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Annotation of rpl/lapack/lapack/dptcon.f, revision 1.1

1.1     ! bertrand    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