Annotation of rpl/lapack/lapack/dlaed2.f, revision 1.18

1.18    ! bertrand    1: *> \brief \b DLAED2 used by DSTEDC. Merges eigenvalues and deflates secular equation. Used when the original matrix is tridiagonal.
1.8       bertrand    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 DLAED2 + dependencies
                     10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaed2.f">
                     11: *> [TGZ]</a>
                     12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaed2.f">
                     13: *> [ZIP]</a>
                     14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaed2.f">
1.8       bertrand   15: *> [TXT]</a>
1.15      bertrand   16: *> \endhtmlonly
1.8       bertrand   17: *
                     18: *  Definition:
                     19: *  ===========
                     20: *
                     21: *       SUBROUTINE DLAED2( K, N, N1, D, Q, LDQ, INDXQ, RHO, Z, DLAMDA, W,
                     22: *                          Q2, INDX, INDXC, INDXP, COLTYP, INFO )
1.15      bertrand   23: *
1.8       bertrand   24: *       .. Scalar Arguments ..
                     25: *       INTEGER            INFO, K, LDQ, N, N1
                     26: *       DOUBLE PRECISION   RHO
                     27: *       ..
                     28: *       .. Array Arguments ..
                     29: *       INTEGER            COLTYP( * ), INDX( * ), INDXC( * ), INDXP( * ),
                     30: *      $                   INDXQ( * )
                     31: *       DOUBLE PRECISION   D( * ), DLAMDA( * ), Q( LDQ, * ), Q2( * ),
                     32: *      $                   W( * ), Z( * )
                     33: *       ..
1.15      bertrand   34: *
1.8       bertrand   35: *
                     36: *> \par Purpose:
                     37: *  =============
                     38: *>
                     39: *> \verbatim
                     40: *>
                     41: *> DLAED2 merges the two sets of eigenvalues together into a single
                     42: *> sorted set.  Then it tries to deflate the size of the problem.
                     43: *> There are two ways in which deflation can occur:  when two or more
                     44: *> eigenvalues are close together or if there is a tiny entry in the
                     45: *> Z vector.  For each such occurrence the order of the related secular
                     46: *> equation problem is reduced by one.
                     47: *> \endverbatim
                     48: *
                     49: *  Arguments:
                     50: *  ==========
                     51: *
                     52: *> \param[out] K
                     53: *> \verbatim
                     54: *>          K is INTEGER
                     55: *>         The number of non-deflated eigenvalues, and the order of the
                     56: *>         related secular equation. 0 <= K <=N.
                     57: *> \endverbatim
                     58: *>
                     59: *> \param[in] N
                     60: *> \verbatim
                     61: *>          N is INTEGER
                     62: *>         The dimension of the symmetric tridiagonal matrix.  N >= 0.
                     63: *> \endverbatim
                     64: *>
                     65: *> \param[in] N1
                     66: *> \verbatim
                     67: *>          N1 is INTEGER
                     68: *>         The location of the last eigenvalue in the leading sub-matrix.
                     69: *>         min(1,N) <= N1 <= N/2.
                     70: *> \endverbatim
                     71: *>
                     72: *> \param[in,out] D
                     73: *> \verbatim
                     74: *>          D is DOUBLE PRECISION array, dimension (N)
                     75: *>         On entry, D contains the eigenvalues of the two submatrices to
                     76: *>         be combined.
                     77: *>         On exit, D contains the trailing (N-K) updated eigenvalues
                     78: *>         (those which were deflated) sorted into increasing order.
                     79: *> \endverbatim
                     80: *>
                     81: *> \param[in,out] Q
                     82: *> \verbatim
                     83: *>          Q is DOUBLE PRECISION array, dimension (LDQ, N)
                     84: *>         On entry, Q contains the eigenvectors of two submatrices in
                     85: *>         the two square blocks with corners at (1,1), (N1,N1)
                     86: *>         and (N1+1, N1+1), (N,N).
                     87: *>         On exit, Q contains the trailing (N-K) updated eigenvectors
                     88: *>         (those which were deflated) in its last N-K columns.
                     89: *> \endverbatim
                     90: *>
                     91: *> \param[in] LDQ
                     92: *> \verbatim
                     93: *>          LDQ is INTEGER
                     94: *>         The leading dimension of the array Q.  LDQ >= max(1,N).
                     95: *> \endverbatim
                     96: *>
                     97: *> \param[in,out] INDXQ
                     98: *> \verbatim
                     99: *>          INDXQ is INTEGER array, dimension (N)
                    100: *>         The permutation which separately sorts the two sub-problems
                    101: *>         in D into ascending order.  Note that elements in the second
                    102: *>         half of this permutation must first have N1 added to their
                    103: *>         values. Destroyed on exit.
                    104: *> \endverbatim
                    105: *>
                    106: *> \param[in,out] RHO
                    107: *> \verbatim
                    108: *>          RHO is DOUBLE PRECISION
                    109: *>         On entry, the off-diagonal element associated with the rank-1
                    110: *>         cut which originally split the two submatrices which are now
                    111: *>         being recombined.
                    112: *>         On exit, RHO has been modified to the value required by
                    113: *>         DLAED3.
                    114: *> \endverbatim
                    115: *>
                    116: *> \param[in] Z
                    117: *> \verbatim
                    118: *>          Z is DOUBLE PRECISION array, dimension (N)
                    119: *>         On entry, Z contains the updating vector (the last
                    120: *>         row of the first sub-eigenvector matrix and the first row of
                    121: *>         the second sub-eigenvector matrix).
                    122: *>         On exit, the contents of Z have been destroyed by the updating
                    123: *>         process.
                    124: *> \endverbatim
                    125: *>
                    126: *> \param[out] DLAMDA
                    127: *> \verbatim
                    128: *>          DLAMDA is DOUBLE PRECISION array, dimension (N)
                    129: *>         A copy of the first K eigenvalues which will be used by
                    130: *>         DLAED3 to form the secular equation.
                    131: *> \endverbatim
                    132: *>
                    133: *> \param[out] W
                    134: *> \verbatim
                    135: *>          W is DOUBLE PRECISION array, dimension (N)
                    136: *>         The first k values of the final deflation-altered z-vector
                    137: *>         which will be passed to DLAED3.
                    138: *> \endverbatim
                    139: *>
                    140: *> \param[out] Q2
                    141: *> \verbatim
                    142: *>          Q2 is DOUBLE PRECISION array, dimension (N1**2+(N-N1)**2)
                    143: *>         A copy of the first K eigenvectors which will be used by
                    144: *>         DLAED3 in a matrix multiply (DGEMM) to solve for the new
                    145: *>         eigenvectors.
                    146: *> \endverbatim
                    147: *>
                    148: *> \param[out] INDX
                    149: *> \verbatim
                    150: *>          INDX is INTEGER array, dimension (N)
                    151: *>         The permutation used to sort the contents of DLAMDA into
                    152: *>         ascending order.
                    153: *> \endverbatim
                    154: *>
                    155: *> \param[out] INDXC
                    156: *> \verbatim
                    157: *>          INDXC is INTEGER array, dimension (N)
                    158: *>         The permutation used to arrange the columns of the deflated
                    159: *>         Q matrix into three groups:  the first group contains non-zero
                    160: *>         elements only at and above N1, the second contains
                    161: *>         non-zero elements only below N1, and the third is dense.
                    162: *> \endverbatim
                    163: *>
                    164: *> \param[out] INDXP
                    165: *> \verbatim
                    166: *>          INDXP is INTEGER array, dimension (N)
                    167: *>         The permutation used to place deflated values of D at the end
                    168: *>         of the array.  INDXP(1:K) points to the nondeflated D-values
                    169: *>         and INDXP(K+1:N) points to the deflated eigenvalues.
                    170: *> \endverbatim
                    171: *>
                    172: *> \param[out] COLTYP
                    173: *> \verbatim
                    174: *>          COLTYP is INTEGER array, dimension (N)
                    175: *>         During execution, a label which will indicate which of the
                    176: *>         following types a column in the Q2 matrix is:
                    177: *>         1 : non-zero in the upper half only;
                    178: *>         2 : dense;
                    179: *>         3 : non-zero in the lower half only;
                    180: *>         4 : deflated.
                    181: *>         On exit, COLTYP(i) is the number of columns of type i,
                    182: *>         for i=1 to 4 only.
                    183: *> \endverbatim
                    184: *>
                    185: *> \param[out] INFO
                    186: *> \verbatim
                    187: *>          INFO is INTEGER
                    188: *>          = 0:  successful exit.
                    189: *>          < 0:  if INFO = -i, the i-th argument had an illegal value.
                    190: *> \endverbatim
                    191: *
                    192: *  Authors:
                    193: *  ========
                    194: *
1.15      bertrand  195: *> \author Univ. of Tennessee
                    196: *> \author Univ. of California Berkeley
                    197: *> \author Univ. of Colorado Denver
                    198: *> \author NAG Ltd.
1.8       bertrand  199: *
                    200: *> \ingroup auxOTHERcomputational
                    201: *
                    202: *> \par Contributors:
                    203: *  ==================
                    204: *>
                    205: *> Jeff Rutter, Computer Science Division, University of California
                    206: *> at Berkeley, USA \n
                    207: *>  Modified by Francoise Tisseur, University of Tennessee
                    208: *>
                    209: *  =====================================================================
1.1       bertrand  210:       SUBROUTINE DLAED2( K, N, N1, D, Q, LDQ, INDXQ, RHO, Z, DLAMDA, W,
                    211:      $                   Q2, INDX, INDXC, INDXP, COLTYP, INFO )
                    212: *
1.18    ! bertrand  213: *  -- LAPACK computational routine --
1.1       bertrand  214: *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
                    215: *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
                    216: *
                    217: *     .. Scalar Arguments ..
                    218:       INTEGER            INFO, K, LDQ, N, N1
                    219:       DOUBLE PRECISION   RHO
                    220: *     ..
                    221: *     .. Array Arguments ..
                    222:       INTEGER            COLTYP( * ), INDX( * ), INDXC( * ), INDXP( * ),
                    223:      $                   INDXQ( * )
                    224:       DOUBLE PRECISION   D( * ), DLAMDA( * ), Q( LDQ, * ), Q2( * ),
                    225:      $                   W( * ), Z( * )
                    226: *     ..
                    227: *
                    228: *  =====================================================================
                    229: *
                    230: *     .. Parameters ..
                    231:       DOUBLE PRECISION   MONE, ZERO, ONE, TWO, EIGHT
                    232:       PARAMETER          ( MONE = -1.0D0, ZERO = 0.0D0, ONE = 1.0D0,
                    233:      $                   TWO = 2.0D0, EIGHT = 8.0D0 )
                    234: *     ..
                    235: *     .. Local Arrays ..
                    236:       INTEGER            CTOT( 4 ), PSM( 4 )
                    237: *     ..
                    238: *     .. Local Scalars ..
                    239:       INTEGER            CT, I, IMAX, IQ1, IQ2, J, JMAX, JS, K2, N1P1,
                    240:      $                   N2, NJ, PJ
                    241:       DOUBLE PRECISION   C, EPS, S, T, TAU, TOL
                    242: *     ..
                    243: *     .. External Functions ..
                    244:       INTEGER            IDAMAX
                    245:       DOUBLE PRECISION   DLAMCH, DLAPY2
                    246:       EXTERNAL           IDAMAX, DLAMCH, DLAPY2
                    247: *     ..
                    248: *     .. External Subroutines ..
                    249:       EXTERNAL           DCOPY, DLACPY, DLAMRG, DROT, DSCAL, XERBLA
                    250: *     ..
                    251: *     .. Intrinsic Functions ..
                    252:       INTRINSIC          ABS, MAX, MIN, SQRT
                    253: *     ..
                    254: *     .. Executable Statements ..
                    255: *
                    256: *     Test the input parameters.
                    257: *
                    258:       INFO = 0
                    259: *
                    260:       IF( N.LT.0 ) THEN
                    261:          INFO = -2
                    262:       ELSE IF( LDQ.LT.MAX( 1, N ) ) THEN
                    263:          INFO = -6
                    264:       ELSE IF( MIN( 1, ( N / 2 ) ).GT.N1 .OR. ( N / 2 ).LT.N1 ) THEN
                    265:          INFO = -3
                    266:       END IF
                    267:       IF( INFO.NE.0 ) THEN
                    268:          CALL XERBLA( 'DLAED2', -INFO )
                    269:          RETURN
                    270:       END IF
                    271: *
                    272: *     Quick return if possible
                    273: *
                    274:       IF( N.EQ.0 )
                    275:      $   RETURN
                    276: *
                    277:       N2 = N - N1
                    278:       N1P1 = N1 + 1
                    279: *
                    280:       IF( RHO.LT.ZERO ) THEN
                    281:          CALL DSCAL( N2, MONE, Z( N1P1 ), 1 )
                    282:       END IF
                    283: *
                    284: *     Normalize z so that norm(z) = 1.  Since z is the concatenation of
                    285: *     two normalized vectors, norm2(z) = sqrt(2).
                    286: *
                    287:       T = ONE / SQRT( TWO )
                    288:       CALL DSCAL( N, T, Z, 1 )
                    289: *
                    290: *     RHO = ABS( norm(z)**2 * RHO )
                    291: *
                    292:       RHO = ABS( TWO*RHO )
                    293: *
                    294: *     Sort the eigenvalues into increasing order
                    295: *
                    296:       DO 10 I = N1P1, N
                    297:          INDXQ( I ) = INDXQ( I ) + N1
                    298:    10 CONTINUE
                    299: *
                    300: *     re-integrate the deflated parts from the last pass
                    301: *
                    302:       DO 20 I = 1, N
                    303:          DLAMDA( I ) = D( INDXQ( I ) )
                    304:    20 CONTINUE
                    305:       CALL DLAMRG( N1, N2, DLAMDA, 1, 1, INDXC )
                    306:       DO 30 I = 1, N
                    307:          INDX( I ) = INDXQ( INDXC( I ) )
                    308:    30 CONTINUE
                    309: *
                    310: *     Calculate the allowable deflation tolerance
                    311: *
                    312:       IMAX = IDAMAX( N, Z, 1 )
                    313:       JMAX = IDAMAX( N, D, 1 )
                    314:       EPS = DLAMCH( 'Epsilon' )
                    315:       TOL = EIGHT*EPS*MAX( ABS( D( JMAX ) ), ABS( Z( IMAX ) ) )
                    316: *
                    317: *     If the rank-1 modifier is small enough, no more needs to be done
                    318: *     except to reorganize Q so that its columns correspond with the
                    319: *     elements in D.
                    320: *
                    321:       IF( RHO*ABS( Z( IMAX ) ).LE.TOL ) THEN
                    322:          K = 0
                    323:          IQ2 = 1
                    324:          DO 40 J = 1, N
                    325:             I = INDX( J )
                    326:             CALL DCOPY( N, Q( 1, I ), 1, Q2( IQ2 ), 1 )
                    327:             DLAMDA( J ) = D( I )
                    328:             IQ2 = IQ2 + N
                    329:    40    CONTINUE
                    330:          CALL DLACPY( 'A', N, N, Q2, N, Q, LDQ )
                    331:          CALL DCOPY( N, DLAMDA, 1, D, 1 )
                    332:          GO TO 190
                    333:       END IF
                    334: *
                    335: *     If there are multiple eigenvalues then the problem deflates.  Here
                    336: *     the number of equal eigenvalues are found.  As each equal
                    337: *     eigenvalue is found, an elementary reflector is computed to rotate
                    338: *     the corresponding eigensubspace so that the corresponding
                    339: *     components of Z are zero in this new basis.
                    340: *
                    341:       DO 50 I = 1, N1
                    342:          COLTYP( I ) = 1
                    343:    50 CONTINUE
                    344:       DO 60 I = N1P1, N
                    345:          COLTYP( I ) = 3
                    346:    60 CONTINUE
                    347: *
                    348: *
                    349:       K = 0
                    350:       K2 = N + 1
                    351:       DO 70 J = 1, N
                    352:          NJ = INDX( J )
                    353:          IF( RHO*ABS( Z( NJ ) ).LE.TOL ) THEN
                    354: *
                    355: *           Deflate due to small z component.
                    356: *
                    357:             K2 = K2 - 1
                    358:             COLTYP( NJ ) = 4
                    359:             INDXP( K2 ) = NJ
                    360:             IF( J.EQ.N )
                    361:      $         GO TO 100
                    362:          ELSE
                    363:             PJ = NJ
                    364:             GO TO 80
                    365:          END IF
                    366:    70 CONTINUE
                    367:    80 CONTINUE
                    368:       J = J + 1
                    369:       NJ = INDX( J )
                    370:       IF( J.GT.N )
                    371:      $   GO TO 100
                    372:       IF( RHO*ABS( Z( NJ ) ).LE.TOL ) THEN
                    373: *
                    374: *        Deflate due to small z component.
                    375: *
                    376:          K2 = K2 - 1
                    377:          COLTYP( NJ ) = 4
                    378:          INDXP( K2 ) = NJ
                    379:       ELSE
                    380: *
                    381: *        Check if eigenvalues are close enough to allow deflation.
                    382: *
                    383:          S = Z( PJ )
                    384:          C = Z( NJ )
                    385: *
                    386: *        Find sqrt(a**2+b**2) without overflow or
                    387: *        destructive underflow.
                    388: *
                    389:          TAU = DLAPY2( C, S )
                    390:          T = D( NJ ) - D( PJ )
                    391:          C = C / TAU
                    392:          S = -S / TAU
                    393:          IF( ABS( T*C*S ).LE.TOL ) THEN
                    394: *
                    395: *           Deflation is possible.
                    396: *
                    397:             Z( NJ ) = TAU
                    398:             Z( PJ ) = ZERO
                    399:             IF( COLTYP( NJ ).NE.COLTYP( PJ ) )
                    400:      $         COLTYP( NJ ) = 2
                    401:             COLTYP( PJ ) = 4
                    402:             CALL DROT( N, Q( 1, PJ ), 1, Q( 1, NJ ), 1, C, S )
                    403:             T = D( PJ )*C**2 + D( NJ )*S**2
                    404:             D( NJ ) = D( PJ )*S**2 + D( NJ )*C**2
                    405:             D( PJ ) = T
                    406:             K2 = K2 - 1
                    407:             I = 1
                    408:    90       CONTINUE
                    409:             IF( K2+I.LE.N ) THEN
                    410:                IF( D( PJ ).LT.D( INDXP( K2+I ) ) ) THEN
                    411:                   INDXP( K2+I-1 ) = INDXP( K2+I )
                    412:                   INDXP( K2+I ) = PJ
                    413:                   I = I + 1
                    414:                   GO TO 90
                    415:                ELSE
                    416:                   INDXP( K2+I-1 ) = PJ
                    417:                END IF
                    418:             ELSE
                    419:                INDXP( K2+I-1 ) = PJ
                    420:             END IF
                    421:             PJ = NJ
                    422:          ELSE
                    423:             K = K + 1
                    424:             DLAMDA( K ) = D( PJ )
                    425:             W( K ) = Z( PJ )
                    426:             INDXP( K ) = PJ
                    427:             PJ = NJ
                    428:          END IF
                    429:       END IF
                    430:       GO TO 80
                    431:   100 CONTINUE
                    432: *
                    433: *     Record the last eigenvalue.
                    434: *
                    435:       K = K + 1
                    436:       DLAMDA( K ) = D( PJ )
                    437:       W( K ) = Z( PJ )
                    438:       INDXP( K ) = PJ
                    439: *
                    440: *     Count up the total number of the various types of columns, then
                    441: *     form a permutation which positions the four column types into
                    442: *     four uniform groups (although one or more of these groups may be
                    443: *     empty).
                    444: *
                    445:       DO 110 J = 1, 4
                    446:          CTOT( J ) = 0
                    447:   110 CONTINUE
                    448:       DO 120 J = 1, N
                    449:          CT = COLTYP( J )
                    450:          CTOT( CT ) = CTOT( CT ) + 1
                    451:   120 CONTINUE
                    452: *
                    453: *     PSM(*) = Position in SubMatrix (of types 1 through 4)
                    454: *
                    455:       PSM( 1 ) = 1
                    456:       PSM( 2 ) = 1 + CTOT( 1 )
                    457:       PSM( 3 ) = PSM( 2 ) + CTOT( 2 )
                    458:       PSM( 4 ) = PSM( 3 ) + CTOT( 3 )
                    459:       K = N - CTOT( 4 )
                    460: *
                    461: *     Fill out the INDXC array so that the permutation which it induces
                    462: *     will place all type-1 columns first, all type-2 columns next,
                    463: *     then all type-3's, and finally all type-4's.
                    464: *
                    465:       DO 130 J = 1, N
                    466:          JS = INDXP( J )
                    467:          CT = COLTYP( JS )
                    468:          INDX( PSM( CT ) ) = JS
                    469:          INDXC( PSM( CT ) ) = J
                    470:          PSM( CT ) = PSM( CT ) + 1
                    471:   130 CONTINUE
                    472: *
                    473: *     Sort the eigenvalues and corresponding eigenvectors into DLAMDA
                    474: *     and Q2 respectively.  The eigenvalues/vectors which were not
                    475: *     deflated go into the first K slots of DLAMDA and Q2 respectively,
                    476: *     while those which were deflated go into the last N - K slots.
                    477: *
                    478:       I = 1
                    479:       IQ1 = 1
                    480:       IQ2 = 1 + ( CTOT( 1 )+CTOT( 2 ) )*N1
                    481:       DO 140 J = 1, CTOT( 1 )
                    482:          JS = INDX( I )
                    483:          CALL DCOPY( N1, Q( 1, JS ), 1, Q2( IQ1 ), 1 )
                    484:          Z( I ) = D( JS )
                    485:          I = I + 1
                    486:          IQ1 = IQ1 + N1
                    487:   140 CONTINUE
                    488: *
                    489:       DO 150 J = 1, CTOT( 2 )
                    490:          JS = INDX( I )
                    491:          CALL DCOPY( N1, Q( 1, JS ), 1, Q2( IQ1 ), 1 )
                    492:          CALL DCOPY( N2, Q( N1+1, JS ), 1, Q2( IQ2 ), 1 )
                    493:          Z( I ) = D( JS )
                    494:          I = I + 1
                    495:          IQ1 = IQ1 + N1
                    496:          IQ2 = IQ2 + N2
                    497:   150 CONTINUE
                    498: *
                    499:       DO 160 J = 1, CTOT( 3 )
                    500:          JS = INDX( I )
                    501:          CALL DCOPY( N2, Q( N1+1, JS ), 1, Q2( IQ2 ), 1 )
                    502:          Z( I ) = D( JS )
                    503:          I = I + 1
                    504:          IQ2 = IQ2 + N2
                    505:   160 CONTINUE
                    506: *
                    507:       IQ1 = IQ2
                    508:       DO 170 J = 1, CTOT( 4 )
                    509:          JS = INDX( I )
                    510:          CALL DCOPY( N, Q( 1, JS ), 1, Q2( IQ2 ), 1 )
                    511:          IQ2 = IQ2 + N
                    512:          Z( I ) = D( JS )
                    513:          I = I + 1
                    514:   170 CONTINUE
                    515: *
                    516: *     The deflated eigenvalues and their corresponding vectors go back
                    517: *     into the last N - K slots of D and Q respectively.
                    518: *
1.8       bertrand  519:       IF( K.LT.N ) THEN
1.15      bertrand  520:          CALL DLACPY( 'A', N, CTOT( 4 ), Q2( IQ1 ), N,
1.8       bertrand  521:      $                Q( 1, K+1 ), LDQ )
                    522:          CALL DCOPY( N-K, Z( K+1 ), 1, D( K+1 ), 1 )
1.15      bertrand  523:       END IF
1.1       bertrand  524: *
                    525: *     Copy CTOT into COLTYP for referencing in DLAED3.
                    526: *
                    527:       DO 180 J = 1, 4
                    528:          COLTYP( J ) = CTOT( J )
                    529:   180 CONTINUE
                    530: *
                    531:   190 CONTINUE
                    532:       RETURN
                    533: *
                    534: *     End of DLAED2
                    535: *
                    536:       END

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