sgtsvx.f

          SUBROUTINE SGTSVX( FACT, TRANS, N, NRHS, DL, D, DU, DLF, DF, DUF,
         $                   DU2, IPIV, B, LDB, X, LDX, RCOND, FERR, BERR,
         $                   WORK, IWORK, INFO )
    *
    *  -- LAPACK routine (version 3.1) --
    *     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
    *     November 2006
    *
    *     .. Scalar Arguments ..
          CHARACTER          FACT, TRANS
          INTEGER            INFO, LDB, LDX, N, NRHS
          REAL               RCOND
    *     ..
    *     .. Array Arguments ..
          INTEGER            IPIV( * ), IWORK( * )
          REAL               B( LDB, * ), BERR( * ), D( * ), DF( * ),
         $                   DL( * ), DLF( * ), DU( * ), DU2( * ), DUF( * ),
         $                   FERR( * ), WORK( * ), X( LDX, * )
    *     ..
    *
    *  Purpose
    *  =======
    *
    *  SGTSVX 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.
    *
    *  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.
    *
    *  Arguments
    *  =========
    *
    *  FACT    (input) 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.
    *
    *  TRANS   (input) 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)
    *
    *  N       (input) INTEGER
    *          The order of the matrix A.  N >= 0.
    *
    *  NRHS    (input) INTEGER
    *          The number of right hand sides, i.e., the number of columns
    *          of the matrix B.  NRHS >= 0.
    *
    *  DL      (input) REAL array, dimension (N-1)
    *          The (n-1) subdiagonal elements of A.
    *
    *  D       (input) REAL array, dimension (N)
    *          The n diagonal elements of A.
    *
    *  DU      (input) REAL array, dimension (N-1)
    *          The (n-1) superdiagonal elements of A.
    *
    *  DLF     (input or output) REAL 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 SGTTRF.
    *
    *          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.
    *
    *  DF      (input or output) REAL 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.
    *
    *  DUF     (input or output) REAL 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.
    *
    *  DU2     (input or output) REAL 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.
    *
    *  IPIV    (input or output) 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 SGTTRF.
    *
    *          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.
    *
    *  B       (input) REAL array, dimension (LDB,NRHS)
    *          The N-by-NRHS right hand side matrix B.
    *
    *  LDB     (input) INTEGER
    *          The leading dimension of the array B.  LDB >= max(1,N).
    *
    *  X       (output) REAL array, dimension (LDX,NRHS)
    *          If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X.
    *
    *  LDX     (input) INTEGER
    *          The leading dimension of the array X.  LDX >= max(1,N).
    *
    *  RCOND   (output) REAL
    *          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.
    *
    *  FERR    (output) REAL 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.
    *
    *  BERR    (output) REAL 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).
    *
    *  WORK    (workspace) REAL array, dimension (3*N)
    *
    *  IWORK   (workspace) INTEGER array, dimension (N)
    *
    *  INFO    (output) 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.
    *
    *  =====================================================================
    *
    *     .. Parameters ..
          REAL               ZERO
          PARAMETER          ( ZERO = 0.0E+0 )
    *     ..
    *     .. Local Scalars ..
          LOGICAL            NOFACT, NOTRAN
          CHARACTER          NORM
          REAL               ANORM
    *     ..
    *     .. External Functions ..
          LOGICAL            LSAME
          REAL               SLAMCH, SLANGT
          EXTERNAL           LSAME, SLAMCH, SLANGT
    *     ..
    *     .. External Subroutines ..
          EXTERNAL           SCOPY, SGTCON, SGTRFS, SGTTRF, SGTTRS, SLACPY,
         $                   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( 'SGTSVX', -INFO )
             RETURN
          END IF
    *
          IF( NOFACT ) THEN
    *
    *        Compute the LU factorization of A.
    *
             CALL SCOPY( N, D, 1, DF, 1 )
             IF( N.GT.1 ) THEN
                CALL SCOPY( N-1, DL, 1, DLF, 1 )
                CALL SCOPY( N-1, DU, 1, DUF, 1 )
             END IF
             CALL SGTTRF( 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 = SLANGT( NORM, N, DL, D, DU )
    *
    *     Compute the reciprocal of the condition number of A.
    *
          CALL SGTCON( NORM, N, DLF, DF, DUF, DU2, IPIV, ANORM, RCOND, WORK,
         $             IWORK, INFO )
    *
    *     Compute the solution vectors X.
    *
          CALL SLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
          CALL SGTTRS( 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 SGTRFS( 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.SLAMCH( 'Epsilon' ) )
         $   INFO = N + 1
    *
          RETURN
    *
    *     End of SGTSVX
    *
          END