summaryrefslogtreecommitdiff
path: root/SRC/slarrr.f
blob: a1e98d2214df853cc41b8008fde711888a6eab7b (plain)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
      SUBROUTINE SLARRR( N, D, E, INFO )
*
*  -- LAPACK auxiliary routine (version 3.2.2) --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*     June 2010
*
*     .. Scalar Arguments ..
      INTEGER            N, INFO
*     ..
*     .. Array Arguments ..
      REAL               D( * ), E( * )
*     ..
*
*
*  Purpose
*  =======
*
*  Perform tests to decide whether the symmetric tridiagonal matrix T
*  warrants expensive computations which guarantee high relative accuracy
*  in the eigenvalues.
*
*  Arguments
*  =========
*
*  N       (input) INTEGER
*          The order of the matrix. N > 0.
*
*  D       (input) REAL             array, dimension (N)
*          The N diagonal elements of the tridiagonal matrix T.
*
*  E       (input/output) REAL array, dimension (N)
*          On entry, the first (N-1) entries contain the subdiagonal
*          elements of the tridiagonal matrix T; E(N) is set to ZERO.
*
*  INFO    (output) INTEGER
*          INFO = 0(default) : the matrix warrants computations preserving
*                              relative accuracy.
*          INFO = 1          : the matrix warrants computations guaranteeing
*                              only absolute accuracy.
*
*  Further Details
*  ===============
*
*  Based on contributions by
*     Beresford Parlett, University of California, Berkeley, USA
*     Jim Demmel, University of California, Berkeley, USA
*     Inderjit Dhillon, University of Texas, Austin, USA
*     Osni Marques, LBNL/NERSC, USA
*     Christof Voemel, University of California, Berkeley, USA
*
*  =====================================================================
*
*     .. Parameters ..
      REAL               ZERO, RELCOND
      PARAMETER          ( ZERO = 0.0E0,
     $                     RELCOND = 0.999E0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I
      LOGICAL            YESREL
      REAL               EPS, SAFMIN, SMLNUM, RMIN, TMP, TMP2,
     $          OFFDIG, OFFDIG2

*     ..
*     .. External Functions ..
      REAL               SLAMCH
      EXTERNAL           SLAMCH
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS
*     ..
*     .. Executable Statements ..
*
*     As a default, do NOT go for relative-accuracy preserving computations.
      INFO = 1

      SAFMIN = SLAMCH( 'Safe minimum' )
      EPS = SLAMCH( 'Precision' )
      SMLNUM = SAFMIN / EPS
      RMIN = SQRT( SMLNUM )

*     Tests for relative accuracy
*
*     Test for scaled diagonal dominance
*     Scale the diagonal entries to one and check whether the sum of the
*     off-diagonals is less than one
*
*     The sdd relative error bounds have a 1/(1- 2*x) factor in them,
*     x = max(OFFDIG + OFFDIG2), so when x is close to 1/2, no relative
*     accuracy is promised.  In the notation of the code fragment below,
*     1/(1 - (OFFDIG + OFFDIG2)) is the condition number.
*     We don't think it is worth going into "sdd mode" unless the relative
*     condition number is reasonable, not 1/macheps.
*     The threshold should be compatible with other thresholds used in the
*     code. We set  OFFDIG + OFFDIG2 <= .999 =: RELCOND, it corresponds
*     to losing at most 3 decimal digits: 1 / (1 - (OFFDIG + OFFDIG2)) <= 1000
*     instead of the current OFFDIG + OFFDIG2 < 1
*
      YESREL = .TRUE.
      OFFDIG = ZERO
      TMP = SQRT(ABS(D(1)))
      IF (TMP.LT.RMIN) YESREL = .FALSE.
      IF(.NOT.YESREL) GOTO 11
      DO 10 I = 2, N
         TMP2 = SQRT(ABS(D(I)))
         IF (TMP2.LT.RMIN) YESREL = .FALSE.
         IF(.NOT.YESREL) GOTO 11
         OFFDIG2 = ABS(E(I-1))/(TMP*TMP2)
         IF(OFFDIG+OFFDIG2.GE.RELCOND) YESREL = .FALSE.
         IF(.NOT.YESREL) GOTO 11
         TMP = TMP2
         OFFDIG = OFFDIG2
 10   CONTINUE
 11   CONTINUE

      IF( YESREL ) THEN
         INFO = 0
         RETURN
      ELSE
      ENDIF
*

*
*     *** MORE TO BE IMPLEMENTED ***
*

*
*     Test if the lower bidiagonal matrix L from T = L D L^T
*     (zero shift facto) is well conditioned
*

*
*     Test if the upper bidiagonal matrix U from T = U D U^T
*     (zero shift facto) is well conditioned.
*     In this case, the matrix needs to be flipped and, at the end
*     of the eigenvector computation, the flip needs to be applied
*     to the computed eigenvectors (and the support)
*

*
      RETURN
*
*     END OF SLARRR
*
      END