*> \brief ZPTSVX computes the solution to system of linear equations A * X = B for PT matrices * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download ZPTSVX + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE ZPTSVX( FACT, N, NRHS, D, E, DF, EF, B, LDB, X, LDX, * RCOND, FERR, BERR, WORK, RWORK, INFO ) * * .. Scalar Arguments .. * CHARACTER FACT * INTEGER INFO, LDB, LDX, N, NRHS * DOUBLE PRECISION RCOND * .. * .. Array Arguments .. * DOUBLE PRECISION BERR( * ), D( * ), DF( * ), FERR( * ), * \$ RWORK( * ) * COMPLEX*16 B( LDB, * ), E( * ), EF( * ), WORK( * ), * \$ X( LDX, * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> ZPTSVX uses the factorization A = L*D*L**H to compute the solution *> to a complex system of linear equations A*X = B, where A is an *> N-by-N Hermitian 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**H, where L *> is a unit lower bidiagonal matrix and D is diagonal. The *> factorization can also be regarded as having the form *> A = U**H*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 the matrix *> A is 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 COMPLEX*16 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**H 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**H factorization of A. *> \endverbatim *> *> \param[in,out] EF *> \verbatim *> EF is COMPLEX*16 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**H 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**H factorization of A. *> \endverbatim *> *> \param[in] B *> \verbatim *> B is COMPLEX*16 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 COMPLEX*16 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 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 COMPLEX*16 array, dimension (N) *> \endverbatim *> *> \param[out] RWORK *> \verbatim *> RWORK is DOUBLE PRECISION 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 December 2016 * *> \ingroup complex16PTsolve * * ===================================================================== SUBROUTINE ZPTSVX( FACT, N, NRHS, D, E, DF, EF, B, LDB, X, LDX, \$ RCOND, FERR, BERR, WORK, RWORK, 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 INTEGER INFO, LDB, LDX, N, NRHS DOUBLE PRECISION RCOND * .. * .. Array Arguments .. DOUBLE PRECISION BERR( * ), D( * ), DF( * ), FERR( * ), \$ RWORK( * ) COMPLEX*16 B( LDB, * ), E( * ), EF( * ), 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, ZLANHT EXTERNAL LSAME, DLAMCH, ZLANHT * .. * .. External Subroutines .. EXTERNAL DCOPY, XERBLA, ZCOPY, ZLACPY, ZPTCON, ZPTRFS, \$ ZPTTRF, ZPTTRS * .. * .. 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( 'ZPTSVX', -INFO ) RETURN END IF * IF( NOFACT ) THEN * * Compute the L*D*L**H (or U**H*D*U) factorization of A. * CALL DCOPY( N, D, 1, DF, 1 ) IF( N.GT.1 ) \$ CALL ZCOPY( N-1, E, 1, EF, 1 ) CALL ZPTTRF( 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 = ZLANHT( '1', N, D, E ) * * Compute the reciprocal of the condition number of A. * CALL ZPTCON( N, DF, EF, ANORM, RCOND, RWORK, INFO ) * * Compute the solution vectors X. * CALL ZLACPY( 'Full', N, NRHS, B, LDB, X, LDX ) CALL ZPTTRS( 'Lower', 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 ZPTRFS( 'Lower', N, NRHS, D, E, DF, EF, B, LDB, X, LDX, FERR, \$ BERR, WORK, RWORK, INFO ) * * Set INFO = N+1 if the matrix is singular to working precision. * IF( RCOND.LT.DLAMCH( 'Epsilon' ) ) \$ INFO = N + 1 * RETURN * * End of ZPTSVX * END