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Annotation of rpl/lapack/lapack/zptcon.f, revision 1.1.1.1

1.1       bertrand    1:       SUBROUTINE ZPTCON( N, D, E, ANORM, RCOND, RWORK, 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( * ), RWORK( * )
                     14:       COMPLEX*16         E( * )
                     15: *     ..
                     16: *
                     17: *  Purpose
                     18: *  =======
                     19: *
                     20: *  ZPTCON computes the reciprocal of the condition number (in the
                     21: *  1-norm) of a complex Hermitian positive definite tridiagonal matrix
                     22: *  using the factorization A = L*D*L**H or A = U**H*D*U computed by
                     23: *  ZPTTRF.
                     24: *
                     25: *  Norm(inv(A)) is computed by a direct method, and the reciprocal of
                     26: *  the condition number is computed as
                     27: *                   RCOND = 1 / (ANORM * norm(inv(A))).
                     28: *
                     29: *  Arguments
                     30: *  =========
                     31: *
                     32: *  N       (input) INTEGER
                     33: *          The order of the matrix A.  N >= 0.
                     34: *
                     35: *  D       (input) DOUBLE PRECISION array, dimension (N)
                     36: *          The n diagonal elements of the diagonal matrix D from the
                     37: *          factorization of A, as computed by ZPTTRF.
                     38: *
                     39: *  E       (input) COMPLEX*16 array, dimension (N-1)
                     40: *          The (n-1) off-diagonal elements of the unit bidiagonal factor
                     41: *          U or L from the factorization of A, as computed by ZPTTRF.
                     42: *
                     43: *  ANORM   (input) DOUBLE PRECISION
                     44: *          The 1-norm of the original matrix A.
                     45: *
                     46: *  RCOND   (output) DOUBLE PRECISION
                     47: *          The reciprocal of the condition number of the matrix A,
                     48: *          computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is the
                     49: *          1-norm of inv(A) computed in this routine.
                     50: *
                     51: *  RWORK   (workspace) DOUBLE PRECISION array, dimension (N)
                     52: *
                     53: *  INFO    (output) INTEGER
                     54: *          = 0:  successful exit
                     55: *          < 0:  if INFO = -i, the i-th argument had an illegal value
                     56: *
                     57: *  Further Details
                     58: *  ===============
                     59: *
                     60: *  The method used is described in Nicholas J. Higham, "Efficient
                     61: *  Algorithms for Computing the Condition Number of a Tridiagonal
                     62: *  Matrix", SIAM J. Sci. Stat. Comput., Vol. 7, No. 1, January 1986.
                     63: *
                     64: *  =====================================================================
                     65: *
                     66: *     .. Parameters ..
                     67:       DOUBLE PRECISION   ONE, ZERO
                     68:       PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
                     69: *     ..
                     70: *     .. Local Scalars ..
                     71:       INTEGER            I, IX
                     72:       DOUBLE PRECISION   AINVNM
                     73: *     ..
                     74: *     .. External Functions ..
                     75:       INTEGER            IDAMAX
                     76:       EXTERNAL           IDAMAX
                     77: *     ..
                     78: *     .. External Subroutines ..
                     79:       EXTERNAL           XERBLA
                     80: *     ..
                     81: *     .. Intrinsic Functions ..
                     82:       INTRINSIC          ABS
                     83: *     ..
                     84: *     .. Executable Statements ..
                     85: *
                     86: *     Test the input arguments.
                     87: *
                     88:       INFO = 0
                     89:       IF( N.LT.0 ) THEN
                     90:          INFO = -1
                     91:       ELSE IF( ANORM.LT.ZERO ) THEN
                     92:          INFO = -4
                     93:       END IF
                     94:       IF( INFO.NE.0 ) THEN
                     95:          CALL XERBLA( 'ZPTCON', -INFO )
                     96:          RETURN
                     97:       END IF
                     98: *
                     99: *     Quick return if possible
                    100: *
                    101:       RCOND = ZERO
                    102:       IF( N.EQ.0 ) THEN
                    103:          RCOND = ONE
                    104:          RETURN
                    105:       ELSE IF( ANORM.EQ.ZERO ) THEN
                    106:          RETURN
                    107:       END IF
                    108: *
                    109: *     Check that D(1:N) is positive.
                    110: *
                    111:       DO 10 I = 1, N
                    112:          IF( D( I ).LE.ZERO )
                    113:      $      RETURN
                    114:    10 CONTINUE
                    115: *
                    116: *     Solve M(A) * x = e, where M(A) = (m(i,j)) is given by
                    117: *
                    118: *        m(i,j) =  abs(A(i,j)), i = j,
                    119: *        m(i,j) = -abs(A(i,j)), i .ne. j,
                    120: *
                    121: *     and e = [ 1, 1, ..., 1 ]'.  Note M(A) = M(L)*D*M(L)'.
                    122: *
                    123: *     Solve M(L) * x = e.
                    124: *
                    125:       RWORK( 1 ) = ONE
                    126:       DO 20 I = 2, N
                    127:          RWORK( I ) = ONE + RWORK( I-1 )*ABS( E( I-1 ) )
                    128:    20 CONTINUE
                    129: *
                    130: *     Solve D * M(L)' * x = b.
                    131: *
                    132:       RWORK( N ) = RWORK( N ) / D( N )
                    133:       DO 30 I = N - 1, 1, -1
                    134:          RWORK( I ) = RWORK( I ) / D( I ) + RWORK( I+1 )*ABS( E( I ) )
                    135:    30 CONTINUE
                    136: *
                    137: *     Compute AINVNM = max(x(i)), 1<=i<=n.
                    138: *
                    139:       IX = IDAMAX( N, RWORK, 1 )
                    140:       AINVNM = ABS( RWORK( IX ) )
                    141: *
                    142: *     Compute the reciprocal condition number.
                    143: *
                    144:       IF( AINVNM.NE.ZERO )
                    145:      $   RCOND = ( ONE / AINVNM ) / ANORM
                    146: *
                    147:       RETURN
                    148: *
                    149: *     End of ZPTCON
                    150: *
                    151:       END