slarrr.f.html

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      SUBROUTINE <a name="SLARRR.1"></a><a href="slarrr.f.html#SLARRR.1">SLARRR</a>( N, D, E, INFO )
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  -- LAPACK auxiliary routine (version 3.1) --
</span><span class="comment">*</span><span class="comment">     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
</span><span class="comment">*</span><span class="comment">     November 2006
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     .. Scalar Arguments ..
</span>      INTEGER            N, INFO
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">     .. Array Arguments ..
</span>      REAL               D( * ), E( * )
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  Purpose
</span><span class="comment">*</span><span class="comment">  =======
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  Perform tests to decide whether the symmetric tridiagonal matrix T
</span><span class="comment">*</span><span class="comment">  warrants expensive computations which guarantee high relative accuracy
</span><span class="comment">*</span><span class="comment">  in the eigenvalues.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  Arguments
</span><span class="comment">*</span><span class="comment">  =========
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  N       (input) INTEGER
</span><span class="comment">*</span><span class="comment">          The order of the matrix. N &gt; 0.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  D       (input) REAL             array, dimension (N)
</span><span class="comment">*</span><span class="comment">          The N diagonal elements of the tridiagonal matrix T.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  E       (input/output) REAL             array, dimension (N)
</span><span class="comment">*</span><span class="comment">          On entry, the first (N-1) entries contain the subdiagonal
</span><span class="comment">*</span><span class="comment">          elements of the tridiagonal matrix T; E(N) is set to ZERO.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  INFO    (output) INTEGER
</span><span class="comment">*</span><span class="comment">          INFO = 0(default) : the matrix warrants computations preserving
</span><span class="comment">*</span><span class="comment">                              relative accuracy.
</span><span class="comment">*</span><span class="comment">          INFO = 1          : the matrix warrants computations guaranteeing
</span><span class="comment">*</span><span class="comment">                              only absolute accuracy.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  Further Details
</span><span class="comment">*</span><span class="comment">  ===============
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  Based on contributions by
</span><span class="comment">*</span><span class="comment">     Beresford Parlett, University of California, Berkeley, USA
</span><span class="comment">*</span><span class="comment">     Jim Demmel, University of California, Berkeley, USA
</span><span class="comment">*</span><span class="comment">     Inderjit Dhillon, University of Texas, Austin, USA
</span><span class="comment">*</span><span class="comment">     Osni Marques, LBNL/NERSC, USA
</span><span class="comment">*</span><span class="comment">     Christof Voemel, University of California, Berkeley, USA
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  =====================================================================
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     .. Parameters ..
</span>      REAL               ZERO, RELCOND
      PARAMETER          ( ZERO = 0.0E0,
     $                     RELCOND = 0.999E0 )
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">     .. Local Scalars ..
</span>      INTEGER            I
      LOGICAL            YESREL
      REAL               EPS, SAFMIN, SMLNUM, RMIN, TMP, TMP2,
     $          OFFDIG, OFFDIG2

<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">     .. External Functions ..
</span>      REAL               <a name="SLAMCH.66"></a><a href="slamch.f.html#SLAMCH.1">SLAMCH</a>
      EXTERNAL           <a name="SLAMCH.67"></a><a href="slamch.f.html#SLAMCH.1">SLAMCH</a>
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">     .. Intrinsic Functions ..
</span>      INTRINSIC          ABS
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">     .. Executable Statements ..
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     As a default, do NOT go for relative-accuracy preserving computations.
</span>      INFO = 1

      SAFMIN = <a name="SLAMCH.77"></a><a href="slamch.f.html#SLAMCH.1">SLAMCH</a>( <span class="string">'Safe minimum'</span> )
      EPS = <a name="SLAMCH.78"></a><a href="slamch.f.html#SLAMCH.1">SLAMCH</a>( <span class="string">'Precision'</span> )
      SMLNUM = SAFMIN / EPS
      RMIN = SQRT( SMLNUM )

<span class="comment">*</span><span class="comment">     Tests for relative accuracy
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     Test for scaled diagonal dominance
</span><span class="comment">*</span><span class="comment">     Scale the diagonal entries to one and check whether the sum of the
</span><span class="comment">*</span><span class="comment">     off-diagonals is less than one
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     The sdd relative error bounds have a 1/(1- 2*x) factor in them,
</span><span class="comment">*</span><span class="comment">     x = max(OFFDIG + OFFDIG2), so when x is close to 1/2, no relative
</span><span class="comment">*</span><span class="comment">     accuracy is promised.  In the notation of the code fragment below,
</span><span class="comment">*</span><span class="comment">     1/(1 - (OFFDIG + OFFDIG2)) is the condition number.
</span><span class="comment">*</span><span class="comment">     We don't think it is worth going into &quot;sdd mode&quot; unless the relative
</span><span class="comment">*</span><span class="comment">     condition number is reasonable, not 1/macheps.
</span><span class="comment">*</span><span class="comment">     The threshold should be compatible with other thresholds used in the
</span><span class="comment">*</span><span class="comment">     code. We set  OFFDIG + OFFDIG2 &lt;= .999 =: RELCOND, it corresponds
</span><span class="comment">*</span><span class="comment">     to losing at most 3 decimal digits: 1 / (1 - (OFFDIG + OFFDIG2)) &lt;= 1000
</span><span class="comment">*</span><span class="comment">     instead of the current OFFDIG + OFFDIG2 &lt; 1
</span><span class="comment">*</span><span class="comment">
</span>      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
<span class="comment">*</span><span class="comment">
</span>
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     *** MORE TO BE IMPLEMENTED ***
</span><span class="comment">*</span><span class="comment">
</span>
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     Test if the lower bidiagonal matrix L from T = L D L^T
</span><span class="comment">*</span><span class="comment">     (zero shift facto) is well conditioned
</span><span class="comment">*</span><span class="comment">
</span>
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     Test if the upper bidiagonal matrix U from T = U D U^T
</span><span class="comment">*</span><span class="comment">     (zero shift facto) is well conditioned.
</span><span class="comment">*</span><span class="comment">     In this case, the matrix needs to be flipped and, at the end
</span><span class="comment">*</span><span class="comment">     of the eigenvector computation, the flip needs to be applied
</span><span class="comment">*</span><span class="comment">     to the computed eigenvectors (and the support)
</span><span class="comment">*</span><span class="comment">
</span>
<span class="comment">*</span><span class="comment">
</span>      RETURN
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     END OF <a name="SLARRR.143"></a><a href="slarrr.f.html#SLARRR.1">SLARRR</a>
</span><span class="comment">*</span><span class="comment">
</span>      END

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