Provided by: scalapack-doc_1.5-11_all 
NAME
PDPOSVX - use the Cholesky factorization A = U**T*U or A = L*L**T to compute the solution to a real
system of linear equations A(IA:IA+N-1,JA:JA+N-1) * X = B(IB:IB+N-1,JB:JB+NRHS-1),
SYNOPSIS
SUBROUTINE PDPOSVX( FACT, UPLO, N, NRHS, A, IA, JA, DESCA, AF, IAF, JAF, DESCAF, EQUED, SR, SC, B, IB,
JB, DESCB, X, IX, JX, DESCX, RCOND, FERR, BERR, WORK, LWORK, IWORK, LIWORK, INFO )
CHARACTER EQUED, FACT, UPLO
INTEGER IA, IAF, IB, INFO, IX, JA, JAF, JB, JX, LIWORK, LWORK, N, NRHS
DOUBLE PRECISION RCOND
INTEGER DESCA( * ), DESCAF( * ), DESCB( * ), DESCX( * ), IWORK( * )
DOUBLE PRECISION A( * ), AF( * ), B( * ), BERR( * ), FERR( * ), SC( * ), SR( * ), WORK( * ),
X( * )
PURPOSE
PDPOSVX uses the Cholesky factorization A = U**T*U or A = L*L**T to compute the solution to a real system
of linear equations
where A(IA:IA+N-1,JA:JA+N-1) is an N-by-N matrix and X and B(IB:IB+N-1,JB:JB+NRHS-1) are N-by-NRHS
matrices.
Error bounds on the solution and a condition estimate are also provided. In the following comments Y
denotes Y(IY:IY+M-1,JY:JY+K-1) a M-by-K matrix where Y can be A, AF, B and X.
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
DESCRIPTION
The following steps are performed:
1. If FACT = 'E', real scaling factors are computed to equilibrate
the system:
diag(SR) * A * diag(SC) * inv(diag(SC)) * X = diag(SR) * B
Whether or not the system will be equilibrated depends on the
scaling of the matrix A, but if equilibration is used, A is
overwritten by diag(SR)*A*diag(SC) and B by diag(SR)*B.
2. If FACT = 'N' or 'E', the Cholesky decomposition is used to
factor the matrix A (after equilibration if FACT = 'E') as
A = U**T* U, if UPLO = 'U', or
A = L * L**T, if UPLO = 'L',
where U is an upper triangular matrix and L is a lower triangular
matrix.
3. The factored form of A is used to estimate the condition number
of the matrix A. If the reciprocal of the condition number is
less than machine precision, steps 4-6 are skipped.
4. The system of equations is solved for X using the factored form
of A.
5. Iterative refinement is applied to improve the computed solution
matrix and calculate error bounds and backward error estimates
for it.
6. If equilibration was used, the matrix X is premultiplied by
diag(SR) so that it solves the original system before
equilibration.
ARGUMENTS
FACT (global input) CHARACTER
Specifies whether or not the factored form of the matrix A is supplied on entry, and if not,
whether the matrix A should be equilibrated before it is factored. = 'F': On entry, AF contains
the factored form of A. If EQUED = 'Y', the matrix A has been equilibrated with scaling factors
given by S. A and AF will not be modified. = 'N': The matrix A will be copied to AF and
factored.
= 'E': The matrix A will be equilibrated if necessary, then copied to AF and factored.
UPLO (global input) CHARACTER
= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored.
N (global input) INTEGER
The number of rows and columns to be operated on, i.e. the order of the distributed submatrix
A(IA:IA+N-1,JA:JA+N-1). N >= 0.
NRHS (global input) INTEGER
The number of right hand sides, i.e., the number of columns of the distributed submatrices B and
X. NRHS >= 0.
A (local input/local output) DOUBLE PRECISION pointer into
the local memory to an array of local dimension ( LLD_A, LOCc(JA+N-1) ). On entry, the symmetric
matrix A, except if FACT = 'F' and EQUED = 'Y', then A must contain the equilibrated matrix
diag(SR)*A*diag(SC). If UPLO = 'U', the leading N-by-N upper triangular part of A contains the
upper triangular part of the matrix A, and the strictly lower triangular part of A is not
referenced. If UPLO = 'L', the leading N-by-N lower triangular part of A contains the lower
triangular part of the matrix A, and the strictly upper triangular part of A is not referenced.
A is not modified if FACT = 'F' or 'N', or if FACT = 'E' and EQUED = 'N' on exit.
On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by diag(SR)*A*diag(SC).
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 or local output) DOUBLE PRECISION pointer
into the local memory to an array of local dimension ( LLD_AF, LOCc(JA+N-1)). If FACT = 'F',
then AF is an input argument and on entry contains the triangular factor U or L from the Cholesky
factorization A = U**T*U or A = L*L**T, in the same storage format as A. If EQUED .ne. 'N', then
AF is the factored form of the equilibrated matrix diag(SR)*A*diag(SC).
If FACT = 'N', then AF is an output argument and on exit returns the triangular factor U or L
from the Cholesky factorization A = U**T*U or A = L*L**T of the original matrix A.
If FACT = 'E', then AF is an output argument and on exit returns the triangular factor U or L
from the Cholesky factorization A = U**T*U or A = L*L**T of the equilibrated matrix A (see the
description of A for the form of the equilibrated matrix).
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.
EQUED (global input/global output) CHARACTER
Specifies the form of equilibration that was done. = 'N': No equilibration (always true if FACT
= 'N').
= 'Y': Equilibration was done, i.e., A has been replaced by diag(SR) * A * diag(SC). EQUED is
an input variable if FACT = 'F'; otherwise, it is an output variable.
SR (local input/local output) DOUBLE PRECISION array,
dimension (LLD_A) The scale factors for A distributed across process rows; not accessed if EQUED
= 'N'. SR is an input variable if FACT = 'F'; otherwise, SR is an output variable. If FACT =
'F' and EQUED = 'Y', each element of SR must be positive.
SC (local input/local output) DOUBLE PRECISION array,
dimension (LOC(N_A)) The scale factors for A distributed across process columns; not accessed if
EQUED = 'N'. SC is an input variable if FACT = 'F'; otherwise, SC is an output variable. If FACT
= 'F' and EQUED = 'Y', each element of SC must be positive.
B (local input/local output) DOUBLE PRECISION pointer into
the local memory to an array of local dimension ( LLD_B, LOCc(JB+NRHS-1) ). On entry, the N-by-
NRHS right-hand side matrix B. On exit, if EQUED = 'N', B is not modified; if TRANS = 'N' and
EQUED = 'R' or 'B', B is overwritten by diag(R)*B; if TRANS = 'T' or 'C' and EQUED = 'C' or 'B',
B is overwritten by diag(C)*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/local output) DOUBLE PRECISION pointer into
the local memory to an array of local dimension ( LLD_X, LOCc(JX+NRHS-1) ). If INFO = 0, the N-
by-NRHS solution matrix X to the original system of equations. Note that A and B are modified on
exit if EQUED .ne. 'N', and the solution to the equilibrated system is inv(diag(SC))*X if TRANS =
'N' and EQUED = 'C' or or 'B'.
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.
RCOND (global output) DOUBLE PRECISION
The estimate of the reciprocal condition number of the matrix A after equilibration (if done).
If RCOND is less than the machine precision (in particular, if RCOND = 0), the matrix is singular
to working precision. This condition is indicated by a return code of INFO > 0, and the solution
and error bounds are not computed.
FERR (local output) DOUBLE PRECISION array, dimension (LOC(N_B))
The estimated forward error bounds for each solution vector X(j) (the j-th column of the solution
matrix X). If XTRUE is the true solution, FERR(j) bounds the magnitude of the largest entry in
(X(j) - XTRUE) divided by the magnitude of the largest entry in X(j). The quality of the error
bound depends on the quality of the estimate of norm(inv(A)) computed in the code; if the
estimate of norm(inv(A)) is accurate, the error bound is guaranteed.
BERR (local output) DOUBLE PRECISION array, dimension (LOC(N_B))
The componentwise relative backward error of each solution vector X(j) (i.e., the smallest
relative change in any entry of A or B that makes X(j) an exact solution).
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 = MAX( PDPOCON(
LWORK ), PDPORFS( LWORK ) ) + LOCr( N_A ). LWORK = 3*DESCA( LLD_ )
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 = DESCA(
LLD_ ) LIWORK = LOCr(N_A).
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 INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, and i is
<= N: if INFO = i, the leading minor of order i of A is not positive definite, so the
factorization could not be completed, and the solution and error bounds could not be computed. =
N+1: RCOND is less than machine precision. The factorization has been completed, but the matrix
is singular to working precision, and the solution and error bounds have not been computed.