table of contents
PSGESVX(l) | LAPACK routine (version 1.5) | PSGESVX(l) |
NAME¶
PSGESVX - use the LU factorization 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 PSGESVX(
- FACT, TRANS, N, NRHS, A, IA, JA, DESCA, AF, IAF, JAF, DESCAF, IPIV, EQUED, R, C, B, IB, JB, DESCB, X, IX, JX, DESCX, RCOND, FERR, BERR, WORK, LWORK, IWORK, LIWORK, INFO )
PURPOSE¶
PSGESVX uses the LU factorization 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.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¶
In the following description, A denotes A(IA:IA+N-1,JA:JA+N-1), B denotes B(IB:IB+N-1,JB:JB+NRHS-1) and X denotesthe system:
TRANS = 'N': diag(R)*A*diag(C) *inv(diag(C))*X = diag(R)*B
TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B
TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*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(R)*A*diag(C) and B by diag(R)*B (if TRANS='N')
or diag(C)*B (if TRANS = 'T' or 'C').
matrix A (after equilibration if FACT = 'E') as
A = P * L * U,
where P is a permutation matrix, L is a unit lower triangular
matrix, and U is upper triangular.
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.
premultiplied by diag(C) (if TRANS = 'N') or diag(R) (if
TRANS = 'T' or 'C') 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(IA:IA+N-1,JA:JA+N-1) is supplied on entry, and if not,
- TRANS (global input) CHARACTER
-
- 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(IB:IB+N-1,JB:JB+NRHS-1) and
- A (local input/local output) REAL pointer into
- the local memory to an array of local dimension (LLD_A,LOCc(JA+N-1)). On
entry, the N-by-N matrix A(IA:IA+N-1,JA:JA+N-1). If FACT = 'F' and EQUED
is not 'N',
- 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) REAL pointer
- into the local memory to an array of local dimension
(LLD_AF,LOCc(JA+N-1)). If FACT = 'F', then AF(IAF:IAF+N-1,JAF:JAF+N-1) is
an input argument and on entry contains the factors L and U from the
factorization A(IA:IA+N-1,JA:JA+N-1) = P*L*U as computed by PSGETRF. If
EQUED .ne. 'N', then AF is the factored form of the equilibrated matrix
A(IA:IA+N-1,JA:JA+N-1).
If FACT = 'N', then AF(IAF:IAF+N-1,JAF:JAF+N-1) is an output argument and on
exit returns the factors L and U from the factorization
A(IA:IA+N-1,JA:JA+N-1) = P*L*U of the original
- 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.
- IPIV (local input or local output) INTEGER array, dimension
- LOCr(M_A)+MB_A. If FACT = 'F', then IPIV is an input argu- ment and on
entry contains the pivot indices from the fac- torization
A(IA:IA+N-1,JA:JA+N-1) = P*L*U as computed by PSGETRF; IPIV(i) -> The
global row local row i was swapped with. This array must be aligned with
A( IA:IA+N-1, * ).
If FACT = 'N', then IPIV is an output argument and on exit contains the
pivot indices from the factorization A(IA:IA+N-1,JA:JA+N-1) = P*L*U of the
original matrix
- EQUED (global input or global output) CHARACTER
- Specifies the form of equilibration that was done. = 'N': No equilibration
(always true if FACT = 'N').
- R (local input or local output) REAL array,
- dimension LOCr(M_A). The row scale factors for A(IA:IA+N-1,JA:JA+N-1).
- C (local input or local output) REAL array,
- dimension LOCc(N_A). The column scale factors for A(IA:IA+N-1,JA:JA+N-1).
- B (local input/local output) REAL 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(IB:IB+N-1,JB:JB+NRHS-1). On exit, if
- 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) REAL 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(IX:IX+N-1,JX:JX+NRHS-1) to the original
- 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(IA:IA+N-1,JA:JA+N-1) 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.
- FERR (local output) REAL array, dimension LOCc(N_B)
- The estimated forward error bounds for each solution vector X(j) (the j-th column of the solution matrix X(IX:IX+N-1,JX:JX+NRHS-1). 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 estimate is as reliable as the estimate for RCOND, and is almost always a slight overestimate of the true error. FERR is replicated in every process row, and is aligned with the matrices B and X.
- BERR (local output) REAL array, dimension LOCc(N_B).
- The componentwise relative backward error of each solution vector X(j)
(i.e., the smallest relative change in any entry of A(IA:IA+N-1,JA:JA+N-1)
or
- 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 = MAX( PSGECON( LWORK ), PSGERFS( LWORK ) ) + LOCr( N_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_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
12 May 1997 | LAPACK version 1.5 |