NAME
PCPOSVX - use the Cholesky factorization A = U**H*U or A = L*L**H to
compute the solution to a complex 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 PCPOSVX( 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,
RWORK, LRWORK, INFO )
CHARACTER EQUED, FACT, UPLO
INTEGER IA, IAF, IB, INFO, IX, JA, JAF, JB, JX, LRWORK,
LWORK, N, NRHS
REAL RCOND
INTEGER DESCA( * ), DESCAF( * ), DESCB( * ), DESCX( * )
REAL BERR( * ), FERR( * ), SC( * ), SR( * ), RWORK( * )
COMPLEX A( * ), AF( * ), B( * ), WORK( * ), X( * )
PURPOSE
PCPOSVX uses the Cholesky factorization A = U**H*U or A = L*L**H to
compute the solution to a complex 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) COMPLEX pointer into
the local memory to an array of local dimension ( LLD_A,
LOCc(JA+N-1) ). On entry, the Hermitian 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) COMPLEX 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) COMPLEX 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) COMPLEX 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) COMPLEX 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) COMPLEX 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) REAL
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) REAL 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) REAL 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) COMPLEX 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( PCPOCON( LWORK ), PCPORFS( 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.
RWORK (local workspace/local output) REAL 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 = 2*LOCc(N_A).
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 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.