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),
- 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( * )
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 A(IA:IA+N-1,JA:JA+N-1) * X =
B(IB:IB+N-1,JB:JB+NRHS-1), 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
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.
- 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.