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*> \brief \b SLARRD computes the eigenvalues of a symmetric tridiagonal matrix to suitable accuracy.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download SLARRD + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/slarrd.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/slarrd.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/slarrd.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE SLARRD( RANGE, ORDER, N, VL, VU, IL, IU, GERS,
* RELTOL, D, E, E2, PIVMIN, NSPLIT, ISPLIT,
* M, W, WERR, WL, WU, IBLOCK, INDEXW,
* WORK, IWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER ORDER, RANGE
* INTEGER IL, INFO, IU, M, N, NSPLIT
* REAL PIVMIN, RELTOL, VL, VU, WL, WU
* ..
* .. Array Arguments ..
* INTEGER IBLOCK( * ), INDEXW( * ),
* $ ISPLIT( * ), IWORK( * )
* REAL D( * ), E( * ), E2( * ),
* $ GERS( * ), W( * ), WERR( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> SLARRD computes the eigenvalues of a symmetric tridiagonal
*> matrix T to suitable accuracy. This is an auxiliary code to be
*> called from SSTEMR.
*> The user may ask for all eigenvalues, all eigenvalues
*> in the half-open interval (VL, VU], or the IL-th through IU-th
*> eigenvalues.
*>
*> To avoid overflow, the matrix must be scaled so that its
*> largest element is no greater than overflow**(1/2) * underflow**(1/4) in absolute value, and for greatest
*> accuracy, it should not be much smaller than that.
*>
*> See W. Kahan "Accurate Eigenvalues of a Symmetric Tridiagonal
*> Matrix", Report CS41, Computer Science Dept., Stanford
*> University, July 21, 1966.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] RANGE
*> \verbatim
*> RANGE is CHARACTER*1
*> = 'A': ("All") all eigenvalues will be found.
*> = 'V': ("Value") all eigenvalues in the half-open interval
*> (VL, VU] will be found.
*> = 'I': ("Index") the IL-th through IU-th eigenvalues (of the
*> entire matrix) will be found.
*> \endverbatim
*>
*> \param[in] ORDER
*> \verbatim
*> ORDER is CHARACTER*1
*> = 'B': ("By Block") the eigenvalues will be grouped by
*> split-off block (see IBLOCK, ISPLIT) and
*> ordered from smallest to largest within
*> the block.
*> = 'E': ("Entire matrix")
*> the eigenvalues for the entire matrix
*> will be ordered from smallest to
*> largest.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the tridiagonal matrix T. N >= 0.
*> \endverbatim
*>
*> \param[in] VL
*> \verbatim
*> VL is REAL
*> If RANGE='V', the lower bound of the interval to
*> be searched for eigenvalues. Eigenvalues less than or equal
*> to VL, or greater than VU, will not be returned. VL < VU.
*> Not referenced if RANGE = 'A' or 'I'.
*> \endverbatim
*>
*> \param[in] VU
*> \verbatim
*> VU is REAL
*> If RANGE='V', the upper bound of the interval to
*> be searched for eigenvalues. Eigenvalues less than or equal
*> to VL, or greater than VU, will not be returned. VL < VU.
*> Not referenced if RANGE = 'A' or 'I'.
*> \endverbatim
*>
*> \param[in] IL
*> \verbatim
*> IL is INTEGER
*> If RANGE='I', the index of the
*> smallest eigenvalue to be returned.
*> 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
*> Not referenced if RANGE = 'A' or 'V'.
*> \endverbatim
*>
*> \param[in] IU
*> \verbatim
*> IU is INTEGER
*> If RANGE='I', the index of the
*> largest eigenvalue to be returned.
*> 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
*> Not referenced if RANGE = 'A' or 'V'.
*> \endverbatim
*>
*> \param[in] GERS
*> \verbatim
*> GERS is REAL array, dimension (2*N)
*> The N Gerschgorin intervals (the i-th Gerschgorin interval
*> is (GERS(2*i-1), GERS(2*i)).
*> \endverbatim
*>
*> \param[in] RELTOL
*> \verbatim
*> RELTOL is REAL
*> The minimum relative width of an interval. When an interval
*> is narrower than RELTOL times the larger (in
*> magnitude) endpoint, then it is considered to be
*> sufficiently small, i.e., converged. Note: this should
*> always be at least radix*machine epsilon.
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*> D is REAL array, dimension (N)
*> The n diagonal elements of the tridiagonal matrix T.
*> \endverbatim
*>
*> \param[in] E
*> \verbatim
*> E is REAL array, dimension (N-1)
*> The (n-1) off-diagonal elements of the tridiagonal matrix T.
*> \endverbatim
*>
*> \param[in] E2
*> \verbatim
*> E2 is REAL array, dimension (N-1)
*> The (n-1) squared off-diagonal elements of the tridiagonal matrix T.
*> \endverbatim
*>
*> \param[in] PIVMIN
*> \verbatim
*> PIVMIN is REAL
*> The minimum pivot allowed in the Sturm sequence for T.
*> \endverbatim
*>
*> \param[in] NSPLIT
*> \verbatim
*> NSPLIT is INTEGER
*> The number of diagonal blocks in the matrix T.
*> 1 <= NSPLIT <= N.
*> \endverbatim
*>
*> \param[in] ISPLIT
*> \verbatim
*> ISPLIT is INTEGER array, dimension (N)
*> The splitting points, at which T breaks up into submatrices.
*> The first submatrix consists of rows/columns 1 to ISPLIT(1),
*> the second of rows/columns ISPLIT(1)+1 through ISPLIT(2),
*> etc., and the NSPLIT-th consists of rows/columns
*> ISPLIT(NSPLIT-1)+1 through ISPLIT(NSPLIT)=N.
*> (Only the first NSPLIT elements will actually be used, but
*> since the user cannot know a priori what value NSPLIT will
*> have, N words must be reserved for ISPLIT.)
*> \endverbatim
*>
*> \param[out] M
*> \verbatim
*> M is INTEGER
*> The actual number of eigenvalues found. 0 <= M <= N.
*> (See also the description of INFO=2,3.)
*> \endverbatim
*>
*> \param[out] W
*> \verbatim
*> W is REAL array, dimension (N)
*> On exit, the first M elements of W will contain the
*> eigenvalue approximations. SLARRD computes an interval
*> I_j = (a_j, b_j] that includes eigenvalue j. The eigenvalue
*> approximation is given as the interval midpoint
*> W(j)= ( a_j + b_j)/2. The corresponding error is bounded by
*> WERR(j) = abs( a_j - b_j)/2
*> \endverbatim
*>
*> \param[out] WERR
*> \verbatim
*> WERR is REAL array, dimension (N)
*> The error bound on the corresponding eigenvalue approximation
*> in W.
*> \endverbatim
*>
*> \param[out] WL
*> \verbatim
*> WL is REAL
*> \endverbatim
*>
*> \param[out] WU
*> \verbatim
*> WU is REAL
*> The interval (WL, WU] contains all the wanted eigenvalues.
*> If RANGE='V', then WL=VL and WU=VU.
*> If RANGE='A', then WL and WU are the global Gerschgorin bounds
*> on the spectrum.
*> If RANGE='I', then WL and WU are computed by SLAEBZ from the
*> index range specified.
*> \endverbatim
*>
*> \param[out] IBLOCK
*> \verbatim
*> IBLOCK is INTEGER array, dimension (N)
*> At each row/column j where E(j) is zero or small, the
*> matrix T is considered to split into a block diagonal
*> matrix. On exit, if INFO = 0, IBLOCK(i) specifies to which
*> block (from 1 to the number of blocks) the eigenvalue W(i)
*> belongs. (SLARRD may use the remaining N-M elements as
*> workspace.)
*> \endverbatim
*>
*> \param[out] INDEXW
*> \verbatim
*> INDEXW is INTEGER array, dimension (N)
*> The indices of the eigenvalues within each block (submatrix);
*> for example, INDEXW(i)= j and IBLOCK(i)=k imply that the
*> i-th eigenvalue W(i) is the j-th eigenvalue in block k.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is REAL array, dimension (4*N)
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (3*N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: some or all of the eigenvalues failed to converge or
*> were not computed:
*> =1 or 3: Bisection failed to converge for some
*> eigenvalues; these eigenvalues are flagged by a
*> negative block number. The effect is that the
*> eigenvalues may not be as accurate as the
*> absolute and relative tolerances. This is
*> generally caused by unexpectedly inaccurate
*> arithmetic.
*> =2 or 3: RANGE='I' only: Not all of the eigenvalues
*> IL:IU were found.
*> Effect: M < IU+1-IL
*> Cause: non-monotonic arithmetic, causing the
*> Sturm sequence to be non-monotonic.
*> Cure: recalculate, using RANGE='A', and pick
*> out eigenvalues IL:IU. In some cases,
*> increasing the PARAMETER "FUDGE" may
*> make things work.
*> = 4: RANGE='I', and the Gershgorin interval
*> initially used was too small. No eigenvalues
*> were computed.
*> Probable cause: your machine has sloppy
*> floating-point arithmetic.
*> Cure: Increase the PARAMETER "FUDGE",
*> recompile, and try again.
*> \endverbatim
*
*> \par Internal Parameters:
* =========================
*>
*> \verbatim
*> FUDGE REAL, default = 2
*> A "fudge factor" to widen the Gershgorin intervals. Ideally,
*> a value of 1 should work, but on machines with sloppy
*> arithmetic, this needs to be larger. The default for
*> publicly released versions should be large enough to handle
*> the worst machine around. Note that this has no effect
*> on accuracy of the solution.
*> \endverbatim
*>
*> \par Contributors:
* ==================
*>
*> W. Kahan, University of California, Berkeley, USA \n
*> Beresford Parlett, University of California, Berkeley, USA \n
*> Jim Demmel, University of California, Berkeley, USA \n
*> Inderjit Dhillon, University of Texas, Austin, USA \n
*> Osni Marques, LBNL/NERSC, USA \n
*> Christof Voemel, University of California, Berkeley, USA \n
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup OTHERauxiliary
*
* =====================================================================
SUBROUTINE SLARRD( RANGE, ORDER, N, VL, VU, IL, IU, GERS,
$ RELTOL, D, E, E2, PIVMIN, NSPLIT, ISPLIT,
$ M, W, WERR, WL, WU, IBLOCK, INDEXW,
$ WORK, IWORK, INFO )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER ORDER, RANGE
INTEGER IL, INFO, IU, M, N, NSPLIT
REAL PIVMIN, RELTOL, VL, VU, WL, WU
* ..
* .. Array Arguments ..
INTEGER IBLOCK( * ), INDEXW( * ),
$ ISPLIT( * ), IWORK( * )
REAL D( * ), E( * ), E2( * ),
$ GERS( * ), W( * ), WERR( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
REAL ZERO, ONE, TWO, HALF, FUDGE
PARAMETER ( ZERO = 0.0E0, ONE = 1.0E0,
$ TWO = 2.0E0, HALF = ONE/TWO,
$ FUDGE = TWO )
INTEGER ALLRNG, VALRNG, INDRNG
PARAMETER ( ALLRNG = 1, VALRNG = 2, INDRNG = 3 )
* ..
* .. Local Scalars ..
LOGICAL NCNVRG, TOOFEW
INTEGER I, IB, IBEGIN, IDISCL, IDISCU, IE, IEND, IINFO,
$ IM, IN, IOFF, IOUT, IRANGE, ITMAX, ITMP1,
$ ITMP2, IW, IWOFF, J, JBLK, JDISC, JE, JEE, NB,
$ NWL, NWU
REAL ATOLI, EPS, GL, GU, RTOLI, TMP1, TMP2,
$ TNORM, UFLOW, WKILL, WLU, WUL
* ..
* .. Local Arrays ..
INTEGER IDUMMA( 1 )
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV
REAL SLAMCH
EXTERNAL LSAME, ILAENV, SLAMCH
* ..
* .. External Subroutines ..
EXTERNAL SLAEBZ
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, INT, LOG, MAX, MIN
* ..
* .. Executable Statements ..
*
INFO = 0
*
* Quick return if possible
*
IF( N.LE.0 ) THEN
RETURN
END IF
*
* Decode RANGE
*
IF( LSAME( RANGE, 'A' ) ) THEN
IRANGE = ALLRNG
ELSE IF( LSAME( RANGE, 'V' ) ) THEN
IRANGE = VALRNG
ELSE IF( LSAME( RANGE, 'I' ) ) THEN
IRANGE = INDRNG
ELSE
IRANGE = 0
END IF
*
* Check for Errors
*
IF( IRANGE.LE.0 ) THEN
INFO = -1
ELSE IF( .NOT.(LSAME(ORDER,'B').OR.LSAME(ORDER,'E')) ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( IRANGE.EQ.VALRNG ) THEN
IF( VL.GE.VU )
$ INFO = -5
ELSE IF( IRANGE.EQ.INDRNG .AND.
$ ( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) ) THEN
INFO = -6
ELSE IF( IRANGE.EQ.INDRNG .AND.
$ ( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) ) THEN
INFO = -7
END IF
*
IF( INFO.NE.0 ) THEN
RETURN
END IF
* Initialize error flags
INFO = 0
NCNVRG = .FALSE.
TOOFEW = .FALSE.
* Quick return if possible
M = 0
IF( N.EQ.0 ) RETURN
* Simplification:
IF( IRANGE.EQ.INDRNG .AND. IL.EQ.1 .AND. IU.EQ.N ) IRANGE = 1
* Get machine constants
EPS = SLAMCH( 'P' )
UFLOW = SLAMCH( 'U' )
* Special Case when N=1
* Treat case of 1x1 matrix for quick return
IF( N.EQ.1 ) THEN
IF( (IRANGE.EQ.ALLRNG).OR.
$ ((IRANGE.EQ.VALRNG).AND.(D(1).GT.VL).AND.(D(1).LE.VU)).OR.
$ ((IRANGE.EQ.INDRNG).AND.(IL.EQ.1).AND.(IU.EQ.1)) ) THEN
M = 1
W(1) = D(1)
* The computation error of the eigenvalue is zero
WERR(1) = ZERO
IBLOCK( 1 ) = 1
INDEXW( 1 ) = 1
ENDIF
RETURN
END IF
* NB is the minimum vector length for vector bisection, or 0
* if only scalar is to be done.
NB = ILAENV( 1, 'SSTEBZ', ' ', N, -1, -1, -1 )
IF( NB.LE.1 ) NB = 0
* Find global spectral radius
GL = D(1)
GU = D(1)
DO 5 I = 1,N
GL = MIN( GL, GERS( 2*I - 1))
GU = MAX( GU, GERS(2*I) )
5 CONTINUE
* Compute global Gerschgorin bounds and spectral diameter
TNORM = MAX( ABS( GL ), ABS( GU ) )
GL = GL - FUDGE*TNORM*EPS*N - FUDGE*TWO*PIVMIN
GU = GU + FUDGE*TNORM*EPS*N + FUDGE*TWO*PIVMIN
* [JAN/28/2009] remove the line below since SPDIAM variable not use
* SPDIAM = GU - GL
* Input arguments for SLAEBZ:
* The relative tolerance. An interval (a,b] lies within
* "relative tolerance" if b-a < RELTOL*max(|a|,|b|),
RTOLI = RELTOL
* Set the absolute tolerance for interval convergence to zero to force
* interval convergence based on relative size of the interval.
* This is dangerous because intervals might not converge when RELTOL is
* small. But at least a very small number should be selected so that for
* strongly graded matrices, the code can get relatively accurate
* eigenvalues.
ATOLI = FUDGE*TWO*UFLOW + FUDGE*TWO*PIVMIN
IF( IRANGE.EQ.INDRNG ) THEN
* RANGE='I': Compute an interval containing eigenvalues
* IL through IU. The initial interval [GL,GU] from the global
* Gerschgorin bounds GL and GU is refined by SLAEBZ.
ITMAX = INT( ( LOG( TNORM+PIVMIN )-LOG( PIVMIN ) ) /
$ LOG( TWO ) ) + 2
WORK( N+1 ) = GL
WORK( N+2 ) = GL
WORK( N+3 ) = GU
WORK( N+4 ) = GU
WORK( N+5 ) = GL
WORK( N+6 ) = GU
IWORK( 1 ) = -1
IWORK( 2 ) = -1
IWORK( 3 ) = N + 1
IWORK( 4 ) = N + 1
IWORK( 5 ) = IL - 1
IWORK( 6 ) = IU
*
CALL SLAEBZ( 3, ITMAX, N, 2, 2, NB, ATOLI, RTOLI, PIVMIN,
$ D, E, E2, IWORK( 5 ), WORK( N+1 ), WORK( N+5 ), IOUT,
$ IWORK, W, IBLOCK, IINFO )
IF( IINFO .NE. 0 ) THEN
INFO = IINFO
RETURN
END IF
* On exit, output intervals may not be ordered by ascending negcount
IF( IWORK( 6 ).EQ.IU ) THEN
WL = WORK( N+1 )
WLU = WORK( N+3 )
NWL = IWORK( 1 )
WU = WORK( N+4 )
WUL = WORK( N+2 )
NWU = IWORK( 4 )
ELSE
WL = WORK( N+2 )
WLU = WORK( N+4 )
NWL = IWORK( 2 )
WU = WORK( N+3 )
WUL = WORK( N+1 )
NWU = IWORK( 3 )
END IF
* On exit, the interval [WL, WLU] contains a value with negcount NWL,
* and [WUL, WU] contains a value with negcount NWU.
IF( NWL.LT.0 .OR. NWL.GE.N .OR. NWU.LT.1 .OR. NWU.GT.N ) THEN
INFO = 4
RETURN
END IF
ELSEIF( IRANGE.EQ.VALRNG ) THEN
WL = VL
WU = VU
ELSEIF( IRANGE.EQ.ALLRNG ) THEN
WL = GL
WU = GU
ENDIF
* Find Eigenvalues -- Loop Over blocks and recompute NWL and NWU.
* NWL accumulates the number of eigenvalues .le. WL,
* NWU accumulates the number of eigenvalues .le. WU
M = 0
IEND = 0
INFO = 0
NWL = 0
NWU = 0
*
DO 70 JBLK = 1, NSPLIT
IOFF = IEND
IBEGIN = IOFF + 1
IEND = ISPLIT( JBLK )
IN = IEND - IOFF
*
IF( IN.EQ.1 ) THEN
* 1x1 block
IF( WL.GE.D( IBEGIN )-PIVMIN )
$ NWL = NWL + 1
IF( WU.GE.D( IBEGIN )-PIVMIN )
$ NWU = NWU + 1
IF( IRANGE.EQ.ALLRNG .OR.
$ ( WL.LT.D( IBEGIN )-PIVMIN
$ .AND. WU.GE. D( IBEGIN )-PIVMIN ) ) THEN
M = M + 1
W( M ) = D( IBEGIN )
WERR(M) = ZERO
* The gap for a single block doesn't matter for the later
* algorithm and is assigned an arbitrary large value
IBLOCK( M ) = JBLK
INDEXW( M ) = 1
END IF
* Disabled 2x2 case because of a failure on the following matrix
* RANGE = 'I', IL = IU = 4
* Original Tridiagonal, d = [
* -0.150102010615740E+00
* -0.849897989384260E+00
* -0.128208148052635E-15
* 0.128257718286320E-15
* ];
* e = [
* -0.357171383266986E+00
* -0.180411241501588E-15
* -0.175152352710251E-15
* ];
*
* ELSE IF( IN.EQ.2 ) THEN
** 2x2 block
* DISC = SQRT( (HALF*(D(IBEGIN)-D(IEND)))**2 + E(IBEGIN)**2 )
* TMP1 = HALF*(D(IBEGIN)+D(IEND))
* L1 = TMP1 - DISC
* IF( WL.GE. L1-PIVMIN )
* $ NWL = NWL + 1
* IF( WU.GE. L1-PIVMIN )
* $ NWU = NWU + 1
* IF( IRANGE.EQ.ALLRNG .OR. ( WL.LT.L1-PIVMIN .AND. WU.GE.
* $ L1-PIVMIN ) ) THEN
* M = M + 1
* W( M ) = L1
** The uncertainty of eigenvalues of a 2x2 matrix is very small
* WERR( M ) = EPS * ABS( W( M ) ) * TWO
* IBLOCK( M ) = JBLK
* INDEXW( M ) = 1
* ENDIF
* L2 = TMP1 + DISC
* IF( WL.GE. L2-PIVMIN )
* $ NWL = NWL + 1
* IF( WU.GE. L2-PIVMIN )
* $ NWU = NWU + 1
* IF( IRANGE.EQ.ALLRNG .OR. ( WL.LT.L2-PIVMIN .AND. WU.GE.
* $ L2-PIVMIN ) ) THEN
* M = M + 1
* W( M ) = L2
** The uncertainty of eigenvalues of a 2x2 matrix is very small
* WERR( M ) = EPS * ABS( W( M ) ) * TWO
* IBLOCK( M ) = JBLK
* INDEXW( M ) = 2
* ENDIF
ELSE
* General Case - block of size IN >= 2
* Compute local Gerschgorin interval and use it as the initial
* interval for SLAEBZ
GU = D( IBEGIN )
GL = D( IBEGIN )
TMP1 = ZERO
DO 40 J = IBEGIN, IEND
GL = MIN( GL, GERS( 2*J - 1))
GU = MAX( GU, GERS(2*J) )
40 CONTINUE
* [JAN/28/2009]
* change SPDIAM by TNORM in lines 2 and 3 thereafter
* line 1: remove computation of SPDIAM (not useful anymore)
* SPDIAM = GU - GL
* GL = GL - FUDGE*SPDIAM*EPS*IN - FUDGE*PIVMIN
* GU = GU + FUDGE*SPDIAM*EPS*IN + FUDGE*PIVMIN
GL = GL - FUDGE*TNORM*EPS*IN - FUDGE*PIVMIN
GU = GU + FUDGE*TNORM*EPS*IN + FUDGE*PIVMIN
*
IF( IRANGE.GT.1 ) THEN
IF( GU.LT.WL ) THEN
* the local block contains none of the wanted eigenvalues
NWL = NWL + IN
NWU = NWU + IN
GO TO 70
END IF
* refine search interval if possible, only range (WL,WU] matters
GL = MAX( GL, WL )
GU = MIN( GU, WU )
IF( GL.GE.GU )
$ GO TO 70
END IF
* Find negcount of initial interval boundaries GL and GU
WORK( N+1 ) = GL
WORK( N+IN+1 ) = GU
CALL SLAEBZ( 1, 0, IN, IN, 1, NB, ATOLI, RTOLI, PIVMIN,
$ D( IBEGIN ), E( IBEGIN ), E2( IBEGIN ),
$ IDUMMA, WORK( N+1 ), WORK( N+2*IN+1 ), IM,
$ IWORK, W( M+1 ), IBLOCK( M+1 ), IINFO )
IF( IINFO .NE. 0 ) THEN
INFO = IINFO
RETURN
END IF
*
NWL = NWL + IWORK( 1 )
NWU = NWU + IWORK( IN+1 )
IWOFF = M - IWORK( 1 )
* Compute Eigenvalues
ITMAX = INT( ( LOG( GU-GL+PIVMIN )-LOG( PIVMIN ) ) /
$ LOG( TWO ) ) + 2
CALL SLAEBZ( 2, ITMAX, IN, IN, 1, NB, ATOLI, RTOLI, PIVMIN,
$ D( IBEGIN ), E( IBEGIN ), E2( IBEGIN ),
$ IDUMMA, WORK( N+1 ), WORK( N+2*IN+1 ), IOUT,
$ IWORK, W( M+1 ), IBLOCK( M+1 ), IINFO )
IF( IINFO .NE. 0 ) THEN
INFO = IINFO
RETURN
END IF
*
* Copy eigenvalues into W and IBLOCK
* Use -JBLK for block number for unconverged eigenvalues.
* Loop over the number of output intervals from SLAEBZ
DO 60 J = 1, IOUT
* eigenvalue approximation is middle point of interval
TMP1 = HALF*( WORK( J+N )+WORK( J+IN+N ) )
* semi length of error interval
TMP2 = HALF*ABS( WORK( J+N )-WORK( J+IN+N ) )
IF( J.GT.IOUT-IINFO ) THEN
* Flag non-convergence.
NCNVRG = .TRUE.
IB = -JBLK
ELSE
IB = JBLK
END IF
DO 50 JE = IWORK( J ) + 1 + IWOFF,
$ IWORK( J+IN ) + IWOFF
W( JE ) = TMP1
WERR( JE ) = TMP2
INDEXW( JE ) = JE - IWOFF
IBLOCK( JE ) = IB
50 CONTINUE
60 CONTINUE
*
M = M + IM
END IF
70 CONTINUE
* If RANGE='I', then (WL,WU) contains eigenvalues NWL+1,...,NWU
* If NWL+1 < IL or NWU > IU, discard extra eigenvalues.
IF( IRANGE.EQ.INDRNG ) THEN
IDISCL = IL - 1 - NWL
IDISCU = NWU - IU
*
IF( IDISCL.GT.0 ) THEN
IM = 0
DO 80 JE = 1, M
* Remove some of the smallest eigenvalues from the left so that
* at the end IDISCL =0. Move all eigenvalues up to the left.
IF( W( JE ).LE.WLU .AND. IDISCL.GT.0 ) THEN
IDISCL = IDISCL - 1
ELSE
IM = IM + 1
W( IM ) = W( JE )
WERR( IM ) = WERR( JE )
INDEXW( IM ) = INDEXW( JE )
IBLOCK( IM ) = IBLOCK( JE )
END IF
80 CONTINUE
M = IM
END IF
IF( IDISCU.GT.0 ) THEN
* Remove some of the largest eigenvalues from the right so that
* at the end IDISCU =0. Move all eigenvalues up to the left.
IM=M+1
DO 81 JE = M, 1, -1
IF( W( JE ).GE.WUL .AND. IDISCU.GT.0 ) THEN
IDISCU = IDISCU - 1
ELSE
IM = IM - 1
W( IM ) = W( JE )
WERR( IM ) = WERR( JE )
INDEXW( IM ) = INDEXW( JE )
IBLOCK( IM ) = IBLOCK( JE )
END IF
81 CONTINUE
JEE = 0
DO 82 JE = IM, M
JEE = JEE + 1
W( JEE ) = W( JE )
WERR( JEE ) = WERR( JE )
INDEXW( JEE ) = INDEXW( JE )
IBLOCK( JEE ) = IBLOCK( JE )
82 CONTINUE
M = M-IM+1
END IF
IF( IDISCL.GT.0 .OR. IDISCU.GT.0 ) THEN
* Code to deal with effects of bad arithmetic. (If N(w) is
* monotone non-decreasing, this should never happen.)
* Some low eigenvalues to be discarded are not in (WL,WLU],
* or high eigenvalues to be discarded are not in (WUL,WU]
* so just kill off the smallest IDISCL/largest IDISCU
* eigenvalues, by marking the corresponding IBLOCK = 0
IF( IDISCL.GT.0 ) THEN
WKILL = WU
DO 100 JDISC = 1, IDISCL
IW = 0
DO 90 JE = 1, M
IF( IBLOCK( JE ).NE.0 .AND.
$ ( W( JE ).LT.WKILL .OR. IW.EQ.0 ) ) THEN
IW = JE
WKILL = W( JE )
END IF
90 CONTINUE
IBLOCK( IW ) = 0
100 CONTINUE
END IF
IF( IDISCU.GT.0 ) THEN
WKILL = WL
DO 120 JDISC = 1, IDISCU
IW = 0
DO 110 JE = 1, M
IF( IBLOCK( JE ).NE.0 .AND.
$ ( W( JE ).GE.WKILL .OR. IW.EQ.0 ) ) THEN
IW = JE
WKILL = W( JE )
END IF
110 CONTINUE
IBLOCK( IW ) = 0
120 CONTINUE
END IF
* Now erase all eigenvalues with IBLOCK set to zero
IM = 0
DO 130 JE = 1, M
IF( IBLOCK( JE ).NE.0 ) THEN
IM = IM + 1
W( IM ) = W( JE )
WERR( IM ) = WERR( JE )
INDEXW( IM ) = INDEXW( JE )
IBLOCK( IM ) = IBLOCK( JE )
END IF
130 CONTINUE
M = IM
END IF
IF( IDISCL.LT.0 .OR. IDISCU.LT.0 ) THEN
TOOFEW = .TRUE.
END IF
END IF
*
IF(( IRANGE.EQ.ALLRNG .AND. M.NE.N ).OR.
$ ( IRANGE.EQ.INDRNG .AND. M.NE.IU-IL+1 ) ) THEN
TOOFEW = .TRUE.
END IF
* If ORDER='B', do nothing the eigenvalues are already sorted by
* block.
* If ORDER='E', sort the eigenvalues from smallest to largest
IF( LSAME(ORDER,'E') .AND. NSPLIT.GT.1 ) THEN
DO 150 JE = 1, M - 1
IE = 0
TMP1 = W( JE )
DO 140 J = JE + 1, M
IF( W( J ).LT.TMP1 ) THEN
IE = J
TMP1 = W( J )
END IF
140 CONTINUE
IF( IE.NE.0 ) THEN
TMP2 = WERR( IE )
ITMP1 = IBLOCK( IE )
ITMP2 = INDEXW( IE )
W( IE ) = W( JE )
WERR( IE ) = WERR( JE )
IBLOCK( IE ) = IBLOCK( JE )
INDEXW( IE ) = INDEXW( JE )
W( JE ) = TMP1
WERR( JE ) = TMP2
IBLOCK( JE ) = ITMP1
INDEXW( JE ) = ITMP2
END IF
150 CONTINUE
END IF
*
INFO = 0
IF( NCNVRG )
$ INFO = INFO + 1
IF( TOOFEW )
$ INFO = INFO + 2
RETURN
*
* End of SLARRD
*
END
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