*> \brief ** DPOSVX computes the solution to system of linear equations A * X = B for PO matrices**
*
* =========== DOCUMENTATION ===========
*
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*
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*
* Definition:
* ===========
*
* SUBROUTINE DPOSVX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, EQUED,
* S, B, LDB, X, LDX, RCOND, FERR, BERR, WORK,
* IWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER EQUED, FACT, UPLO
* INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS
* DOUBLE PRECISION RCOND
* ..
* .. Array Arguments ..
* INTEGER IWORK( * )
* DOUBLE PRECISION A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
* $ BERR( * ), FERR( * ), S( * ), WORK( * ),
* $ X( LDX, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DPOSVX 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 * X = B,
*> where A is an N-by-N symmetric positive definite matrix and X and B
*> are N-by-NRHS matrices.
*>
*> Error bounds on the solution and a condition estimate are also
*> provided.
*> \endverbatim
*
*> \par Description:
* =================
*>
*> \verbatim
*>
*> The following steps are performed:
*>
*> 1. If FACT = 'E', real scaling factors are computed to equilibrate
*> the system:
*> diag(S) * A * diag(S) * inv(diag(S)) * X = diag(S) * 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(S)*A*diag(S) and B by diag(S)*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. If the leading i-by-i principal minor is not positive definite,
*> then the routine returns with INFO = i. Otherwise, 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, INFO = N+1 is returned as a warning, but the routine
*> still goes on to solve for X and compute error bounds as
*> described below.
*>
*> 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(S) so that it solves the original system before
*> equilibration.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] FACT
*> \verbatim
*> FACT is CHARACTER*1
*> 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.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': Upper triangle of A is stored;
*> = 'L': Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of linear equations, i.e., the order of the
*> matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*> NRHS is INTEGER
*> The number of right hand sides, i.e., the number of columns
*> of the matrices B and X. NRHS >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the symmetric matrix A, except if FACT = 'F' and
*> EQUED = 'Y', then A must contain the equilibrated matrix
*> diag(S)*A*diag(S). 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(S)*A*diag(S).
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] AF
*> \verbatim
*> AF is DOUBLE PRECISION array, dimension (LDAF,N)
*> 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(S)*A*diag(S).
*>
*> 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).
*> \endverbatim
*>
*> \param[in] LDAF
*> \verbatim
*> LDAF is INTEGER
*> The leading dimension of the array AF. LDAF >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] EQUED
*> \verbatim
*> EQUED is CHARACTER*1
*> 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(S) * A * diag(S).
*> EQUED is an input argument if FACT = 'F'; otherwise, it is an
*> output argument.
*> \endverbatim
*>
*> \param[in,out] S
*> \verbatim
*> S is DOUBLE PRECISION array, dimension (N)
*> The scale factors for A; not accessed if EQUED = 'N'. S is
*> an input argument if FACT = 'F'; otherwise, S is an output
*> argument. If FACT = 'F' and EQUED = 'Y', each element of S
*> must be positive.
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*> On entry, the N-by-NRHS right hand side matrix B.
*> On exit, if EQUED = 'N', B is not modified; if EQUED = 'Y',
*> B is overwritten by diag(S) * B.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] X
*> \verbatim
*> X is DOUBLE PRECISION array, dimension (LDX,NRHS)
*> If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X to
*> the original system of equations. Note that if EQUED = 'Y',
*> A and B are modified on exit, and the solution to the
*> equilibrated system is inv(diag(S))*X.
*> \endverbatim
*>
*> \param[in] LDX
*> \verbatim
*> LDX is INTEGER
*> The leading dimension of the array X. LDX >= max(1,N).
*> \endverbatim
*>
*> \param[out] RCOND
*> \verbatim
*> RCOND is 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.
*> \endverbatim
*>
*> \param[out] FERR
*> \verbatim
*> FERR is DOUBLE PRECISION array, dimension (NRHS)
*> The estimated forward error bound for each solution vector
*> X(j) (the j-th column of the solution matrix X).
*> If XTRUE is the true solution corresponding to X(j), FERR(j)
*> is an estimated upper bound for the magnitude of the largest
*> element in (X(j) - XTRUE) divided by the magnitude of the
*> largest element in X(j). The estimate is as reliable as
*> the estimate for RCOND, and is almost always a slight
*> overestimate of the true error.
*> \endverbatim
*>
*> \param[out] BERR
*> \verbatim
*> BERR is DOUBLE PRECISION array, dimension (NRHS)
*> The componentwise relative backward error of each solution
*> vector X(j) (i.e., the smallest relative change in
*> any element of A or B that makes X(j) an exact solution).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (3*N)
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: if INFO = i, and i is
*> <= N: the leading minor of order i of A is
*> not positive definite, so the factorization
*> could not be completed, and the solution has not
*> been computed. RCOND = 0 is returned.
*> = N+1: U is nonsingular, but RCOND is less than machine
*> precision, meaning that the matrix is singular
*> to working precision. Nevertheless, the
*> solution and error bounds are computed because
*> there are a number of situations where the
*> computed solution can be more accurate than the
*> value of RCOND would suggest.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date April 2012
*
*> \ingroup doublePOsolve
*
* =====================================================================
SUBROUTINE DPOSVX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, EQUED,
$ S, B, LDB, X, LDX, RCOND, FERR, BERR, WORK,
$ IWORK, INFO )
*
* -- LAPACK driver routine (version 3.7.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* April 2012
*
* .. Scalar Arguments ..
CHARACTER EQUED, FACT, UPLO
INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS
DOUBLE PRECISION RCOND
* ..
* .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
$ BERR( * ), FERR( * ), S( * ), WORK( * ),
$ X( LDX, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL EQUIL, NOFACT, RCEQU
INTEGER I, INFEQU, J
DOUBLE PRECISION AMAX, ANORM, BIGNUM, SCOND, SMAX, SMIN, SMLNUM
* ..
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DLAMCH, DLANSY
EXTERNAL LSAME, DLAMCH, DLANSY
* ..
* .. External Subroutines ..
EXTERNAL DLACPY, DLAQSY, DPOCON, DPOEQU, DPORFS, DPOTRF,
$ DPOTRS, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. Executable Statements ..
*
INFO = 0
NOFACT = LSAME( FACT, 'N' )
EQUIL = LSAME( FACT, 'E' )
IF( NOFACT .OR. EQUIL ) THEN
EQUED = 'N'
RCEQU = .FALSE.
ELSE
RCEQU = LSAME( EQUED, 'Y' )
SMLNUM = DLAMCH( 'Safe minimum' )
BIGNUM = ONE / SMLNUM
END IF
*
* Test the input parameters.
*
IF( .NOT.NOFACT .AND. .NOT.EQUIL .AND. .NOT.LSAME( FACT, 'F' ) )
$ THEN
INFO = -1
ELSE IF( .NOT.LSAME( UPLO, 'U' ) .AND. .NOT.LSAME( UPLO, 'L' ) )
$ THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( NRHS.LT.0 ) THEN
INFO = -4
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -6
ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN
INFO = -8
ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT.
$ ( RCEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN
INFO = -9
ELSE
IF( RCEQU ) THEN
SMIN = BIGNUM
SMAX = ZERO
DO 10 J = 1, N
SMIN = MIN( SMIN, S( J ) )
SMAX = MAX( SMAX, S( J ) )
10 CONTINUE
IF( SMIN.LE.ZERO ) THEN
INFO = -10
ELSE IF( N.GT.0 ) THEN
SCOND = MAX( SMIN, SMLNUM ) / MIN( SMAX, BIGNUM )
ELSE
SCOND = ONE
END IF
END IF
IF( INFO.EQ.0 ) THEN
IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -12
ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
INFO = -14
END IF
END IF
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DPOSVX', -INFO )
RETURN
END IF
*
IF( EQUIL ) THEN
*
* Compute row and column scalings to equilibrate the matrix A.
*
CALL DPOEQU( N, A, LDA, S, SCOND, AMAX, INFEQU )
IF( INFEQU.EQ.0 ) THEN
*
* Equilibrate the matrix.
*
CALL DLAQSY( UPLO, N, A, LDA, S, SCOND, AMAX, EQUED )
RCEQU = LSAME( EQUED, 'Y' )
END IF
END IF
*
* Scale the right hand side.
*
IF( RCEQU ) THEN
DO 30 J = 1, NRHS
DO 20 I = 1, N
B( I, J ) = S( I )*B( I, J )
20 CONTINUE
30 CONTINUE
END IF
*
IF( NOFACT .OR. EQUIL ) THEN
*
* Compute the Cholesky factorization A = U**T *U or A = L*L**T.
*
CALL DLACPY( UPLO, N, N, A, LDA, AF, LDAF )
CALL DPOTRF( UPLO, N, AF, LDAF, INFO )
*
* Return if INFO is non-zero.
*
IF( INFO.GT.0 )THEN
RCOND = ZERO
RETURN
END IF
END IF
*
* Compute the norm of the matrix A.
*
ANORM = DLANSY( '1', UPLO, N, A, LDA, WORK )
*
* Compute the reciprocal of the condition number of A.
*
CALL DPOCON( UPLO, N, AF, LDAF, ANORM, RCOND, WORK, IWORK, INFO )
*
* Compute the solution matrix X.
*
CALL DLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
CALL DPOTRS( UPLO, N, NRHS, AF, LDAF, X, LDX, INFO )
*
* Use iterative refinement to improve the computed solution and
* compute error bounds and backward error estimates for it.
*
CALL DPORFS( UPLO, N, NRHS, A, LDA, AF, LDAF, B, LDB, X, LDX,
$ FERR, BERR, WORK, IWORK, INFO )
*
* Transform the solution matrix X to a solution of the original
* system.
*
IF( RCEQU ) THEN
DO 50 J = 1, NRHS
DO 40 I = 1, N
X( I, J ) = S( I )*X( I, J )
40 CONTINUE
50 CONTINUE
DO 60 J = 1, NRHS
FERR( J ) = FERR( J ) / SCOND
60 CONTINUE
END IF
*
* Set INFO = N+1 if the matrix is singular to working precision.
*
IF( RCOND.LT.DLAMCH( 'Epsilon' ) )
$ INFO = N + 1
*
RETURN
*
* End of DPOSVX
*
END