table of contents
PZPOSVX(l) | LAPACK routine (version 1.5) | PZPOSVX(l) |
NAME¶
PZPOSVX - 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 PZPOSVX(
- 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 )
PURPOSE¶
PZPOSVX 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.DTYPE_A = 1.
the BLACS process grid A is distribu-
ted over. The context itself is glo-
bal, but the handle (the integer
value) may vary.
array A.
array A.
the rows of the array.
the columns of the array.
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.
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( 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: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.
of the matrix A. If the reciprocal of the condition number is
less than machine precision, steps 4-6 are skipped.
of A.
matrix and calculate error bounds and backward error estimates
for it.
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.
- UPLO (global input) CHARACTER
- = 'U': Upper 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*16 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*16 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').
- SR (local input/local output) COMPLEX*16 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*16 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*16 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*16 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) COMPLEX*16 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( PZPOCON( LWORK ), PZPORFS( 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) DOUBLE PRECISION 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
12 May 1997 | LAPACK version 1.5 |