NAME
PSPORFS - 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
SYNOPSIS
SUBROUTINE PSPORFS( 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( * )
REAL A( * ), AF( * ), B( * ), BERR( * ), FERR( * ),
WORK( * ), X( * )
PURPOSE
PSPORFS 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).
ARGUMENTS
UPLO (global input) CHARACTER*1
Specifies whether the upper or lower triangular part of the
symmetric 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) REAL 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 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.
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) REAL 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**T or U**T*U, as computed by
PSPOTRF.
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) REAL 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) REAL 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) REAL 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) 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) REAL 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.
PARAMETERS
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.