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

1.1       bertrand    1:       SUBROUTINE DLARRR( N, D, E, INFO )
                      2: *
                      3: *  -- LAPACK auxiliary 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            N, INFO
                     10: *     ..
                     11: *     .. Array Arguments ..
                     12:       DOUBLE PRECISION   D( * ), E( * )
                     13: *     ..
                     14: *
                     15: *
                     16: *  Purpose
                     17: *  =======
                     18: *
                     19: *  Perform tests to decide whether the symmetric tridiagonal matrix T
                     20: *  warrants expensive computations which guarantee high relative accuracy
                     21: *  in the eigenvalues.
                     22: *
                     23: *  Arguments
                     24: *  =========
                     25: *
                     26: *  N       (input) INTEGER
                     27: *          The order of the matrix. N > 0.
                     28: *
                     29: *  D       (input) DOUBLE PRECISION array, dimension (N)
                     30: *          The N diagonal elements of the tridiagonal matrix T.
                     31: *
                     32: *  E       (input/output) DOUBLE PRECISION array, dimension (N)
                     33: *          On entry, the first (N-1) entries contain the subdiagonal
                     34: *          elements of the tridiagonal matrix T; E(N) is set to ZERO.
                     35: *
                     36: *  INFO    (output) INTEGER
                     37: *          INFO = 0(default) : the matrix warrants computations preserving
                     38: *                              relative accuracy.
                     39: *          INFO = 1          : the matrix warrants computations guaranteeing
                     40: *                              only absolute accuracy.
                     41: *
                     42: *  Further Details
                     43: *  ===============
                     44: *
                     45: *  Based on contributions by
                     46: *     Beresford Parlett, University of California, Berkeley, USA
                     47: *     Jim Demmel, University of California, Berkeley, USA
                     48: *     Inderjit Dhillon, University of Texas, Austin, USA
                     49: *     Osni Marques, LBNL/NERSC, USA
                     50: *     Christof Voemel, University of California, Berkeley, USA
                     51: *
                     52: *  =====================================================================
                     53: *
                     54: *     .. Parameters ..
                     55:       DOUBLE PRECISION   ZERO, RELCOND
                     56:       PARAMETER          ( ZERO = 0.0D0,
                     57:      $                     RELCOND = 0.999D0 )
                     58: *     ..
                     59: *     .. Local Scalars ..
                     60:       INTEGER            I
                     61:       LOGICAL            YESREL
                     62:       DOUBLE PRECISION   EPS, SAFMIN, SMLNUM, RMIN, TMP, TMP2,
                     63:      $          OFFDIG, OFFDIG2
                     64: 
                     65: *     ..
                     66: *     .. External Functions ..
                     67:       DOUBLE PRECISION   DLAMCH
                     68:       EXTERNAL           DLAMCH
                     69: *     ..
                     70: *     .. Intrinsic Functions ..
                     71:       INTRINSIC          ABS
                     72: *     ..
                     73: *     .. Executable Statements ..
                     74: *
                     75: *     As a default, do NOT go for relative-accuracy preserving computations.
                     76:       INFO = 1
                     77: 
                     78:       SAFMIN = DLAMCH( 'Safe minimum' )
                     79:       EPS = DLAMCH( 'Precision' )
                     80:       SMLNUM = SAFMIN / EPS
                     81:       RMIN = SQRT( SMLNUM )
                     82: 
                     83: *     Tests for relative accuracy
                     84: *
                     85: *     Test for scaled diagonal dominance
                     86: *     Scale the diagonal entries to one and check whether the sum of the
                     87: *     off-diagonals is less than one
                     88: *
                     89: *     The sdd relative error bounds have a 1/(1- 2*x) factor in them,
                     90: *     x = max(OFFDIG + OFFDIG2), so when x is close to 1/2, no relative
                     91: *     accuracy is promised.  In the notation of the code fragment below,
                     92: *     1/(1 - (OFFDIG + OFFDIG2)) is the condition number.
                     93: *     We don't think it is worth going into "sdd mode" unless the relative
                     94: *     condition number is reasonable, not 1/macheps.
                     95: *     The threshold should be compatible with other thresholds used in the
                     96: *     code. We set  OFFDIG + OFFDIG2 <= .999 =: RELCOND, it corresponds
                     97: *     to losing at most 3 decimal digits: 1 / (1 - (OFFDIG + OFFDIG2)) <= 1000
                     98: *     instead of the current OFFDIG + OFFDIG2 < 1
                     99: *
                    100:       YESREL = .TRUE.
                    101:       OFFDIG = ZERO
                    102:       TMP = SQRT(ABS(D(1)))
                    103:       IF (TMP.LT.RMIN) YESREL = .FALSE.
                    104:       IF(.NOT.YESREL) GOTO 11
                    105:       DO 10 I = 2, N
                    106:          TMP2 = SQRT(ABS(D(I)))
                    107:          IF (TMP2.LT.RMIN) YESREL = .FALSE.
                    108:          IF(.NOT.YESREL) GOTO 11
                    109:          OFFDIG2 = ABS(E(I-1))/(TMP*TMP2)
                    110:          IF(OFFDIG+OFFDIG2.GE.RELCOND) YESREL = .FALSE.
                    111:          IF(.NOT.YESREL) GOTO 11
                    112:          TMP = TMP2
                    113:          OFFDIG = OFFDIG2
                    114:  10   CONTINUE
                    115:  11   CONTINUE
                    116: 
                    117:       IF( YESREL ) THEN
                    118:          INFO = 0
                    119:          RETURN
                    120:       ELSE
                    121:       ENDIF
                    122: *
                    123: 
                    124: *
                    125: *     *** MORE TO BE IMPLEMENTED ***
                    126: *
                    127: 
                    128: *
                    129: *     Test if the lower bidiagonal matrix L from T = L D L^T
                    130: *     (zero shift facto) is well conditioned
                    131: *
                    132: 
                    133: *
                    134: *     Test if the upper bidiagonal matrix U from T = U D U^T
                    135: *     (zero shift facto) is well conditioned.
                    136: *     In this case, the matrix needs to be flipped and, at the end
                    137: *     of the eigenvector computation, the flip needs to be applied
                    138: *     to the computed eigenvectors (and the support)
                    139: *
                    140: 
                    141: *
                    142:       RETURN
                    143: *
                    144: *     END OF DLARRR
                    145: *
                    146:       END