*> \brief ** DPTSVX computes the solution to system of linear equations A * X = B for PT matrices**
*
* =========== DOCUMENTATION ===========
*
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*
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*
* Definition:
* ===========
*
* SUBROUTINE DPTSVX( FACT, N, NRHS, D, E, DF, EF, B, LDB, X, LDX,
* RCOND, FERR, BERR, WORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER FACT
* INTEGER INFO, LDB, LDX, N, NRHS
* DOUBLE PRECISION RCOND
* ..
* .. Array Arguments ..
* DOUBLE PRECISION B( LDB, * ), BERR( * ), D( * ), DF( * ),
* $ E( * ), EF( * ), FERR( * ), WORK( * ),
* $ X( LDX, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DPTSVX uses the factorization A = L*D*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 tridiagonal 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 = 'N', the matrix A is factored as A = L*D*L**T, where L
*> is a unit lower bidiagonal matrix and D is diagonal. The
*> factorization can also be regarded as having the form
*> A = U**T*D*U.
*>
*> 2. 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.
*>
*> 3. The system of equations is solved for X using the factored form
*> of A.
*>
*> 4. Iterative refinement is applied to improve the computed solution
*> matrix and calculate error bounds and backward error estimates
*> for it.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] FACT
*> \verbatim
*> FACT is CHARACTER*1
*> Specifies whether or not the factored form of A has been
*> supplied on entry.
*> = 'F': On entry, DF and EF contain the factored form of A.
*> D, E, DF, and EF will not be modified.
*> = 'N': The matrix A will be copied to DF and EF and
*> factored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> 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] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> The n diagonal elements of the tridiagonal matrix A.
*> \endverbatim
*>
*> \param[in] E
*> \verbatim
*> E is DOUBLE PRECISION array, dimension (N-1)
*> The (n-1) subdiagonal elements of the tridiagonal matrix A.
*> \endverbatim
*>
*> \param[in,out] DF
*> \verbatim
*> DF is DOUBLE PRECISION array, dimension (N)
*> If FACT = 'F', then DF is an input argument and on entry
*> contains the n diagonal elements of the diagonal matrix D
*> from the L*D*L**T factorization of A.
*> If FACT = 'N', then DF is an output argument and on exit
*> contains the n diagonal elements of the diagonal matrix D
*> from the L*D*L**T factorization of A.
*> \endverbatim
*>
*> \param[in,out] EF
*> \verbatim
*> EF is DOUBLE PRECISION array, dimension (N-1)
*> If FACT = 'F', then EF is an input argument and on entry
*> contains the (n-1) subdiagonal elements of the unit
*> bidiagonal factor L from the L*D*L**T factorization of A.
*> If FACT = 'N', then EF is an output argument and on exit
*> contains the (n-1) subdiagonal elements of the unit
*> bidiagonal factor L from the L*D*L**T factorization of A.
*> \endverbatim
*>
*> \param[in] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*> The N-by-NRHS right hand side matrix 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 of INFO = N+1, the N-by-NRHS solution matrix 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 reciprocal condition number of the matrix A. 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 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).
*> \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 (2*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.
*
*> \ingroup doublePTsolve
*
* =====================================================================
SUBROUTINE DPTSVX( FACT, N, NRHS, D, E, DF, EF, B, LDB, X, LDX,
$ RCOND, FERR, BERR, WORK, INFO )
*
* -- LAPACK driver routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER FACT
INTEGER INFO, LDB, LDX, N, NRHS
DOUBLE PRECISION RCOND
* ..
* .. Array Arguments ..
DOUBLE PRECISION B( LDB, * ), BERR( * ), D( * ), DF( * ),
$ E( * ), EF( * ), FERR( * ), WORK( * ),
$ X( LDX, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL NOFACT
DOUBLE PRECISION ANORM
* ..
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DLAMCH, DLANST
EXTERNAL LSAME, DLAMCH, DLANST
* ..
* .. External Subroutines ..
EXTERNAL DCOPY, DLACPY, DPTCON, DPTRFS, DPTTRF, DPTTRS,
$ XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
NOFACT = LSAME( FACT, 'N' )
IF( .NOT.NOFACT .AND. .NOT.LSAME( FACT, 'F' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( NRHS.LT.0 ) THEN
INFO = -3
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -9
ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
INFO = -11
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DPTSVX', -INFO )
RETURN
END IF
*
IF( NOFACT ) THEN
*
* Compute the L*D*L**T (or U**T*D*U) factorization of A.
*
CALL DCOPY( N, D, 1, DF, 1 )
IF( N.GT.1 )
$ CALL DCOPY( N-1, E, 1, EF, 1 )
CALL DPTTRF( N, DF, EF, 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 = DLANST( '1', N, D, E )
*
* Compute the reciprocal of the condition number of A.
*
CALL DPTCON( N, DF, EF, ANORM, RCOND, WORK, INFO )
*
* Compute the solution vectors X.
*
CALL DLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
CALL DPTTRS( N, NRHS, DF, EF, X, LDX, INFO )
*
* Use iterative refinement to improve the computed solutions and
* compute error bounds and backward error estimates for them.
*
CALL DPTRFS( N, NRHS, D, E, DF, EF, B, LDB, X, LDX, FERR, BERR,
$ WORK, INFO )
*
* Set INFO = N+1 if the matrix is singular to working precision.
*
IF( RCOND.LT.DLAMCH( 'Epsilon' ) )
$ INFO = N + 1
*
RETURN
*
* End of DPTSVX
*
END