*> \brief ** DGTSVX computes the solution to system of linear equations A * X = B for GT matrices **
*
* =========== DOCUMENTATION ===========
*
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*
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*
* Definition:
* ===========
*
* SUBROUTINE DGTSVX( FACT, TRANS, N, NRHS, DL, D, DU, DLF, DF, DUF,
* DU2, IPIV, B, LDB, X, LDX, RCOND, FERR, BERR,
* WORK, IWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER FACT, TRANS
* INTEGER INFO, LDB, LDX, N, NRHS
* DOUBLE PRECISION RCOND
* ..
* .. Array Arguments ..
* INTEGER IPIV( * ), IWORK( * )
* DOUBLE PRECISION B( LDB, * ), BERR( * ), D( * ), DF( * ),
* $ DL( * ), DLF( * ), DU( * ), DU2( * ), DUF( * ),
* $ FERR( * ), WORK( * ), X( LDX, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGTSVX uses the LU factorization to compute the solution to a real
*> system of linear equations A * X = B or A**T * X = B,
*> where A is a tridiagonal matrix of order N 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 LU decomposition is used to factor the matrix A
*> as A = L * U, where L is a product of permutation and unit lower
*> bidiagonal matrices and U is upper triangular with nonzeros in
*> only the main diagonal and first two superdiagonals.
*>
*> 2. If some U(i,i)=0, so that U is exactly singular, 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': DLF, DF, DUF, DU2, and IPIV contain the factored
*> form of A; DL, D, DU, DLF, DF, DUF, DU2 and IPIV
*> will not be modified.
*> = 'N': The matrix will be copied to DLF, DF, and DUF
*> and factored.
*> \endverbatim
*>
*> \param[in] TRANS
*> \verbatim
*> TRANS is CHARACTER*1
*> Specifies the form of the system of equations:
*> = 'N': A * X = B (No transpose)
*> = 'T': A**T * X = B (Transpose)
*> = 'C': A**H * X = B (Conjugate transpose = Transpose)
*> \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 matrix B. NRHS >= 0.
*> \endverbatim
*>
*> \param[in] DL
*> \verbatim
*> DL is DOUBLE PRECISION array, dimension (N-1)
*> The (n-1) subdiagonal elements of A.
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> The n diagonal elements of A.
*> \endverbatim
*>
*> \param[in] DU
*> \verbatim
*> DU is DOUBLE PRECISION array, dimension (N-1)
*> The (n-1) superdiagonal elements of A.
*> \endverbatim
*>
*> \param[in,out] DLF
*> \verbatim
*> DLF is DOUBLE PRECISION array, dimension (N-1)
*> If FACT = 'F', then DLF is an input argument and on entry
*> contains the (n-1) multipliers that define the matrix L from
*> the LU factorization of A as computed by DGTTRF.
*>
*> If FACT = 'N', then DLF is an output argument and on exit
*> contains the (n-1) multipliers that define the matrix L from
*> the LU factorization of 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 upper triangular
*> matrix U from the LU factorization of A.
*>
*> If FACT = 'N', then DF is an output argument and on exit
*> contains the n diagonal elements of the upper triangular
*> matrix U from the LU factorization of A.
*> \endverbatim
*>
*> \param[in,out] DUF
*> \verbatim
*> DUF is DOUBLE PRECISION array, dimension (N-1)
*> If FACT = 'F', then DUF is an input argument and on entry
*> contains the (n-1) elements of the first superdiagonal of U.
*>
*> If FACT = 'N', then DUF is an output argument and on exit
*> contains the (n-1) elements of the first superdiagonal of U.
*> \endverbatim
*>
*> \param[in,out] DU2
*> \verbatim
*> DU2 is DOUBLE PRECISION array, dimension (N-2)
*> If FACT = 'F', then DU2 is an input argument and on entry
*> contains the (n-2) elements of the second superdiagonal of
*> U.
*>
*> If FACT = 'N', then DU2 is an output argument and on exit
*> contains the (n-2) elements of the second superdiagonal of
*> U.
*> \endverbatim
*>
*> \param[in,out] IPIV
*> \verbatim
*> IPIV is INTEGER array, dimension (N)
*> If FACT = 'F', then IPIV is an input argument and on entry
*> contains the pivot indices from the LU factorization of A as
*> computed by DGTTRF.
*>
*> If FACT = 'N', then IPIV is an output argument and on exit
*> contains the pivot indices from the LU factorization of A;
*> row i of the matrix was interchanged with row IPIV(i).
*> IPIV(i) will always be either i or i+1; IPIV(i) = i indicates
*> a row interchange was not required.
*> \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 or 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 estimate of 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 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: U(i,i) is exactly zero. The factorization
*> has not been completed unless i = N, but the
*> factor U is exactly singular, so the solution
*> and error bounds could not be 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 December 2016
*
*> \ingroup doubleGTsolve
*
* =====================================================================
SUBROUTINE DGTSVX( FACT, TRANS, N, NRHS, DL, D, DU, DLF, DF, DUF,
$ DU2, IPIV, 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..--
* December 2016
*
* .. Scalar Arguments ..
CHARACTER FACT, TRANS
INTEGER INFO, LDB, LDX, N, NRHS
DOUBLE PRECISION RCOND
* ..
* .. Array Arguments ..
INTEGER IPIV( * ), IWORK( * )
DOUBLE PRECISION B( LDB, * ), BERR( * ), D( * ), DF( * ),
$ DL( * ), DLF( * ), DU( * ), DU2( * ), DUF( * ),
$ FERR( * ), WORK( * ), X( LDX, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL NOFACT, NOTRAN
CHARACTER NORM
DOUBLE PRECISION ANORM
* ..
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DLAMCH, DLANGT
EXTERNAL LSAME, DLAMCH, DLANGT
* ..
* .. External Subroutines ..
EXTERNAL DCOPY, DGTCON, DGTRFS, DGTTRF, DGTTRS, DLACPY,
$ XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. Executable Statements ..
*
INFO = 0
NOFACT = LSAME( FACT, 'N' )
NOTRAN = LSAME( TRANS, 'N' )
IF( .NOT.NOFACT .AND. .NOT.LSAME( FACT, 'F' ) ) THEN
INFO = -1
ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
$ LSAME( TRANS, 'C' ) ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( NRHS.LT.0 ) THEN
INFO = -4
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -14
ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
INFO = -16
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGTSVX', -INFO )
RETURN
END IF
*
IF( NOFACT ) THEN
*
* Compute the LU factorization of A.
*
CALL DCOPY( N, D, 1, DF, 1 )
IF( N.GT.1 ) THEN
CALL DCOPY( N-1, DL, 1, DLF, 1 )
CALL DCOPY( N-1, DU, 1, DUF, 1 )
END IF
CALL DGTTRF( N, DLF, DF, DUF, DU2, IPIV, 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.
*
IF( NOTRAN ) THEN
NORM = '1'
ELSE
NORM = 'I'
END IF
ANORM = DLANGT( NORM, N, DL, D, DU )
*
* Compute the reciprocal of the condition number of A.
*
CALL DGTCON( NORM, N, DLF, DF, DUF, DU2, IPIV, ANORM, RCOND, WORK,
$ IWORK, INFO )
*
* Compute the solution vectors X.
*
CALL DLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
CALL DGTTRS( TRANS, N, NRHS, DLF, DF, DUF, DU2, IPIV, X, LDX,
$ INFO )
*
* Use iterative refinement to improve the computed solutions and
* compute error bounds and backward error estimates for them.
*
CALL DGTRFS( TRANS, N, NRHS, DL, D, DU, DLF, DF, DUF, DU2, IPIV,
$ B, LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO )
*
* Set INFO = N+1 if the matrix is singular to working precision.
*
IF( RCOND.LT.DLAMCH( 'Epsilon' ) )
$ INFO = N + 1
*
RETURN
*
* End of DGTSVX
*
END