 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.
The row index in the global array A indicating the first row of sub( A ).
The column index in the global array A indicating the first column of sub( A ).
The array descriptor for the distributed matrix A.
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**T or U**T*U, as computed by PDPOTRF.
The row index in the global array AF indicating the first row of sub( AF ).
The column index in the global array AF indicating the first column of sub( AF ).
The array descriptor for the distributed matrix AF.
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 ).
The row index in the global array B indicating the first row of sub( B ).
The column index in the global array B indicating the first column of sub( B ).
The array descriptor for the distributed matrix B.
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.
The row index in the global array X indicating the first row of sub( X ).
The column index in the global array X indicating the first column of sub( X ).
The array descriptor for the distributed matrix X.
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.
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.
dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK.
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.
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.

= 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.