zgtsvx()

subroutine zgtsvx	(	character	fact,
		character	trans,
		integer	n,
		integer	nrhs,
		complex*16, dimension( * )	dl,
		complex*16, dimension( * )	d,
		complex*16, dimension( * )	du,
		complex*16, dimension( * )	dlf,
		complex*16, dimension( * )	df,
		complex*16, dimension( * )	duf,
		complex*16, dimension( * )	du2,
		integer, dimension( * )	ipiv,
		complex*16, dimension( ldb, * )	b,
		integer	ldb,
		complex*16, dimension( ldx, * )	x,
		integer	ldx,
		double precision	rcond,
		double precision, dimension( * )	ferr,
		double precision, dimension( * )	berr,
		complex*16, dimension( * )	work,
		double precision, dimension( * )	rwork,
		integer	info )

ZGTSVX computes the solution to system of linear equations A * X = B for GT matrices

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Purpose:

!>
!> ZGTSVX uses the LU factorization to compute the solution to a complex
!> system of linear equations A * X = B, A**T * X = B, or A**H * 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.
!> 

Description:

!>
!> 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.
!> 

Parameters

[in]	FACT	!> 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. !>
[in]	TRANS	!> 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) !>
[in]	N	!> N is INTEGER !> The order of the matrix A. N >= 0. !>
[in]	NRHS	!> NRHS is INTEGER !> The number of right hand sides, i.e., the number of columns !> of the matrix B. NRHS >= 0. !>
[in]	DL	!> DL is COMPLEX*16 array, dimension (N-1) !> The (n-1) subdiagonal elements of A. !>
[in]	D	!> D is COMPLEX*16 array, dimension (N) !> The n diagonal elements of A. !>
[in]	DU	!> DU is COMPLEX*16 array, dimension (N-1) !> The (n-1) superdiagonal elements of A. !>
[in,out]	DLF	!> DLF is COMPLEX*16 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 ZGTTRF. !> !> 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. !>
[in,out]	DF	!> DF is COMPLEX*16 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. !>
[in,out]	DUF	!> DUF is COMPLEX*16 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. !>
[in,out]	DU2	!> DU2 is COMPLEX*16 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. !>
[in,out]	IPIV	!> 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 ZGTTRF. !> !> 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. !>
[in]	B	!> B is COMPLEX*16 array, dimension (LDB,NRHS) !> The N-by-NRHS right hand side matrix B. !>
[in]	LDB	!> LDB is INTEGER !> The leading dimension of the array B. LDB >= max(1,N). !>
[out]	X	!> X is COMPLEX*16 array, dimension (LDX,NRHS) !> If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X. !>
[in]	LDX	!> LDX is INTEGER !> The leading dimension of the array X. LDX >= max(1,N). !>
[out]	RCOND	!> 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. !>
[out]	FERR	!> 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. !>
[out]	BERR	!> 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). !>
[out]	WORK	!> WORK is COMPLEX*16 array, dimension (2*N) !>
[out]	RWORK	!> RWORK is DOUBLE PRECISION array, dimension (N) !>
[out]	INFO	!> 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. !>

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 289 of file zgtsvx.f.

*
*  -- 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, TRANS
      INTEGER            INFO, LDB, LDX, N, NRHS
      DOUBLE PRECISION   RCOND
*     ..
*     .. Array Arguments ..
      INTEGER            IPIV( * )
      DOUBLE PRECISION   BERR( * ), FERR( * ), RWORK( * )
      COMPLEX*16         B( LDB, * ), D( * ), DF( * ), DL( * ),
     $                   DLF( * ), DU( * ), DU2( * ), DUF( * ),
     $                   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, ZLANGT
      EXTERNAL           lsame, dlamch, zlangt
*     ..
*     .. External Subroutines ..
      EXTERNAL           xerbla, zcopy, zgtcon, zgtrfs, zgttrf,
     $                   zgttrs,
     $                   zlacpy
*     ..
*     .. 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( 'ZGTSVX', -info )
         RETURN
      END IF
*
      IF( nofact ) THEN
*
*        Compute the LU factorization of A.
*
         CALL zcopy( n, d, 1, df, 1 )
         IF( n.GT.1 ) THEN
            CALL zcopy( n-1, dl, 1, dlf, 1 )
            CALL zcopy( n-1, du, 1, duf, 1 )
         END IF
         CALL zgttrf( 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 = zlangt( norm, n, dl, d, du )
*
*     Compute the reciprocal of the condition number of A.
*
      CALL zgtcon( norm, n, dlf, df, duf, du2, ipiv, anorm, rcond,
     $             work,
     $             info )
*
*     Compute the solution vectors X.
*
      CALL zlacpy( 'Full', n, nrhs, b, ldb, x, ldx )
      CALL zgttrs( 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 zgtrfs( trans, n, nrhs, dl, d, du, dlf, df, duf, du2,
     $             ipiv,
     $             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 ZGTSVX
*

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