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Annotation of rpl/lapack/lapack/dptcon.f, revision 1.4

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