Annotation of rpl/lapack/lapack/zstedc.f, revision 1.17

1.8       bertrand    1: *> \brief \b ZSTEDC
                      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 ZSTEDC + dependencies
                     10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zstedc.f">
                     11: *> [TGZ]</a>
                     12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zstedc.f">
                     13: *> [ZIP]</a>
                     14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zstedc.f">
1.8       bertrand   15: *> [TXT]</a>
1.15      bertrand   16: *> \endhtmlonly
1.8       bertrand   17: *
                     18: *  Definition:
                     19: *  ===========
                     20: *
                     21: *       SUBROUTINE ZSTEDC( COMPZ, N, D, E, Z, LDZ, WORK, LWORK, RWORK,
                     22: *                          LRWORK, IWORK, LIWORK, INFO )
1.15      bertrand   23: *
1.8       bertrand   24: *       .. Scalar Arguments ..
                     25: *       CHARACTER          COMPZ
                     26: *       INTEGER            INFO, LDZ, LIWORK, LRWORK, LWORK, N
                     27: *       ..
                     28: *       .. Array Arguments ..
                     29: *       INTEGER            IWORK( * )
                     30: *       DOUBLE PRECISION   D( * ), E( * ), RWORK( * )
                     31: *       COMPLEX*16         WORK( * ), Z( LDZ, * )
                     32: *       ..
1.15      bertrand   33: *
1.8       bertrand   34: *
                     35: *> \par Purpose:
                     36: *  =============
                     37: *>
                     38: *> \verbatim
                     39: *>
                     40: *> ZSTEDC computes all eigenvalues and, optionally, eigenvectors of a
                     41: *> symmetric tridiagonal matrix using the divide and conquer method.
                     42: *> The eigenvectors of a full or band complex Hermitian matrix can also
                     43: *> be found if ZHETRD or ZHPTRD or ZHBTRD has been used to reduce this
                     44: *> matrix to tridiagonal form.
                     45: *>
                     46: *> This code makes very mild assumptions about floating point
                     47: *> arithmetic. It will work on machines with a guard digit in
                     48: *> add/subtract, or on those binary machines without guard digits
                     49: *> which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2.
                     50: *> It could conceivably fail on hexadecimal or decimal machines
                     51: *> without guard digits, but we know of none.  See DLAED3 for details.
                     52: *> \endverbatim
                     53: *
                     54: *  Arguments:
                     55: *  ==========
                     56: *
                     57: *> \param[in] COMPZ
                     58: *> \verbatim
                     59: *>          COMPZ is CHARACTER*1
                     60: *>          = 'N':  Compute eigenvalues only.
                     61: *>          = 'I':  Compute eigenvectors of tridiagonal matrix also.
                     62: *>          = 'V':  Compute eigenvectors of original Hermitian matrix
                     63: *>                  also.  On entry, Z contains the unitary matrix used
                     64: *>                  to reduce the original matrix to tridiagonal form.
                     65: *> \endverbatim
                     66: *>
                     67: *> \param[in] N
                     68: *> \verbatim
                     69: *>          N is INTEGER
                     70: *>          The dimension of the symmetric tridiagonal matrix.  N >= 0.
                     71: *> \endverbatim
                     72: *>
                     73: *> \param[in,out] D
                     74: *> \verbatim
                     75: *>          D is DOUBLE PRECISION array, dimension (N)
                     76: *>          On entry, the diagonal elements of the tridiagonal matrix.
                     77: *>          On exit, if INFO = 0, the eigenvalues in ascending order.
                     78: *> \endverbatim
                     79: *>
                     80: *> \param[in,out] E
                     81: *> \verbatim
                     82: *>          E is DOUBLE PRECISION array, dimension (N-1)
                     83: *>          On entry, the subdiagonal elements of the tridiagonal matrix.
                     84: *>          On exit, E has been destroyed.
                     85: *> \endverbatim
                     86: *>
                     87: *> \param[in,out] Z
                     88: *> \verbatim
                     89: *>          Z is COMPLEX*16 array, dimension (LDZ,N)
                     90: *>          On entry, if COMPZ = 'V', then Z contains the unitary
                     91: *>          matrix used in the reduction to tridiagonal form.
                     92: *>          On exit, if INFO = 0, then if COMPZ = 'V', Z contains the
                     93: *>          orthonormal eigenvectors of the original Hermitian matrix,
                     94: *>          and if COMPZ = 'I', Z contains the orthonormal eigenvectors
                     95: *>          of the symmetric tridiagonal matrix.
                     96: *>          If  COMPZ = 'N', then Z is not referenced.
                     97: *> \endverbatim
                     98: *>
                     99: *> \param[in] LDZ
                    100: *> \verbatim
                    101: *>          LDZ is INTEGER
                    102: *>          The leading dimension of the array Z.  LDZ >= 1.
                    103: *>          If eigenvectors are desired, then LDZ >= max(1,N).
                    104: *> \endverbatim
                    105: *>
                    106: *> \param[out] WORK
                    107: *> \verbatim
                    108: *>          WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
                    109: *>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
                    110: *> \endverbatim
                    111: *>
                    112: *> \param[in] LWORK
                    113: *> \verbatim
                    114: *>          LWORK is INTEGER
                    115: *>          The dimension of the array WORK.
                    116: *>          If COMPZ = 'N' or 'I', or N <= 1, LWORK must be at least 1.
                    117: *>          If COMPZ = 'V' and N > 1, LWORK must be at least N*N.
                    118: *>          Note that for COMPZ = 'V', then if N is less than or
                    119: *>          equal to the minimum divide size, usually 25, then LWORK need
                    120: *>          only be 1.
                    121: *>
                    122: *>          If LWORK = -1, then a workspace query is assumed; the routine
                    123: *>          only calculates the optimal sizes of the WORK, RWORK and
                    124: *>          IWORK arrays, returns these values as the first entries of
                    125: *>          the WORK, RWORK and IWORK arrays, and no error message
                    126: *>          related to LWORK or LRWORK or LIWORK is issued by XERBLA.
                    127: *> \endverbatim
                    128: *>
                    129: *> \param[out] RWORK
                    130: *> \verbatim
1.17    ! bertrand  131: *>          RWORK is DOUBLE PRECISION array, dimension (MAX(1,LRWORK))
1.8       bertrand  132: *>          On exit, if INFO = 0, RWORK(1) returns the optimal LRWORK.
                    133: *> \endverbatim
                    134: *>
                    135: *> \param[in] LRWORK
                    136: *> \verbatim
                    137: *>          LRWORK is INTEGER
                    138: *>          The dimension of the array RWORK.
                    139: *>          If COMPZ = 'N' or N <= 1, LRWORK must be at least 1.
                    140: *>          If COMPZ = 'V' and N > 1, LRWORK must be at least
                    141: *>                         1 + 3*N + 2*N*lg N + 4*N**2 ,
                    142: *>                         where lg( N ) = smallest integer k such
                    143: *>                         that 2**k >= N.
                    144: *>          If COMPZ = 'I' and N > 1, LRWORK must be at least
                    145: *>                         1 + 4*N + 2*N**2 .
                    146: *>          Note that for COMPZ = 'I' or 'V', then if N is less than or
                    147: *>          equal to the minimum divide size, usually 25, then LRWORK
                    148: *>          need only be max(1,2*(N-1)).
                    149: *>
                    150: *>          If LRWORK = -1, then a workspace query is assumed; the
                    151: *>          routine only calculates the optimal sizes of the WORK, RWORK
                    152: *>          and IWORK arrays, returns these values as the first entries
                    153: *>          of the WORK, RWORK and IWORK arrays, and no error message
                    154: *>          related to LWORK or LRWORK or LIWORK is issued by XERBLA.
                    155: *> \endverbatim
                    156: *>
                    157: *> \param[out] IWORK
                    158: *> \verbatim
                    159: *>          IWORK is INTEGER array, dimension (MAX(1,LIWORK))
                    160: *>          On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
                    161: *> \endverbatim
                    162: *>
                    163: *> \param[in] LIWORK
                    164: *> \verbatim
                    165: *>          LIWORK is INTEGER
                    166: *>          The dimension of the array IWORK.
                    167: *>          If COMPZ = 'N' or N <= 1, LIWORK must be at least 1.
                    168: *>          If COMPZ = 'V' or N > 1,  LIWORK must be at least
                    169: *>                                    6 + 6*N + 5*N*lg N.
                    170: *>          If COMPZ = 'I' or N > 1,  LIWORK must be at least
                    171: *>                                    3 + 5*N .
                    172: *>          Note that for COMPZ = 'I' or 'V', then if N is less than or
                    173: *>          equal to the minimum divide size, usually 25, then LIWORK
                    174: *>          need only be 1.
                    175: *>
                    176: *>          If LIWORK = -1, then a workspace query is assumed; the
                    177: *>          routine only calculates the optimal sizes of the WORK, RWORK
                    178: *>          and IWORK arrays, returns these values as the first entries
                    179: *>          of the WORK, RWORK and IWORK arrays, and no error message
                    180: *>          related to LWORK or LRWORK or LIWORK is issued by XERBLA.
                    181: *> \endverbatim
                    182: *>
                    183: *> \param[out] INFO
                    184: *> \verbatim
                    185: *>          INFO is INTEGER
                    186: *>          = 0:  successful exit.
                    187: *>          < 0:  if INFO = -i, the i-th argument had an illegal value.
                    188: *>          > 0:  The algorithm failed to compute an eigenvalue while
                    189: *>                working on the submatrix lying in rows and columns
                    190: *>                INFO/(N+1) through mod(INFO,N+1).
                    191: *> \endverbatim
                    192: *
                    193: *  Authors:
                    194: *  ========
                    195: *
1.15      bertrand  196: *> \author Univ. of Tennessee
                    197: *> \author Univ. of California Berkeley
                    198: *> \author Univ. of Colorado Denver
                    199: *> \author NAG Ltd.
1.8       bertrand  200: *
1.17    ! bertrand  201: *> \date June 2017
1.8       bertrand  202: *
                    203: *> \ingroup complex16OTHERcomputational
                    204: *
                    205: *> \par Contributors:
                    206: *  ==================
                    207: *>
                    208: *> Jeff Rutter, Computer Science Division, University of California
                    209: *> at Berkeley, USA
                    210: *
                    211: *  =====================================================================
1.1       bertrand  212:       SUBROUTINE ZSTEDC( COMPZ, N, D, E, Z, LDZ, WORK, LWORK, RWORK,
                    213:      $                   LRWORK, IWORK, LIWORK, INFO )
                    214: *
1.17    ! bertrand  215: *  -- LAPACK computational routine (version 3.7.1) --
1.1       bertrand  216: *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
                    217: *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
1.17    ! bertrand  218: *     June 2017
1.1       bertrand  219: *
                    220: *     .. Scalar Arguments ..
                    221:       CHARACTER          COMPZ
                    222:       INTEGER            INFO, LDZ, LIWORK, LRWORK, LWORK, N
                    223: *     ..
                    224: *     .. Array Arguments ..
                    225:       INTEGER            IWORK( * )
                    226:       DOUBLE PRECISION   D( * ), E( * ), RWORK( * )
                    227:       COMPLEX*16         WORK( * ), Z( LDZ, * )
                    228: *     ..
                    229: *
                    230: *  =====================================================================
                    231: *
                    232: *     .. Parameters ..
                    233:       DOUBLE PRECISION   ZERO, ONE, TWO
                    234:       PARAMETER          ( ZERO = 0.0D0, ONE = 1.0D0, TWO = 2.0D0 )
                    235: *     ..
                    236: *     .. Local Scalars ..
                    237:       LOGICAL            LQUERY
                    238:       INTEGER            FINISH, I, ICOMPZ, II, J, K, LGN, LIWMIN, LL,
                    239:      $                   LRWMIN, LWMIN, M, SMLSIZ, START
                    240:       DOUBLE PRECISION   EPS, ORGNRM, P, TINY
                    241: *     ..
                    242: *     .. External Functions ..
                    243:       LOGICAL            LSAME
                    244:       INTEGER            ILAENV
                    245:       DOUBLE PRECISION   DLAMCH, DLANST
                    246:       EXTERNAL           LSAME, ILAENV, DLAMCH, DLANST
                    247: *     ..
                    248: *     .. External Subroutines ..
                    249:       EXTERNAL           DLASCL, DLASET, DSTEDC, DSTEQR, DSTERF, XERBLA,
                    250:      $                   ZLACPY, ZLACRM, ZLAED0, ZSTEQR, ZSWAP
                    251: *     ..
                    252: *     .. Intrinsic Functions ..
                    253:       INTRINSIC          ABS, DBLE, INT, LOG, MAX, MOD, SQRT
                    254: *     ..
                    255: *     .. Executable Statements ..
                    256: *
                    257: *     Test the input parameters.
                    258: *
                    259:       INFO = 0
                    260:       LQUERY = ( LWORK.EQ.-1 .OR. LRWORK.EQ.-1 .OR. LIWORK.EQ.-1 )
                    261: *
                    262:       IF( LSAME( COMPZ, 'N' ) ) THEN
                    263:          ICOMPZ = 0
                    264:       ELSE IF( LSAME( COMPZ, 'V' ) ) THEN
                    265:          ICOMPZ = 1
                    266:       ELSE IF( LSAME( COMPZ, 'I' ) ) THEN
                    267:          ICOMPZ = 2
                    268:       ELSE
                    269:          ICOMPZ = -1
                    270:       END IF
                    271:       IF( ICOMPZ.LT.0 ) THEN
                    272:          INFO = -1
                    273:       ELSE IF( N.LT.0 ) THEN
                    274:          INFO = -2
                    275:       ELSE IF( ( LDZ.LT.1 ) .OR.
                    276:      $         ( ICOMPZ.GT.0 .AND. LDZ.LT.MAX( 1, N ) ) ) THEN
                    277:          INFO = -6
                    278:       END IF
                    279: *
                    280:       IF( INFO.EQ.0 ) THEN
                    281: *
                    282: *        Compute the workspace requirements
                    283: *
                    284:          SMLSIZ = ILAENV( 9, 'ZSTEDC', ' ', 0, 0, 0, 0 )
                    285:          IF( N.LE.1 .OR. ICOMPZ.EQ.0 ) THEN
                    286:             LWMIN = 1
                    287:             LIWMIN = 1
                    288:             LRWMIN = 1
                    289:          ELSE IF( N.LE.SMLSIZ ) THEN
                    290:             LWMIN = 1
                    291:             LIWMIN = 1
                    292:             LRWMIN = 2*( N - 1 )
                    293:          ELSE IF( ICOMPZ.EQ.1 ) THEN
                    294:             LGN = INT( LOG( DBLE( N ) ) / LOG( TWO ) )
                    295:             IF( 2**LGN.LT.N )
                    296:      $         LGN = LGN + 1
                    297:             IF( 2**LGN.LT.N )
                    298:      $         LGN = LGN + 1
                    299:             LWMIN = N*N
1.8       bertrand  300:             LRWMIN = 1 + 3*N + 2*N*LGN + 4*N**2
1.1       bertrand  301:             LIWMIN = 6 + 6*N + 5*N*LGN
                    302:          ELSE IF( ICOMPZ.EQ.2 ) THEN
                    303:             LWMIN = 1
                    304:             LRWMIN = 1 + 4*N + 2*N**2
                    305:             LIWMIN = 3 + 5*N
                    306:          END IF
                    307:          WORK( 1 ) = LWMIN
                    308:          RWORK( 1 ) = LRWMIN
                    309:          IWORK( 1 ) = LIWMIN
                    310: *
                    311:          IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
                    312:             INFO = -8
                    313:          ELSE IF( LRWORK.LT.LRWMIN .AND. .NOT.LQUERY ) THEN
                    314:             INFO = -10
                    315:          ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN
                    316:             INFO = -12
                    317:          END IF
                    318:       END IF
                    319: *
                    320:       IF( INFO.NE.0 ) THEN
                    321:          CALL XERBLA( 'ZSTEDC', -INFO )
                    322:          RETURN
                    323:       ELSE IF( LQUERY ) THEN
                    324:          RETURN
                    325:       END IF
                    326: *
                    327: *     Quick return if possible
                    328: *
                    329:       IF( N.EQ.0 )
                    330:      $   RETURN
                    331:       IF( N.EQ.1 ) THEN
                    332:          IF( ICOMPZ.NE.0 )
                    333:      $      Z( 1, 1 ) = ONE
                    334:          RETURN
                    335:       END IF
                    336: *
                    337: *     If the following conditional clause is removed, then the routine
                    338: *     will use the Divide and Conquer routine to compute only the
                    339: *     eigenvalues, which requires (3N + 3N**2) real workspace and
                    340: *     (2 + 5N + 2N lg(N)) integer workspace.
                    341: *     Since on many architectures DSTERF is much faster than any other
                    342: *     algorithm for finding eigenvalues only, it is used here
                    343: *     as the default. If the conditional clause is removed, then
                    344: *     information on the size of workspace needs to be changed.
                    345: *
                    346: *     If COMPZ = 'N', use DSTERF to compute the eigenvalues.
                    347: *
                    348:       IF( ICOMPZ.EQ.0 ) THEN
                    349:          CALL DSTERF( N, D, E, INFO )
                    350:          GO TO 70
                    351:       END IF
                    352: *
                    353: *     If N is smaller than the minimum divide size (SMLSIZ+1), then
                    354: *     solve the problem with another solver.
                    355: *
                    356:       IF( N.LE.SMLSIZ ) THEN
                    357: *
                    358:          CALL ZSTEQR( COMPZ, N, D, E, Z, LDZ, RWORK, INFO )
                    359: *
                    360:       ELSE
                    361: *
                    362: *        If COMPZ = 'I', we simply call DSTEDC instead.
                    363: *
                    364:          IF( ICOMPZ.EQ.2 ) THEN
                    365:             CALL DLASET( 'Full', N, N, ZERO, ONE, RWORK, N )
                    366:             LL = N*N + 1
                    367:             CALL DSTEDC( 'I', N, D, E, RWORK, N,
                    368:      $                   RWORK( LL ), LRWORK-LL+1, IWORK, LIWORK, INFO )
                    369:             DO 20 J = 1, N
                    370:                DO 10 I = 1, N
                    371:                   Z( I, J ) = RWORK( ( J-1 )*N+I )
                    372:    10          CONTINUE
                    373:    20       CONTINUE
                    374:             GO TO 70
                    375:          END IF
                    376: *
                    377: *        From now on, only option left to be handled is COMPZ = 'V',
                    378: *        i.e. ICOMPZ = 1.
                    379: *
                    380: *        Scale.
                    381: *
                    382:          ORGNRM = DLANST( 'M', N, D, E )
                    383:          IF( ORGNRM.EQ.ZERO )
                    384:      $      GO TO 70
                    385: *
                    386:          EPS = DLAMCH( 'Epsilon' )
                    387: *
                    388:          START = 1
                    389: *
                    390: *        while ( START <= N )
                    391: *
                    392:    30    CONTINUE
                    393:          IF( START.LE.N ) THEN
                    394: *
                    395: *           Let FINISH be the position of the next subdiagonal entry
                    396: *           such that E( FINISH ) <= TINY or FINISH = N if no such
                    397: *           subdiagonal exists.  The matrix identified by the elements
                    398: *           between START and FINISH constitutes an independent
                    399: *           sub-problem.
                    400: *
                    401:             FINISH = START
                    402:    40       CONTINUE
                    403:             IF( FINISH.LT.N ) THEN
                    404:                TINY = EPS*SQRT( ABS( D( FINISH ) ) )*
                    405:      $                    SQRT( ABS( D( FINISH+1 ) ) )
                    406:                IF( ABS( E( FINISH ) ).GT.TINY ) THEN
                    407:                   FINISH = FINISH + 1
                    408:                   GO TO 40
                    409:                END IF
                    410:             END IF
                    411: *
                    412: *           (Sub) Problem determined.  Compute its size and solve it.
                    413: *
                    414:             M = FINISH - START + 1
                    415:             IF( M.GT.SMLSIZ ) THEN
                    416: *
                    417: *              Scale.
                    418: *
                    419:                ORGNRM = DLANST( 'M', M, D( START ), E( START ) )
                    420:                CALL DLASCL( 'G', 0, 0, ORGNRM, ONE, M, 1, D( START ), M,
                    421:      $                      INFO )
                    422:                CALL DLASCL( 'G', 0, 0, ORGNRM, ONE, M-1, 1, E( START ),
                    423:      $                      M-1, INFO )
                    424: *
                    425:                CALL ZLAED0( N, M, D( START ), E( START ), Z( 1, START ),
                    426:      $                      LDZ, WORK, N, RWORK, IWORK, INFO )
                    427:                IF( INFO.GT.0 ) THEN
                    428:                   INFO = ( INFO / ( M+1 )+START-1 )*( N+1 ) +
                    429:      $                   MOD( INFO, ( M+1 ) ) + START - 1
                    430:                   GO TO 70
                    431:                END IF
                    432: *
                    433: *              Scale back.
                    434: *
                    435:                CALL DLASCL( 'G', 0, 0, ONE, ORGNRM, M, 1, D( START ), M,
                    436:      $                      INFO )
                    437: *
                    438:             ELSE
                    439:                CALL DSTEQR( 'I', M, D( START ), E( START ), RWORK, M,
                    440:      $                      RWORK( M*M+1 ), INFO )
                    441:                CALL ZLACRM( N, M, Z( 1, START ), LDZ, RWORK, M, WORK, N,
                    442:      $                      RWORK( M*M+1 ) )
                    443:                CALL ZLACPY( 'A', N, M, WORK, N, Z( 1, START ), LDZ )
                    444:                IF( INFO.GT.0 ) THEN
                    445:                   INFO = START*( N+1 ) + FINISH
                    446:                   GO TO 70
                    447:                END IF
                    448:             END IF
                    449: *
                    450:             START = FINISH + 1
                    451:             GO TO 30
                    452:          END IF
                    453: *
                    454: *        endwhile
                    455: *
1.13      bertrand  456: *
                    457: *        Use Selection Sort to minimize swaps of eigenvectors
                    458: *
                    459:          DO 60 II = 2, N
                    460:            I = II - 1
                    461:            K = I
                    462:            P = D( I )
                    463:            DO 50 J = II, N
                    464:               IF( D( J ).LT.P ) THEN
                    465:                  K = J
                    466:                  P = D( J )
                    467:               END IF
                    468:    50      CONTINUE
                    469:            IF( K.NE.I ) THEN
                    470:               D( K ) = D( I )
                    471:               D( I ) = P
                    472:               CALL ZSWAP( N, Z( 1, I ), 1, Z( 1, K ), 1 )
                    473:            END IF
                    474:    60    CONTINUE
1.1       bertrand  475:       END IF
                    476: *
                    477:    70 CONTINUE
                    478:       WORK( 1 ) = LWMIN
                    479:       RWORK( 1 ) = LRWMIN
                    480:       IWORK( 1 ) = LIWMIN
                    481: *
                    482:       RETURN
                    483: *
                    484: *     End of ZSTEDC
                    485: *
                    486:       END

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