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    1:       SUBROUTINE DPTCON( N, D, E, ANORM, RCOND, WORK, INFO )
    2: *
    3: *  -- LAPACK routine (version 3.2) --
    4: *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
    5: *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
    6: *     November 2006
    7: *
    8: *     .. Scalar Arguments ..
    9:       INTEGER            INFO, N
   10:       DOUBLE PRECISION   ANORM, RCOND
   11: *     ..
   12: *     .. Array Arguments ..
   13:       DOUBLE PRECISION   D( * ), E( * ), WORK( * )
   14: *     ..
   15: *
   16: *  Purpose
   17: *  =======
   18: *
   19: *  DPTCON computes the reciprocal of the condition number (in the
   20: *  1-norm) of a real symmetric positive definite tridiagonal matrix
   21: *  using the factorization A = L*D*L**T or A = U**T*D*U computed by
   22: *  DPTTRF.
   23: *
   24: *  Norm(inv(A)) is computed by a direct method, and the reciprocal of
   25: *  the condition number is computed as
   26: *               RCOND = 1 / (ANORM * norm(inv(A))).
   27: *
   28: *  Arguments
   29: *  =========
   30: *
   31: *  N       (input) INTEGER
   32: *          The order of the matrix A.  N >= 0.
   33: *
   34: *  D       (input) DOUBLE PRECISION array, dimension (N)
   35: *          The n diagonal elements of the diagonal matrix D from the
   36: *          factorization of A, as computed by DPTTRF.
   37: *
   38: *  E       (input) DOUBLE PRECISION array, dimension (N-1)
   39: *          The (n-1) off-diagonal elements of the unit bidiagonal factor
   40: *          U or L from the factorization of A,  as computed by DPTTRF.
   41: *
   42: *  ANORM   (input) DOUBLE PRECISION
   43: *          The 1-norm of the original matrix A.
   44: *
   45: *  RCOND   (output) DOUBLE PRECISION
   46: *          The reciprocal of the condition number of the matrix A,
   47: *          computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is the
   48: *          1-norm of inv(A) computed in this routine.
   49: *
   50: *  WORK    (workspace) DOUBLE PRECISION array, dimension (N)
   51: *
   52: *  INFO    (output) INTEGER
   53: *          = 0:  successful exit
   54: *          < 0:  if INFO = -i, the i-th argument had an illegal value
   55: *
   56: *  Further Details
   57: *  ===============
   58: *
   59: *  The method used is described in Nicholas J. Higham, "Efficient
   60: *  Algorithms for Computing the Condition Number of a Tridiagonal
   61: *  Matrix", SIAM J. Sci. Stat. Comput., Vol. 7, No. 1, January 1986.
   62: *
   63: *  =====================================================================
   64: *
   65: *     .. Parameters ..
   66:       DOUBLE PRECISION   ONE, ZERO
   67:       PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
   68: *     ..
   69: *     .. Local Scalars ..
   70:       INTEGER            I, IX
   71:       DOUBLE PRECISION   AINVNM
   72: *     ..
   73: *     .. External Functions ..
   74:       INTEGER            IDAMAX
   75:       EXTERNAL           IDAMAX
   76: *     ..
   77: *     .. External Subroutines ..
   78:       EXTERNAL           XERBLA
   79: *     ..
   80: *     .. Intrinsic Functions ..
   81:       INTRINSIC          ABS
   82: *     ..
   83: *     .. Executable Statements ..
   84: *
   85: *     Test the input arguments.
   86: *
   87:       INFO = 0
   88:       IF( N.LT.0 ) THEN
   89:          INFO = -1
   90:       ELSE IF( ANORM.LT.ZERO ) THEN
   91:          INFO = -4
   92:       END IF
   93:       IF( INFO.NE.0 ) THEN
   94:          CALL XERBLA( 'DPTCON', -INFO )
   95:          RETURN
   96:       END IF
   97: *
   98: *     Quick return if possible
   99: *
  100:       RCOND = ZERO
  101:       IF( N.EQ.0 ) THEN
  102:          RCOND = ONE
  103:          RETURN
  104:       ELSE IF( ANORM.EQ.ZERO ) THEN
  105:          RETURN
  106:       END IF
  107: *
  108: *     Check that D(1:N) is positive.
  109: *
  110:       DO 10 I = 1, N
  111:          IF( D( I ).LE.ZERO )
  112:      $      RETURN
  113:    10 CONTINUE
  114: *
  115: *     Solve M(A) * x = e, where M(A) = (m(i,j)) is given by
  116: *
  117: *        m(i,j) =  abs(A(i,j)), i = j,
  118: *        m(i,j) = -abs(A(i,j)), i .ne. j,
  119: *
  120: *     and e = [ 1, 1, ..., 1 ]'.  Note M(A) = M(L)*D*M(L)'.
  121: *
  122: *     Solve M(L) * x = e.
  123: *
  124:       WORK( 1 ) = ONE
  125:       DO 20 I = 2, N
  126:          WORK( I ) = ONE + WORK( I-1 )*ABS( E( I-1 ) )
  127:    20 CONTINUE
  128: *
  129: *     Solve D * M(L)' * x = b.
  130: *
  131:       WORK( N ) = WORK( N ) / D( N )
  132:       DO 30 I = N - 1, 1, -1
  133:          WORK( I ) = WORK( I ) / D( I ) + WORK( I+1 )*ABS( E( I ) )
  134:    30 CONTINUE
  135: *
  136: *     Compute AINVNM = max(x(i)), 1<=i<=n.
  137: *
  138:       IX = IDAMAX( N, WORK, 1 )
  139:       AINVNM = ABS( WORK( IX ) )
  140: *
  141: *     Compute the reciprocal condition number.
  142: *
  143:       IF( AINVNM.NE.ZERO )
  144:      $   RCOND = ( ONE / AINVNM ) / ANORM
  145: *
  146:       RETURN
  147: *
  148: *     End of DPTCON
  149: *
  150:       END