Annotation of rpl/lapack/lapack/dsbevx.f, revision 1.18

1.8       bertrand    1: *> \brief <b> DSBEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices</b>
                      2: *
                      3: *  =========== DOCUMENTATION ===========
                      4: *
1.15      bertrand    5: * Online html documentation available at
                      6: *            http://www.netlib.org/lapack/explore-html/
1.8       bertrand    7: *
                      8: *> \htmlonly
1.15      bertrand    9: *> Download DSBEVX + dependencies
                     10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsbevx.f">
                     11: *> [TGZ]</a>
                     12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsbevx.f">
                     13: *> [ZIP]</a>
                     14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsbevx.f">
1.8       bertrand   15: *> [TXT]</a>
1.15      bertrand   16: *> \endhtmlonly
1.8       bertrand   17: *
                     18: *  Definition:
                     19: *  ===========
                     20: *
                     21: *       SUBROUTINE DSBEVX( JOBZ, RANGE, UPLO, N, KD, AB, LDAB, Q, LDQ, VL,
                     22: *                          VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, IWORK,
                     23: *                          IFAIL, INFO )
1.15      bertrand   24: *
1.8       bertrand   25: *       .. Scalar Arguments ..
                     26: *       CHARACTER          JOBZ, RANGE, UPLO
                     27: *       INTEGER            IL, INFO, IU, KD, LDAB, LDQ, LDZ, M, N
                     28: *       DOUBLE PRECISION   ABSTOL, VL, VU
                     29: *       ..
                     30: *       .. Array Arguments ..
                     31: *       INTEGER            IFAIL( * ), IWORK( * )
                     32: *       DOUBLE PRECISION   AB( LDAB, * ), Q( LDQ, * ), W( * ), WORK( * ),
                     33: *      $                   Z( LDZ, * )
                     34: *       ..
1.15      bertrand   35: *
1.8       bertrand   36: *
                     37: *> \par Purpose:
                     38: *  =============
                     39: *>
                     40: *> \verbatim
                     41: *>
                     42: *> DSBEVX computes selected eigenvalues and, optionally, eigenvectors
                     43: *> of a real symmetric band matrix A.  Eigenvalues and eigenvectors can
                     44: *> be selected by specifying either a range of values or a range of
                     45: *> indices for the desired eigenvalues.
                     46: *> \endverbatim
                     47: *
                     48: *  Arguments:
                     49: *  ==========
                     50: *
                     51: *> \param[in] JOBZ
                     52: *> \verbatim
                     53: *>          JOBZ is CHARACTER*1
                     54: *>          = 'N':  Compute eigenvalues only;
                     55: *>          = 'V':  Compute eigenvalues and eigenvectors.
                     56: *> \endverbatim
                     57: *>
                     58: *> \param[in] RANGE
                     59: *> \verbatim
                     60: *>          RANGE is CHARACTER*1
                     61: *>          = 'A': all eigenvalues will be found;
                     62: *>          = 'V': all eigenvalues in the half-open interval (VL,VU]
                     63: *>                 will be found;
                     64: *>          = 'I': the IL-th through IU-th eigenvalues will be found.
                     65: *> \endverbatim
                     66: *>
                     67: *> \param[in] UPLO
                     68: *> \verbatim
                     69: *>          UPLO is CHARACTER*1
                     70: *>          = 'U':  Upper triangle of A is stored;
                     71: *>          = 'L':  Lower triangle of A is stored.
                     72: *> \endverbatim
                     73: *>
                     74: *> \param[in] N
                     75: *> \verbatim
                     76: *>          N is INTEGER
                     77: *>          The order of the matrix A.  N >= 0.
                     78: *> \endverbatim
                     79: *>
                     80: *> \param[in] KD
                     81: *> \verbatim
                     82: *>          KD is INTEGER
                     83: *>          The number of superdiagonals of the matrix A if UPLO = 'U',
                     84: *>          or the number of subdiagonals if UPLO = 'L'.  KD >= 0.
                     85: *> \endverbatim
                     86: *>
                     87: *> \param[in,out] AB
                     88: *> \verbatim
                     89: *>          AB is DOUBLE PRECISION array, dimension (LDAB, N)
                     90: *>          On entry, the upper or lower triangle of the symmetric band
                     91: *>          matrix A, stored in the first KD+1 rows of the array.  The
                     92: *>          j-th column of A is stored in the j-th column of the array AB
                     93: *>          as follows:
                     94: *>          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
                     95: *>          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
                     96: *>
                     97: *>          On exit, AB is overwritten by values generated during the
                     98: *>          reduction to tridiagonal form.  If UPLO = 'U', the first
                     99: *>          superdiagonal and the diagonal of the tridiagonal matrix T
                    100: *>          are returned in rows KD and KD+1 of AB, and if UPLO = 'L',
                    101: *>          the diagonal and first subdiagonal of T are returned in the
                    102: *>          first two rows of AB.
                    103: *> \endverbatim
                    104: *>
                    105: *> \param[in] LDAB
                    106: *> \verbatim
                    107: *>          LDAB is INTEGER
                    108: *>          The leading dimension of the array AB.  LDAB >= KD + 1.
                    109: *> \endverbatim
                    110: *>
                    111: *> \param[out] Q
                    112: *> \verbatim
                    113: *>          Q is DOUBLE PRECISION array, dimension (LDQ, N)
                    114: *>          If JOBZ = 'V', the N-by-N orthogonal matrix used in the
                    115: *>                         reduction to tridiagonal form.
                    116: *>          If JOBZ = 'N', the array Q is not referenced.
                    117: *> \endverbatim
                    118: *>
                    119: *> \param[in] LDQ
                    120: *> \verbatim
                    121: *>          LDQ is INTEGER
                    122: *>          The leading dimension of the array Q.  If JOBZ = 'V', then
                    123: *>          LDQ >= max(1,N).
                    124: *> \endverbatim
                    125: *>
                    126: *> \param[in] VL
                    127: *> \verbatim
                    128: *>          VL is DOUBLE PRECISION
1.13      bertrand  129: *>          If RANGE='V', the lower bound of the interval to
                    130: *>          be searched for eigenvalues. VL < VU.
                    131: *>          Not referenced if RANGE = 'A' or 'I'.
1.8       bertrand  132: *> \endverbatim
                    133: *>
                    134: *> \param[in] VU
                    135: *> \verbatim
                    136: *>          VU is DOUBLE PRECISION
1.13      bertrand  137: *>          If RANGE='V', the upper bound of the interval to
1.8       bertrand  138: *>          be searched for eigenvalues. VL < VU.
                    139: *>          Not referenced if RANGE = 'A' or 'I'.
                    140: *> \endverbatim
                    141: *>
                    142: *> \param[in] IL
                    143: *> \verbatim
                    144: *>          IL is INTEGER
1.13      bertrand  145: *>          If RANGE='I', the index of the
                    146: *>          smallest eigenvalue to be returned.
                    147: *>          1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
                    148: *>          Not referenced if RANGE = 'A' or 'V'.
1.8       bertrand  149: *> \endverbatim
                    150: *>
                    151: *> \param[in] IU
                    152: *> \verbatim
                    153: *>          IU is INTEGER
1.13      bertrand  154: *>          If RANGE='I', the index of the
                    155: *>          largest eigenvalue to be returned.
1.8       bertrand  156: *>          1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
                    157: *>          Not referenced if RANGE = 'A' or 'V'.
                    158: *> \endverbatim
                    159: *>
                    160: *> \param[in] ABSTOL
                    161: *> \verbatim
                    162: *>          ABSTOL is DOUBLE PRECISION
                    163: *>          The absolute error tolerance for the eigenvalues.
                    164: *>          An approximate eigenvalue is accepted as converged
                    165: *>          when it is determined to lie in an interval [a,b]
                    166: *>          of width less than or equal to
                    167: *>
                    168: *>                  ABSTOL + EPS *   max( |a|,|b| ) ,
                    169: *>
                    170: *>          where EPS is the machine precision.  If ABSTOL is less than
                    171: *>          or equal to zero, then  EPS*|T|  will be used in its place,
                    172: *>          where |T| is the 1-norm of the tridiagonal matrix obtained
                    173: *>          by reducing AB to tridiagonal form.
                    174: *>
                    175: *>          Eigenvalues will be computed most accurately when ABSTOL is
                    176: *>          set to twice the underflow threshold 2*DLAMCH('S'), not zero.
                    177: *>          If this routine returns with INFO>0, indicating that some
                    178: *>          eigenvectors did not converge, try setting ABSTOL to
                    179: *>          2*DLAMCH('S').
                    180: *>
                    181: *>          See "Computing Small Singular Values of Bidiagonal Matrices
                    182: *>          with Guaranteed High Relative Accuracy," by Demmel and
                    183: *>          Kahan, LAPACK Working Note #3.
                    184: *> \endverbatim
                    185: *>
                    186: *> \param[out] M
                    187: *> \verbatim
                    188: *>          M is INTEGER
                    189: *>          The total number of eigenvalues found.  0 <= M <= N.
                    190: *>          If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
                    191: *> \endverbatim
                    192: *>
                    193: *> \param[out] W
                    194: *> \verbatim
                    195: *>          W is DOUBLE PRECISION array, dimension (N)
                    196: *>          The first M elements contain the selected eigenvalues in
                    197: *>          ascending order.
                    198: *> \endverbatim
                    199: *>
                    200: *> \param[out] Z
                    201: *> \verbatim
                    202: *>          Z is DOUBLE PRECISION array, dimension (LDZ, max(1,M))
                    203: *>          If JOBZ = 'V', then if INFO = 0, the first M columns of Z
                    204: *>          contain the orthonormal eigenvectors of the matrix A
                    205: *>          corresponding to the selected eigenvalues, with the i-th
                    206: *>          column of Z holding the eigenvector associated with W(i).
                    207: *>          If an eigenvector fails to converge, then that column of Z
                    208: *>          contains the latest approximation to the eigenvector, and the
                    209: *>          index of the eigenvector is returned in IFAIL.
                    210: *>          If JOBZ = 'N', then Z is not referenced.
                    211: *>          Note: the user must ensure that at least max(1,M) columns are
                    212: *>          supplied in the array Z; if RANGE = 'V', the exact value of M
                    213: *>          is not known in advance and an upper bound must be used.
                    214: *> \endverbatim
                    215: *>
                    216: *> \param[in] LDZ
                    217: *> \verbatim
                    218: *>          LDZ is INTEGER
                    219: *>          The leading dimension of the array Z.  LDZ >= 1, and if
                    220: *>          JOBZ = 'V', LDZ >= max(1,N).
                    221: *> \endverbatim
                    222: *>
                    223: *> \param[out] WORK
                    224: *> \verbatim
                    225: *>          WORK is DOUBLE PRECISION array, dimension (7*N)
                    226: *> \endverbatim
                    227: *>
                    228: *> \param[out] IWORK
                    229: *> \verbatim
                    230: *>          IWORK is INTEGER array, dimension (5*N)
                    231: *> \endverbatim
                    232: *>
                    233: *> \param[out] IFAIL
                    234: *> \verbatim
                    235: *>          IFAIL is INTEGER array, dimension (N)
                    236: *>          If JOBZ = 'V', then if INFO = 0, the first M elements of
                    237: *>          IFAIL are zero.  If INFO > 0, then IFAIL contains the
                    238: *>          indices of the eigenvectors that failed to converge.
                    239: *>          If JOBZ = 'N', then IFAIL is not referenced.
                    240: *> \endverbatim
                    241: *>
                    242: *> \param[out] INFO
                    243: *> \verbatim
                    244: *>          INFO is INTEGER
                    245: *>          = 0:  successful exit.
                    246: *>          < 0:  if INFO = -i, the i-th argument had an illegal value.
                    247: *>          > 0:  if INFO = i, then i eigenvectors failed to converge.
                    248: *>                Their indices are stored in array IFAIL.
                    249: *> \endverbatim
                    250: *
                    251: *  Authors:
                    252: *  ========
                    253: *
1.15      bertrand  254: *> \author Univ. of Tennessee
                    255: *> \author Univ. of California Berkeley
                    256: *> \author Univ. of Colorado Denver
                    257: *> \author NAG Ltd.
1.8       bertrand  258: *
                    259: *> \ingroup doubleOTHEReigen
                    260: *
                    261: *  =====================================================================
1.1       bertrand  262:       SUBROUTINE DSBEVX( JOBZ, RANGE, UPLO, N, KD, AB, LDAB, Q, LDQ, VL,
                    263:      $                   VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, IWORK,
                    264:      $                   IFAIL, INFO )
                    265: *
1.18    ! bertrand  266: *  -- LAPACK driver routine --
1.1       bertrand  267: *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
                    268: *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
                    269: *
                    270: *     .. Scalar Arguments ..
                    271:       CHARACTER          JOBZ, RANGE, UPLO
                    272:       INTEGER            IL, INFO, IU, KD, LDAB, LDQ, LDZ, M, N
                    273:       DOUBLE PRECISION   ABSTOL, VL, VU
                    274: *     ..
                    275: *     .. Array Arguments ..
                    276:       INTEGER            IFAIL( * ), IWORK( * )
                    277:       DOUBLE PRECISION   AB( LDAB, * ), Q( LDQ, * ), W( * ), WORK( * ),
                    278:      $                   Z( LDZ, * )
                    279: *     ..
                    280: *
                    281: *  =====================================================================
                    282: *
                    283: *     .. Parameters ..
                    284:       DOUBLE PRECISION   ZERO, ONE
                    285:       PARAMETER          ( ZERO = 0.0D0, ONE = 1.0D0 )
                    286: *     ..
                    287: *     .. Local Scalars ..
                    288:       LOGICAL            ALLEIG, INDEIG, LOWER, TEST, VALEIG, WANTZ
                    289:       CHARACTER          ORDER
                    290:       INTEGER            I, IINFO, IMAX, INDD, INDE, INDEE, INDIBL,
                    291:      $                   INDISP, INDIWO, INDWRK, ISCALE, ITMP1, J, JJ,
                    292:      $                   NSPLIT
                    293:       DOUBLE PRECISION   ABSTLL, ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN,
                    294:      $                   SIGMA, SMLNUM, TMP1, VLL, VUU
                    295: *     ..
                    296: *     .. External Functions ..
                    297:       LOGICAL            LSAME
                    298:       DOUBLE PRECISION   DLAMCH, DLANSB
                    299:       EXTERNAL           LSAME, DLAMCH, DLANSB
                    300: *     ..
                    301: *     .. External Subroutines ..
                    302:       EXTERNAL           DCOPY, DGEMV, DLACPY, DLASCL, DSBTRD, DSCAL,
                    303:      $                   DSTEBZ, DSTEIN, DSTEQR, DSTERF, DSWAP, XERBLA
                    304: *     ..
                    305: *     .. Intrinsic Functions ..
                    306:       INTRINSIC          MAX, MIN, SQRT
                    307: *     ..
                    308: *     .. Executable Statements ..
                    309: *
                    310: *     Test the input parameters.
                    311: *
                    312:       WANTZ = LSAME( JOBZ, 'V' )
                    313:       ALLEIG = LSAME( RANGE, 'A' )
                    314:       VALEIG = LSAME( RANGE, 'V' )
                    315:       INDEIG = LSAME( RANGE, 'I' )
                    316:       LOWER = LSAME( UPLO, 'L' )
                    317: *
                    318:       INFO = 0
                    319:       IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
                    320:          INFO = -1
                    321:       ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN
                    322:          INFO = -2
                    323:       ELSE IF( .NOT.( LOWER .OR. LSAME( UPLO, 'U' ) ) ) THEN
                    324:          INFO = -3
                    325:       ELSE IF( N.LT.0 ) THEN
                    326:          INFO = -4
                    327:       ELSE IF( KD.LT.0 ) THEN
                    328:          INFO = -5
                    329:       ELSE IF( LDAB.LT.KD+1 ) THEN
                    330:          INFO = -7
                    331:       ELSE IF( WANTZ .AND. LDQ.LT.MAX( 1, N ) ) THEN
                    332:          INFO = -9
                    333:       ELSE
                    334:          IF( VALEIG ) THEN
                    335:             IF( N.GT.0 .AND. VU.LE.VL )
                    336:      $         INFO = -11
                    337:          ELSE IF( INDEIG ) THEN
                    338:             IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN
                    339:                INFO = -12
                    340:             ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN
                    341:                INFO = -13
                    342:             END IF
                    343:          END IF
                    344:       END IF
                    345:       IF( INFO.EQ.0 ) THEN
                    346:          IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) )
                    347:      $      INFO = -18
                    348:       END IF
                    349: *
                    350:       IF( INFO.NE.0 ) THEN
                    351:          CALL XERBLA( 'DSBEVX', -INFO )
                    352:          RETURN
                    353:       END IF
                    354: *
                    355: *     Quick return if possible
                    356: *
                    357:       M = 0
                    358:       IF( N.EQ.0 )
                    359:      $   RETURN
                    360: *
                    361:       IF( N.EQ.1 ) THEN
                    362:          M = 1
                    363:          IF( LOWER ) THEN
                    364:             TMP1 = AB( 1, 1 )
                    365:          ELSE
                    366:             TMP1 = AB( KD+1, 1 )
                    367:          END IF
                    368:          IF( VALEIG ) THEN
                    369:             IF( .NOT.( VL.LT.TMP1 .AND. VU.GE.TMP1 ) )
                    370:      $         M = 0
                    371:          END IF
                    372:          IF( M.EQ.1 ) THEN
                    373:             W( 1 ) = TMP1
                    374:             IF( WANTZ )
                    375:      $         Z( 1, 1 ) = ONE
                    376:          END IF
                    377:          RETURN
                    378:       END IF
                    379: *
                    380: *     Get machine constants.
                    381: *
                    382:       SAFMIN = DLAMCH( 'Safe minimum' )
                    383:       EPS = DLAMCH( 'Precision' )
                    384:       SMLNUM = SAFMIN / EPS
                    385:       BIGNUM = ONE / SMLNUM
                    386:       RMIN = SQRT( SMLNUM )
                    387:       RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) )
                    388: *
                    389: *     Scale matrix to allowable range, if necessary.
                    390: *
                    391:       ISCALE = 0
                    392:       ABSTLL = ABSTOL
                    393:       IF( VALEIG ) THEN
                    394:          VLL = VL
                    395:          VUU = VU
                    396:       ELSE
                    397:          VLL = ZERO
                    398:          VUU = ZERO
                    399:       END IF
                    400:       ANRM = DLANSB( 'M', UPLO, N, KD, AB, LDAB, WORK )
                    401:       IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN
                    402:          ISCALE = 1
                    403:          SIGMA = RMIN / ANRM
                    404:       ELSE IF( ANRM.GT.RMAX ) THEN
                    405:          ISCALE = 1
                    406:          SIGMA = RMAX / ANRM
                    407:       END IF
                    408:       IF( ISCALE.EQ.1 ) THEN
                    409:          IF( LOWER ) THEN
                    410:             CALL DLASCL( 'B', KD, KD, ONE, SIGMA, N, N, AB, LDAB, INFO )
                    411:          ELSE
                    412:             CALL DLASCL( 'Q', KD, KD, ONE, SIGMA, N, N, AB, LDAB, INFO )
                    413:          END IF
                    414:          IF( ABSTOL.GT.0 )
                    415:      $      ABSTLL = ABSTOL*SIGMA
                    416:          IF( VALEIG ) THEN
                    417:             VLL = VL*SIGMA
                    418:             VUU = VU*SIGMA
                    419:          END IF
                    420:       END IF
                    421: *
                    422: *     Call DSBTRD to reduce symmetric band matrix to tridiagonal form.
                    423: *
                    424:       INDD = 1
                    425:       INDE = INDD + N
                    426:       INDWRK = INDE + N
                    427:       CALL DSBTRD( JOBZ, UPLO, N, KD, AB, LDAB, WORK( INDD ),
                    428:      $             WORK( INDE ), Q, LDQ, WORK( INDWRK ), IINFO )
                    429: *
                    430: *     If all eigenvalues are desired and ABSTOL is less than or equal
                    431: *     to zero, then call DSTERF or SSTEQR.  If this fails for some
                    432: *     eigenvalue, then try DSTEBZ.
                    433: *
                    434:       TEST = .FALSE.
                    435:       IF (INDEIG) THEN
                    436:          IF (IL.EQ.1 .AND. IU.EQ.N) THEN
                    437:             TEST = .TRUE.
                    438:          END IF
                    439:       END IF
                    440:       IF ((ALLEIG .OR. TEST) .AND. (ABSTOL.LE.ZERO)) THEN
                    441:          CALL DCOPY( N, WORK( INDD ), 1, W, 1 )
                    442:          INDEE = INDWRK + 2*N
                    443:          IF( .NOT.WANTZ ) THEN
                    444:             CALL DCOPY( N-1, WORK( INDE ), 1, WORK( INDEE ), 1 )
                    445:             CALL DSTERF( N, W, WORK( INDEE ), INFO )
                    446:          ELSE
                    447:             CALL DLACPY( 'A', N, N, Q, LDQ, Z, LDZ )
                    448:             CALL DCOPY( N-1, WORK( INDE ), 1, WORK( INDEE ), 1 )
                    449:             CALL DSTEQR( JOBZ, N, W, WORK( INDEE ), Z, LDZ,
                    450:      $                   WORK( INDWRK ), INFO )
                    451:             IF( INFO.EQ.0 ) THEN
                    452:                DO 10 I = 1, N
                    453:                   IFAIL( I ) = 0
                    454:    10          CONTINUE
                    455:             END IF
                    456:          END IF
                    457:          IF( INFO.EQ.0 ) THEN
                    458:             M = N
                    459:             GO TO 30
                    460:          END IF
                    461:          INFO = 0
                    462:       END IF
                    463: *
                    464: *     Otherwise, call DSTEBZ and, if eigenvectors are desired, SSTEIN.
                    465: *
                    466:       IF( WANTZ ) THEN
                    467:          ORDER = 'B'
                    468:       ELSE
                    469:          ORDER = 'E'
                    470:       END IF
                    471:       INDIBL = 1
                    472:       INDISP = INDIBL + N
                    473:       INDIWO = INDISP + N
                    474:       CALL DSTEBZ( RANGE, ORDER, N, VLL, VUU, IL, IU, ABSTLL,
                    475:      $             WORK( INDD ), WORK( INDE ), M, NSPLIT, W,
                    476:      $             IWORK( INDIBL ), IWORK( INDISP ), WORK( INDWRK ),
                    477:      $             IWORK( INDIWO ), INFO )
                    478: *
                    479:       IF( WANTZ ) THEN
                    480:          CALL DSTEIN( N, WORK( INDD ), WORK( INDE ), M, W,
                    481:      $                IWORK( INDIBL ), IWORK( INDISP ), Z, LDZ,
                    482:      $                WORK( INDWRK ), IWORK( INDIWO ), IFAIL, INFO )
                    483: *
                    484: *        Apply orthogonal matrix used in reduction to tridiagonal
                    485: *        form to eigenvectors returned by DSTEIN.
                    486: *
                    487:          DO 20 J = 1, M
                    488:             CALL DCOPY( N, Z( 1, J ), 1, WORK( 1 ), 1 )
                    489:             CALL DGEMV( 'N', N, N, ONE, Q, LDQ, WORK, 1, ZERO,
                    490:      $                  Z( 1, J ), 1 )
                    491:    20    CONTINUE
                    492:       END IF
                    493: *
                    494: *     If matrix was scaled, then rescale eigenvalues appropriately.
                    495: *
                    496:    30 CONTINUE
                    497:       IF( ISCALE.EQ.1 ) THEN
                    498:          IF( INFO.EQ.0 ) THEN
                    499:             IMAX = M
                    500:          ELSE
                    501:             IMAX = INFO - 1
                    502:          END IF
                    503:          CALL DSCAL( IMAX, ONE / SIGMA, W, 1 )
                    504:       END IF
                    505: *
                    506: *     If eigenvalues are not in order, then sort them, along with
                    507: *     eigenvectors.
                    508: *
                    509:       IF( WANTZ ) THEN
                    510:          DO 50 J = 1, M - 1
                    511:             I = 0
                    512:             TMP1 = W( J )
                    513:             DO 40 JJ = J + 1, M
                    514:                IF( W( JJ ).LT.TMP1 ) THEN
                    515:                   I = JJ
                    516:                   TMP1 = W( JJ )
                    517:                END IF
                    518:    40       CONTINUE
                    519: *
                    520:             IF( I.NE.0 ) THEN
                    521:                ITMP1 = IWORK( INDIBL+I-1 )
                    522:                W( I ) = W( J )
                    523:                IWORK( INDIBL+I-1 ) = IWORK( INDIBL+J-1 )
                    524:                W( J ) = TMP1
                    525:                IWORK( INDIBL+J-1 ) = ITMP1
                    526:                CALL DSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 )
                    527:                IF( INFO.NE.0 ) THEN
                    528:                   ITMP1 = IFAIL( I )
                    529:                   IFAIL( I ) = IFAIL( J )
                    530:                   IFAIL( J ) = ITMP1
                    531:                END IF
                    532:             END IF
                    533:    50    CONTINUE
                    534:       END IF
                    535: *
                    536:       RETURN
                    537: *
                    538: *     End of DSBEVX
                    539: *
                    540:       END

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