cptsvx.f

          SUBROUTINE CPTSVX( FACT, N, NRHS, D, E, DF, EF, B, LDB, X, LDX,
         $                   RCOND, FERR, BERR, WORK, RWORK, INFO )
    *
    *  -- LAPACK routine (version 3.1) --
    *     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
    *     November 2006
    *
    *     .. Scalar Arguments ..
          CHARACTER          FACT
          INTEGER            INFO, LDB, LDX, N, NRHS
          REAL               RCOND
    *     ..
    *     .. Array Arguments ..
          REAL               BERR( * ), D( * ), DF( * ), FERR( * ),
         $                   RWORK( * )
          COMPLEX            B( LDB, * ), E( * ), EF( * ), WORK( * ),
         $                   X( LDX, * )
    *     ..
    *
    *  Purpose
    *  =======
    *
    *  CPTSVX 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.
    *
    *  Description
    *  ===========
    *
    *  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.
    *
    *  Arguments
    *  =========
    *
    *  FACT    (input) 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.
    *
    *  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 matrices B and X.  NRHS >= 0.
    *
    *  D       (input) REAL array, dimension (N)
    *          The n diagonal elements of the tridiagonal matrix A.
    *
    *  E       (input) COMPLEX array, dimension (N-1)
    *          The (n-1) subdiagonal elements of the tridiagonal matrix 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 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.
    *
    *  EF      (input or output) COMPLEX 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.
    *
    *  B       (input) COMPLEX 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) COMPLEX 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 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 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).
    *
    *  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) COMPLEX array, dimension (N)
    *
    *  RWORK   (workspace) REAL 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:  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.
    *
    *  =====================================================================
    *
    *     .. Parameters ..
          REAL               ZERO
          PARAMETER          ( ZERO = 0.0E+0 )
    *     ..
    *     .. Local Scalars ..
          LOGICAL            NOFACT
          REAL               ANORM
    *     ..
    *     .. External Functions ..
          LOGICAL            LSAME
          REAL               CLANHT, SLAMCH
          EXTERNAL           LSAME, CLANHT, SLAMCH
    *     ..
    *     .. External Subroutines ..
          EXTERNAL           CCOPY, CLACPY, CPTCON, CPTRFS, CPTTRF, CPTTRS,
         $                   SCOPY, 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( 'CPTSVX', -INFO )
             RETURN
          END IF
    *
          IF( NOFACT ) THEN
    *
    *        Compute the L*D*L' (or U'*D*U) factorization of A.
    *
             CALL SCOPY( N, D, 1, DF, 1 )
             IF( N.GT.1 )
         $      CALL CCOPY( N-1, E, 1, EF, 1 )
             CALL CPTTRF( 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 = CLANHT( '1', N, D, E )
    *
    *     Compute the reciprocal of the condition number of A.
    *
          CALL CPTCON( N, DF, EF, ANORM, RCOND, RWORK, INFO )
    *
    *     Compute the solution vectors X.
    *
          CALL CLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
          CALL CPTTRS( '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 CPTRFS( '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.SLAMCH( 'Epsilon' ) )
         $   INFO = N + 1
    *
          RETURN
    *
    *     End of CPTSVX
    *
          END