SUBROUTINE ZSTEMR( JOBZ, RANGE, N, D, E, VL, VU, IL, IU,
$ M, W, Z, LDZ, NZC, ISUPPZ, TRYRAC, WORK, LWORK,
$ IWORK, LIWORK, INFO )
IMPLICIT NONE
*
* -- LAPACK computational routine (version 3.1) --
* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
* November 2006
*
* .. Scalar Arguments ..
CHARACTER JOBZ, RANGE
LOGICAL TRYRAC
INTEGER IL, INFO, IU, LDZ, NZC, LIWORK, LWORK, M, N
DOUBLE PRECISION VL, VU
* ..
* .. Array Arguments ..
INTEGER ISUPPZ( * ), IWORK( * )
DOUBLE PRECISION D( * ), E( * ), W( * ), WORK( * )
COMPLEX*16 Z( LDZ, * )
* ..
*
* Purpose
* =======
*
* ZSTEMR computes selected eigenvalues and, optionally, eigenvectors
* of a real symmetric tridiagonal matrix T. Any such unreduced matrix has
* a well defined set of pairwise different real eigenvalues, the corresponding
* real eigenvectors are pairwise orthogonal.
*
* The spectrum may be computed either completely or partially by specifying
* either an interval (VL,VU] or a range of indices IL:IU for the desired
* eigenvalues.
*
* Depending on the number of desired eigenvalues, these are computed either
* by bisection or the dqds algorithm. Numerically orthogonal eigenvectors are
* computed by the use of various suitable L D L^T factorizations near clusters
* of close eigenvalues (referred to as RRRs, Relatively Robust
* Representations). An informal sketch of the algorithm follows.
*
* For each unreduced block (submatrix) of T,
* (a) Compute T - sigma I = L D L^T, so that L and D
* define all the wanted eigenvalues to high relative accuracy.
* This means that small relative changes in the entries of D and L
* cause only small relative changes in the eigenvalues and
* eigenvectors. The standard (unfactored) representation of the
* tridiagonal matrix T does not have this property in general.
* (b) Compute the eigenvalues to suitable accuracy.
* If the eigenvectors are desired, the algorithm attains full
* accuracy of the computed eigenvalues only right before
* the corresponding vectors have to be computed, see steps c) and d).
* (c) For each cluster of close eigenvalues, select a new
* shift close to the cluster, find a new factorization, and refine
* the shifted eigenvalues to suitable accuracy.
* (d) For each eigenvalue with a large enough relative separation compute
* the corresponding eigenvector by forming a rank revealing twisted
* factorization. Go back to (c) for any clusters that remain.
*
* For more details, see:
* - Inderjit S. Dhillon and Beresford N. Parlett: "Multiple representations
* to compute orthogonal eigenvectors of symmetric tridiagonal matrices,"
* Linear Algebra and its Applications, 387(1), pp. 1-28, August 2004.
* - Inderjit Dhillon and Beresford Parlett: "Orthogonal Eigenvectors and
* Relative Gaps," SIAM Journal on Matrix Analysis and Applications, Vol. 25,
* 2004. Also LAPACK Working Note 154.
* - Inderjit Dhillon: "A new O(n^2) algorithm for the symmetric
* tridiagonal eigenvalue/eigenvector problem",
* Computer Science Division Technical Report No. UCB/CSD-97-971,
* UC Berkeley, May 1997.
*
* Notes:
* 1.ZSTEMR works only on machines which follow IEEE-754
* floating-point standard in their handling of infinities and NaNs.
* This permits the use of efficient inner loops avoiding a check for
* zero divisors.
*
* 2. LAPACK routines can be used to reduce a complex Hermitean matrix to
* real symmetric tridiagonal form.
*
* (Any complex Hermitean tridiagonal matrix has real values on its diagonal
* and potentially complex numbers on its off-diagonals. By applying a
* similarity transform with an appropriate diagonal matrix
* diag(1,e^{i \phy_1}, ... , e^{i \phy_{n-1}}), the complex Hermitean
* matrix can be transformed into a real symmetric matrix and complex
* arithmetic can be entirely avoided.)
*
* While the eigenvectors of the real symmetric tridiagonal matrix are real,
* the eigenvectors of original complex Hermitean matrix have complex entries
* in general.
* Since LAPACK drivers overwrite the matrix data with the eigenvectors,
* ZSTEMR accepts complex workspace to facilitate interoperability
* with ZUNMTR or ZUPMTR.
*
* Arguments
* =========
*
* JOBZ (input) CHARACTER*1
* = 'N': Compute eigenvalues only;
* = 'V': Compute eigenvalues and eigenvectors.
*
* RANGE (input) CHARACTER*1
* = 'A': all eigenvalues will be found.
* = 'V': all eigenvalues in the half-open interval (VL,VU]
* will be found.
* = 'I': the IL-th through IU-th eigenvalues will be found.
*
* N (input) INTEGER
* The order of the matrix. N >= 0.
*
* D (input/output) DOUBLE PRECISION array, dimension (N)
* On entry, the N diagonal elements of the tridiagonal matrix
* T. On exit, D is overwritten.
*
* E (input/output) DOUBLE PRECISION array, dimension (N)
* On entry, the (N-1) subdiagonal elements of the tridiagonal
* matrix T in elements 1 to N-1 of E. E(N) need not be set on
* input, but is used internally as workspace.
* On exit, E is overwritten.
*
* VL (input) DOUBLE PRECISION
* VU (input) DOUBLE PRECISION
* If RANGE='V', the lower and upper bounds of the interval to
* be searched for eigenvalues. VL < VU.
* Not referenced if RANGE = 'A' or 'I'.
*
* IL (input) INTEGER
* IU (input) INTEGER
* If RANGE='I', the indices (in ascending order) of the
* smallest and largest eigenvalues to be returned.
* 1 <= IL <= IU <= N, if N > 0.
* Not referenced if RANGE = 'A' or 'V'.
*
* M (output) INTEGER
* The total number of eigenvalues found. 0 <= M <= N.
* If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
*
* W (output) DOUBLE PRECISION array, dimension (N)
* The first M elements contain the selected eigenvalues in
* ascending order.
*
* Z (output) COMPLEX*16 array, dimension (LDZ, max(1,M) )
* If JOBZ = 'V', and if INFO = 0, then the first M columns of Z
* contain the orthonormal eigenvectors of the matrix T
* corresponding to the selected eigenvalues, with the i-th
* column of Z holding the eigenvector associated with W(i).
* If JOBZ = 'N', then Z is not referenced.
* Note: the user must ensure that at least max(1,M) columns are
* supplied in the array Z; if RANGE = 'V', the exact value of M
* is not known in advance and can be computed with a workspace
* query by setting NZC = -1, see below.
*
* LDZ (input) INTEGER
* The leading dimension of the array Z. LDZ >= 1, and if
* JOBZ = 'V', then LDZ >= max(1,N).
*
* NZC (input) INTEGER
* The number of eigenvectors to be held in the array Z.
* If RANGE = 'A', then NZC >= max(1,N).
* If RANGE = 'V', then NZC >= the number of eigenvalues in (VL,VU].
* If RANGE = 'I', then NZC >= IU-IL+1.
* If NZC = -1, then a workspace query is assumed; the
* routine calculates the number of columns of the array Z that
* are needed to hold the eigenvectors.
* This value is returned as the first entry of the Z array, and
* no error message related to NZC is issued by XERBLA.
*
* ISUPPZ (output) INTEGER ARRAY, dimension ( 2*max(1,M) )
* The support of the eigenvectors in Z, i.e., the indices
* indicating the nonzero elements in Z. The i-th computed eigenvector
* is nonzero only in elements ISUPPZ( 2*i-1 ) through
* ISUPPZ( 2*i ). This is relevant in the case when the matrix
* is split. ISUPPZ is only accessed when JOBZ is 'V' and N > 0.
*
* TRYRAC (input/output) LOGICAL
* If TRYRAC.EQ..TRUE., indicates that the code should check whether
* the tridiagonal matrix defines its eigenvalues to high relative
* accuracy. If so, the code uses relative-accuracy preserving
* algorithms that might be (a bit) slower depending on the matrix.
* If the matrix does not define its eigenvalues to high relative
* accuracy, the code can uses possibly faster algorithms.
* If TRYRAC.EQ..FALSE., the code is not required to guarantee
* relatively accurate eigenvalues and can use the fastest possible
* techniques.
* On exit, a .TRUE. TRYRAC will be set to .FALSE. if the matrix
* does not define its eigenvalues to high relative accuracy.
*
* WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK)
* On exit, if INFO = 0, WORK(1) returns the optimal
* (and minimal) LWORK.
*
* LWORK (input) INTEGER
* The dimension of the array WORK. LWORK >= max(1,18*N)
* if JOBZ = 'V', and LWORK >= max(1,12*N) if JOBZ = 'N'.
* If LWORK = -1, then a workspace query is assumed; the routine
* only calculates the optimal size of the WORK array, returns
* this value as the first entry of the WORK array, and no error
* message related to LWORK is issued by XERBLA.
*
* IWORK (workspace/output) INTEGER array, dimension (LIWORK)
* On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
*
* LIWORK (input) INTEGER
* The dimension of the array IWORK. LIWORK >= max(1,10*N)
* if the eigenvectors are desired, and LIWORK >= max(1,8*N)
* if only the eigenvalues are to be computed.
* If LIWORK = -1, then a workspace query is assumed; the
* routine only calculates the optimal size of the IWORK array,
* returns this value as the first entry of the IWORK array, and
* no error message related to LIWORK is issued by XERBLA.
*
* INFO (output) INTEGER
* On exit, INFO
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value
* > 0: if INFO = 1X, internal error in DLARRE,
* if INFO = 2X, internal error in ZLARRV.
* Here, the digit X = ABS( IINFO ) < 10, where IINFO is
* the nonzero error code returned by DLARRE or
* ZLARRV, respectively.
*
*
* 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 ..
DOUBLE PRECISION ZERO, ONE, FOUR, MINRGP
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0,
$ FOUR = 4.0D0,
$ MINRGP = 1.0D-3 )
* ..
* .. Local Scalars ..
LOGICAL ALLEIG, INDEIG, LQUERY, VALEIG, WANTZ, ZQUERY
INTEGER I, IBEGIN, IEND, IFIRST, IIL, IINDBL, IINDW,
$ IINDWK, IINFO, IINSPL, IIU, ILAST, IN, INDD,
$ INDE2, INDERR, INDGP, INDGRS, INDWRK, ITMP,
$ ITMP2, J, JBLK, JJ, LIWMIN, LWMIN, NSPLIT,
$ NZCMIN, OFFSET, WBEGIN, WEND
DOUBLE PRECISION BIGNUM, CS, EPS, PIVMIN, R1, R2, RMAX, RMIN,
$ RTOL1, RTOL2, SAFMIN, SCALE, SMLNUM, SN,
$ THRESH, TMP, TNRM, WL, WU
* ..
* ..
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DLAMCH, DLANST
EXTERNAL LSAME, DLAMCH, DLANST
* ..
* .. External Subroutines ..
EXTERNAL DCOPY, DLAE2, DLAEV2, DLARRC, DLARRE, DLARRJ,
$ DLARRR, DLASRT, DSCAL, XERBLA, ZLARRV, ZSWAP
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN, SQRT
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
WANTZ = LSAME( JOBZ, 'V' )
ALLEIG = LSAME( RANGE, 'A' )
VALEIG = LSAME( RANGE, 'V' )
INDEIG = LSAME( RANGE, 'I' )
*
LQUERY = ( ( LWORK.EQ.-1 ).OR.( LIWORK.EQ.-1 ) )
ZQUERY = ( NZC.EQ.-1 )
TRYRAC = ( INFO.NE.0 )
* DSTEMR needs WORK of size 6*N, IWORK of size 3*N.
* In addition, DLARRE needs WORK of size 6*N, IWORK of size 5*N.
* Furthermore, ZLARRV needs WORK of size 12*N, IWORK of size 7*N.
IF( WANTZ ) THEN
LWMIN = 18*N
LIWMIN = 10*N
ELSE
* need less workspace if only the eigenvalues are wanted
LWMIN = 12*N
LIWMIN = 8*N
ENDIF
WL = ZERO
WU = ZERO
IIL = 0
IIU = 0
IF( VALEIG ) THEN
* We do not reference VL, VU in the cases RANGE = 'I','A'
* The interval (WL, WU] contains all the wanted eigenvalues.
* It is either given by the user or computed in DLARRE.
WL = VL
WU = VU
ELSEIF( INDEIG ) THEN
* We do not reference IL, IU in the cases RANGE = 'V','A'
IIL = IL
IIU = IU
ENDIF
*
INFO = 0
IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
INFO = -1
ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( VALEIG .AND. N.GT.0 .AND. WU.LE.WL ) THEN
INFO = -7
ELSE IF( INDEIG .AND. ( IIL.LT.1 .OR. IIL.GT.N ) ) THEN
INFO = -8
ELSE IF( INDEIG .AND. ( IIU.LT.IIL .OR. IIU.GT.N ) ) THEN
INFO = -9
ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
INFO = -13
ELSE IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
INFO = -17
ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN
INFO = -19
END IF
*
* Get machine constants.
*
SAFMIN = DLAMCH( 'Safe minimum' )
EPS = DLAMCH( 'Precision' )
SMLNUM = SAFMIN / EPS
BIGNUM = ONE / SMLNUM
RMIN = SQRT( SMLNUM )
RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) )
*
IF( INFO.EQ.0 ) THEN
WORK( 1 ) = LWMIN
IWORK( 1 ) = LIWMIN
*
IF( WANTZ .AND. ALLEIG ) THEN
NZCMIN = N
ELSE IF( WANTZ .AND. VALEIG ) THEN
CALL DLARRC( 'T', N, VL, VU, D, E, SAFMIN,
$ NZCMIN, ITMP, ITMP2, INFO )
ELSE IF( WANTZ .AND. INDEIG ) THEN
NZCMIN = IIU-IIL+1
ELSE
* WANTZ .EQ. FALSE.
NZCMIN = 0
ENDIF
IF( ZQUERY .AND. INFO.EQ.0 ) THEN
Z( 1,1 ) = NZCMIN
ELSE IF( NZC.LT.NZCMIN .AND. .NOT.ZQUERY ) THEN
INFO = -14
END IF
END IF
IF( INFO.NE.0 ) THEN
*
CALL XERBLA( 'ZSTEMR', -INFO )
*
RETURN
ELSE IF( LQUERY .OR. ZQUERY ) THEN
RETURN
END IF
*
* Handle N = 0, 1, and 2 cases immediately
*
M = 0
IF( N.EQ.0 )
$ RETURN
*
IF( N.EQ.1 ) THEN
IF( ALLEIG .OR. INDEIG ) THEN
M = 1
W( 1 ) = D( 1 )
ELSE
IF( WL.LT.D( 1 ) .AND. WU.GE.D( 1 ) ) THEN
M = 1
W( 1 ) = D( 1 )
END IF
END IF
IF( WANTZ.AND.(.NOT.ZQUERY) ) THEN
Z( 1, 1 ) = ONE
ISUPPZ(1) = 1
ISUPPZ(2) = 1
END IF
RETURN
END IF
*
IF( N.EQ.2 ) THEN
IF( .NOT.WANTZ ) THEN
CALL DLAE2( D(1), E(1), D(2), R1, R2 )
ELSE IF( WANTZ.AND.(.NOT.ZQUERY) ) THEN
CALL DLAEV2( D(1), E(1), D(2), R1, R2, CS, SN )
END IF
IF( ALLEIG.OR.
$ (VALEIG.AND.(R2.GT.WL).AND.
$ (R2.LE.WU)).OR.
$ (INDEIG.AND.(IIL.EQ.1)) ) THEN
M = M+1
W( M ) = R2
IF( WANTZ.AND.(.NOT.ZQUERY) ) THEN
Z( 1, M ) = -SN
Z( 2, M ) = CS
* Note: At most one of SN and CS can be zero.
IF (SN.NE.ZERO) THEN
IF (CS.NE.ZERO) THEN
ISUPPZ(2*M-1) = 1
ISUPPZ(2*M-1) = 2
ELSE
ISUPPZ(2*M-1) = 1
ISUPPZ(2*M-1) = 1
END IF
ELSE
ISUPPZ(2*M-1) = 2
ISUPPZ(2*M) = 2
END IF
ENDIF
ENDIF
IF( ALLEIG.OR.
$ (VALEIG.AND.(R1.GT.WL).AND.
$ (R1.LE.WU)).OR.
$ (INDEIG.AND.(IIU.EQ.2)) ) THEN
M = M+1
W( M ) = R1
IF( WANTZ.AND.(.NOT.ZQUERY) ) THEN
Z( 1, M ) = CS
Z( 2, M ) = SN
* Note: At most one of SN and CS can be zero.
IF (SN.NE.ZERO) THEN
IF (CS.NE.ZERO) THEN
ISUPPZ(2*M-1) = 1
ISUPPZ(2*M-1) = 2
ELSE
ISUPPZ(2*M-1) = 1
ISUPPZ(2*M-1) = 1
END IF
ELSE
ISUPPZ(2*M-1) = 2
ISUPPZ(2*M) = 2
END IF
ENDIF
ENDIF
RETURN
END IF
* Continue with general N
INDGRS = 1
INDERR = 2*N + 1
INDGP = 3*N + 1
INDD = 4*N + 1
INDE2 = 5*N + 1
INDWRK = 6*N + 1
*
IINSPL = 1
IINDBL = N + 1
IINDW = 2*N + 1
IINDWK = 3*N + 1
*
* Scale matrix to allowable range, if necessary.
* The allowable range is related to the PIVMIN parameter; see the
* comments in DLARRD. The preference for scaling small values
* up is heuristic; we expect users' matrices not to be close to the
* RMAX threshold.
*
SCALE = ONE
TNRM = DLANST( 'M', N, D, E )
IF( TNRM.GT.ZERO .AND. TNRM.LT.RMIN ) THEN
SCALE = RMIN / TNRM
ELSE IF( TNRM.GT.RMAX ) THEN
SCALE = RMAX / TNRM
END IF
IF( SCALE.NE.ONE ) THEN
CALL DSCAL( N, SCALE, D, 1 )
CALL DSCAL( N-1, SCALE, E, 1 )
TNRM = TNRM*SCALE
IF( VALEIG ) THEN
* If eigenvalues in interval have to be found,
* scale (WL, WU] accordingly
WL = WL*SCALE
WU = WU*SCALE
ENDIF
END IF
*
* Compute the desired eigenvalues of the tridiagonal after splitting
* into smaller subblocks if the corresponding off-diagonal elements
* are small
* THRESH is the splitting parameter for DLARRE
* A negative THRESH forces the old splitting criterion based on the
* size of the off-diagonal. A positive THRESH switches to splitting
* which preserves relative accuracy.
*
IF( TRYRAC ) THEN
* Test whether the matrix warrants the more expensive relative approach.
CALL DLARRR( N, D, E, IINFO )
ELSE
* The user does not care about relative accurately eigenvalues
IINFO = -1
ENDIF
* Set the splitting criterion
IF (IINFO.EQ.0) THEN
THRESH = EPS
ELSE
THRESH = -EPS
* relative accuracy is desired but T does not guarantee it
TRYRAC = .FALSE.
ENDIF
*
IF( TRYRAC ) THEN
* Copy original diagonal, needed to guarantee relative accuracy
CALL DCOPY(N,D,1,WORK(INDD),1)
ENDIF
* Store the squares of the offdiagonal values of T
DO 5 J = 1, N-1
WORK( INDE2+J-1 ) = E(J)**2
5 CONTINUE
* Set the tolerance parameters for bisection
IF( .NOT.WANTZ ) THEN
* DLARRE computes the eigenvalues to full precision.
RTOL1 = FOUR * EPS
RTOL2 = FOUR * EPS
ELSE
* DLARRE computes the eigenvalues to less than full precision.
* ZLARRV will refine the eigenvalue approximations, and we only
* need less accurate initial bisection in DLARRE.
* Note: these settings do only affect the subset case and DLARRE
RTOL1 = SQRT(EPS)
RTOL2 = MAX( SQRT(EPS)*5.0D-3, FOUR * EPS )
ENDIF
CALL DLARRE( RANGE, N, WL, WU, IIL, IIU, D, E,
$ WORK(INDE2), RTOL1, RTOL2, THRESH, NSPLIT,
$ IWORK( IINSPL ), M, W, WORK( INDERR ),
$ WORK( INDGP ), IWORK( IINDBL ),
$ IWORK( IINDW ), WORK( INDGRS ), PIVMIN,
$ WORK( INDWRK ), IWORK( IINDWK ), IINFO )
IF( IINFO.NE.0 ) THEN
INFO = 10 + ABS( IINFO )
RETURN
END IF
* Note that if RANGE .NE. 'V', DLARRE computes bounds on the desired
* part of the spectrum. All desired eigenvalues are contained in
* (WL,WU]
IF( WANTZ ) THEN
*
* Compute the desired eigenvectors corresponding to the computed
* eigenvalues
*
CALL ZLARRV( N, WL, WU, D, E,
$ PIVMIN, IWORK( IINSPL ), M,
$ 1, M, MINRGP, RTOL1, RTOL2,
$ W, WORK( INDERR ), WORK( INDGP ), IWORK( IINDBL ),
$ IWORK( IINDW ), WORK( INDGRS ), Z, LDZ,
$ ISUPPZ, WORK( INDWRK ), IWORK( IINDWK ), IINFO )
IF( IINFO.NE.0 ) THEN
INFO = 20 + ABS( IINFO )
RETURN
END IF
ELSE
* DLARRE computes eigenvalues of the (shifted) root representation
* ZLARRV returns the eigenvalues of the unshifted matrix.
* However, if the eigenvectors are not desired by the user, we need
* to apply the corresponding shifts from DLARRE to obtain the
* eigenvalues of the original matrix.
DO 20 J = 1, M
ITMP = IWORK( IINDBL+J-1 )
W( J ) = W( J ) + E( IWORK( IINSPL+ITMP-1 ) )
20 CONTINUE
END IF
*
IF ( TRYRAC ) THEN
* Refine computed eigenvalues so that they are relatively accurate
* with respect to the original matrix T.
IBEGIN = 1
WBEGIN = 1
DO 39 JBLK = 1, IWORK( IINDBL+M-1 )
IEND = IWORK( IINSPL+JBLK-1 )
IN = IEND - IBEGIN + 1
WEND = WBEGIN - 1
* check if any eigenvalues have to be refined in this block
36 CONTINUE
IF( WEND.LT.M ) THEN
IF( IWORK( IINDBL+WEND ).EQ.JBLK ) THEN
WEND = WEND + 1
GO TO 36
END IF
END IF
IF( WEND.LT.WBEGIN ) THEN
IBEGIN = IEND + 1
GO TO 39
END IF
OFFSET = IWORK(IINDW+WBEGIN-1)-1
IFIRST = IWORK(IINDW+WBEGIN-1)
ILAST = IWORK(IINDW+WEND-1)
RTOL2 = FOUR * EPS
CALL DLARRJ( IN,
$ WORK(INDD+IBEGIN-1), WORK(INDE2+IBEGIN-1),
$ IFIRST, ILAST, RTOL2, OFFSET, W(WBEGIN),
$ WORK( INDERR+WBEGIN-1 ),
$ WORK( INDWRK ), IWORK( IINDWK ), PIVMIN,
$ TNRM, IINFO )
IBEGIN = IEND + 1
WBEGIN = WEND + 1
39 CONTINUE
ENDIF
*
* If matrix was scaled, then rescale eigenvalues appropriately.
*
IF( SCALE.NE.ONE ) THEN
CALL DSCAL( M, ONE / SCALE, W, 1 )
END IF
*
* If eigenvalues are not in increasing order, then sort them,
* possibly along with eigenvectors.
*
IF( NSPLIT.GT.1 ) THEN
IF( .NOT. WANTZ ) THEN
CALL DLASRT( 'I', M, W, IINFO )
IF( IINFO.NE.0 ) THEN
INFO = 3
RETURN
END IF
ELSE
DO 60 J = 1, M - 1
I = 0
TMP = W( J )
DO 50 JJ = J + 1, M
IF( W( JJ ).LT.TMP ) THEN
I = JJ
TMP = W( JJ )
END IF
50 CONTINUE
IF( I.NE.0 ) THEN
W( I ) = W( J )
W( J ) = TMP
IF( WANTZ ) THEN
CALL ZSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 )
ITMP = ISUPPZ( 2*I-1 )
ISUPPZ( 2*I-1 ) = ISUPPZ( 2*J-1 )
ISUPPZ( 2*J-1 ) = ITMP
ITMP = ISUPPZ( 2*I )
ISUPPZ( 2*I ) = ISUPPZ( 2*J )
ISUPPZ( 2*J ) = ITMP
END IF
END IF
60 CONTINUE
END IF
ENDIF
*
*
WORK( 1 ) = LWMIN
IWORK( 1 ) = LIWMIN
RETURN
*
* End of ZSTEMR
*
END