LAPACK 3.3.1 Linear Algebra PACKage

# slarrj.f

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```00001       SUBROUTINE SLARRJ( N, D, E2, IFIRST, ILAST,
00002      \$                   RTOL, OFFSET, W, WERR, WORK, IWORK,
00003      \$                   PIVMIN, SPDIAM, INFO )
00004 *
00005 *  -- LAPACK auxiliary routine (version 3.2.2) --
00006 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00007 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
00008 *     June 2010
00009 *
00010 *     .. Scalar Arguments ..
00011       INTEGER            IFIRST, ILAST, INFO, N, OFFSET
00012       REAL               PIVMIN, RTOL, SPDIAM
00013 *     ..
00014 *     .. Array Arguments ..
00015       INTEGER            IWORK( * )
00016       REAL               D( * ), E2( * ), W( * ),
00017      \$                   WERR( * ), WORK( * )
00018 *     ..
00019 *
00020 *  Purpose
00021 *  =======
00022 *
00023 *  Given the initial eigenvalue approximations of T, SLARRJ
00024 *  does  bisection to refine the eigenvalues of T,
00025 *  W( IFIRST-OFFSET ) through W( ILAST-OFFSET ), to more accuracy. Initial
00026 *  guesses for these eigenvalues are input in W, the corresponding estimate
00027 *  of the error in these guesses in WERR. During bisection, intervals
00028 *  [left, right] are maintained by storing their mid-points and
00029 *  semi-widths in the arrays W and WERR respectively.
00030 *
00031 *  Arguments
00032 *  =========
00033 *
00034 *  N       (input) INTEGER
00035 *          The order of the matrix.
00036 *
00037 *  D       (input) REAL             array, dimension (N)
00038 *          The N diagonal elements of T.
00039 *
00040 *  E2      (input) REAL             array, dimension (N-1)
00041 *          The Squares of the (N-1) subdiagonal elements of T.
00042 *
00043 *  IFIRST  (input) INTEGER
00044 *          The index of the first eigenvalue to be computed.
00045 *
00046 *  ILAST   (input) INTEGER
00047 *          The index of the last eigenvalue to be computed.
00048 *
00049 *  RTOL   (input) REAL
00050 *          Tolerance for the convergence of the bisection intervals.
00051 *          An interval [LEFT,RIGHT] has converged if
00052 *          RIGHT-LEFT.LT.RTOL*MAX(|LEFT|,|RIGHT|).
00053 *
00054 *  OFFSET  (input) INTEGER
00055 *          Offset for the arrays W and WERR, i.e., the IFIRST-OFFSET
00056 *          through ILAST-OFFSET elements of these arrays are to be used.
00057 *
00058 *  W       (input/output) REAL             array, dimension (N)
00059 *          On input, W( IFIRST-OFFSET ) through W( ILAST-OFFSET ) are
00060 *          estimates of the eigenvalues of L D L^T indexed IFIRST through
00061 *          ILAST.
00062 *          On output, these estimates are refined.
00063 *
00064 *  WERR    (input/output) REAL             array, dimension (N)
00065 *          On input, WERR( IFIRST-OFFSET ) through WERR( ILAST-OFFSET ) are
00066 *          the errors in the estimates of the corresponding elements in W.
00067 *          On output, these errors are refined.
00068 *
00069 *  WORK    (workspace) REAL             array, dimension (2*N)
00070 *          Workspace.
00071 *
00072 *  IWORK   (workspace) INTEGER array, dimension (2*N)
00073 *          Workspace.
00074 *
00075 *  PIVMIN  (input) REAL
00076 *          The minimum pivot in the Sturm sequence for T.
00077 *
00078 *  SPDIAM  (input) REAL
00079 *          The spectral diameter of T.
00080 *
00081 *  INFO    (output) INTEGER
00082 *          Error flag.
00083 *
00084 *  Further Details
00085 *  ===============
00086 *
00087 *  Based on contributions by
00088 *     Beresford Parlett, University of California, Berkeley, USA
00089 *     Jim Demmel, University of California, Berkeley, USA
00090 *     Inderjit Dhillon, University of Texas, Austin, USA
00091 *     Osni Marques, LBNL/NERSC, USA
00092 *     Christof Voemel, University of California, Berkeley, USA
00093 *
00094 *  =====================================================================
00095 *
00096 *     .. Parameters ..
00097       REAL               ZERO, ONE, TWO, HALF
00098       PARAMETER        ( ZERO = 0.0E0, ONE = 1.0E0, TWO = 2.0E0,
00099      \$                   HALF = 0.5E0 )
00100       INTEGER   MAXITR
00101 *     ..
00102 *     .. Local Scalars ..
00103       INTEGER            CNT, I, I1, I2, II, ITER, J, K, NEXT, NINT,
00104      \$                   OLNINT, P, PREV, SAVI1
00105       REAL               DPLUS, FAC, LEFT, MID, RIGHT, S, TMP, WIDTH
00106 *
00107 *     ..
00108 *     .. Intrinsic Functions ..
00109       INTRINSIC          ABS, MAX
00110 *     ..
00111 *     .. Executable Statements ..
00112 *
00113       INFO = 0
00114 *
00115       MAXITR = INT( ( LOG( SPDIAM+PIVMIN )-LOG( PIVMIN ) ) /
00116      \$           LOG( TWO ) ) + 2
00117 *
00118 *     Initialize unconverged intervals in [ WORK(2*I-1), WORK(2*I) ].
00119 *     The Sturm Count, Count( WORK(2*I-1) ) is arranged to be I-1, while
00120 *     Count( WORK(2*I) ) is stored in IWORK( 2*I ). The integer IWORK( 2*I-1 )
00121 *     for an unconverged interval is set to the index of the next unconverged
00122 *     interval, and is -1 or 0 for a converged interval. Thus a linked
00123 *     list of unconverged intervals is set up.
00124 *
00125
00126       I1 = IFIRST
00127       I2 = ILAST
00128 *     The number of unconverged intervals
00129       NINT = 0
00130 *     The last unconverged interval found
00131       PREV = 0
00132       DO 75 I = I1, I2
00133          K = 2*I
00134          II = I - OFFSET
00135          LEFT = W( II ) - WERR( II )
00136          MID = W(II)
00137          RIGHT = W( II ) + WERR( II )
00138          WIDTH = RIGHT - MID
00139          TMP = MAX( ABS( LEFT ), ABS( RIGHT ) )
00140
00141 *        The following test prevents the test of converged intervals
00142          IF( WIDTH.LT.RTOL*TMP ) THEN
00143 *           This interval has already converged and does not need refinement.
00144 *           (Note that the gaps might change through refining the
00145 *            eigenvalues, however, they can only get bigger.)
00146 *           Remove it from the list.
00147             IWORK( K-1 ) = -1
00148 *           Make sure that I1 always points to the first unconverged interval
00149             IF((I.EQ.I1).AND.(I.LT.I2)) I1 = I + 1
00150             IF((PREV.GE.I1).AND.(I.LE.I2)) IWORK( 2*PREV-1 ) = I + 1
00151          ELSE
00152 *           unconverged interval found
00153             PREV = I
00154 *           Make sure that [LEFT,RIGHT] contains the desired eigenvalue
00155 *
00156 *           Do while( CNT(LEFT).GT.I-1 )
00157 *
00158             FAC = ONE
00159  20         CONTINUE
00160             CNT = 0
00161             S = LEFT
00162             DPLUS = D( 1 ) - S
00163             IF( DPLUS.LT.ZERO ) CNT = CNT + 1
00164             DO 30 J = 2, N
00165                DPLUS = D( J ) - S - E2( J-1 )/DPLUS
00166                IF( DPLUS.LT.ZERO ) CNT = CNT + 1
00167  30         CONTINUE
00168             IF( CNT.GT.I-1 ) THEN
00169                LEFT = LEFT - WERR( II )*FAC
00170                FAC = TWO*FAC
00171                GO TO 20
00172             END IF
00173 *
00174 *           Do while( CNT(RIGHT).LT.I )
00175 *
00176             FAC = ONE
00177  50         CONTINUE
00178             CNT = 0
00179             S = RIGHT
00180             DPLUS = D( 1 ) - S
00181             IF( DPLUS.LT.ZERO ) CNT = CNT + 1
00182             DO 60 J = 2, N
00183                DPLUS = D( J ) - S - E2( J-1 )/DPLUS
00184                IF( DPLUS.LT.ZERO ) CNT = CNT + 1
00185  60         CONTINUE
00186             IF( CNT.LT.I ) THEN
00187                RIGHT = RIGHT + WERR( II )*FAC
00188                FAC = TWO*FAC
00189                GO TO 50
00190             END IF
00191             NINT = NINT + 1
00192             IWORK( K-1 ) = I + 1
00193             IWORK( K ) = CNT
00194          END IF
00195          WORK( K-1 ) = LEFT
00196          WORK( K ) = RIGHT
00197  75   CONTINUE
00198
00199
00200       SAVI1 = I1
00201 *
00202 *     Do while( NINT.GT.0 ), i.e. there are still unconverged intervals
00203 *     and while (ITER.LT.MAXITR)
00204 *
00205       ITER = 0
00206  80   CONTINUE
00207       PREV = I1 - 1
00208       I = I1
00209       OLNINT = NINT
00210
00211       DO 100 P = 1, OLNINT
00212          K = 2*I
00213          II = I - OFFSET
00214          NEXT = IWORK( K-1 )
00215          LEFT = WORK( K-1 )
00216          RIGHT = WORK( K )
00217          MID = HALF*( LEFT + RIGHT )
00218
00219 *        semiwidth of interval
00220          WIDTH = RIGHT - MID
00221          TMP = MAX( ABS( LEFT ), ABS( RIGHT ) )
00222
00223          IF( ( WIDTH.LT.RTOL*TMP ) .OR.
00224      \$      (ITER.EQ.MAXITR) )THEN
00225 *           reduce number of unconverged intervals
00226             NINT = NINT - 1
00227 *           Mark interval as converged.
00228             IWORK( K-1 ) = 0
00229             IF( I1.EQ.I ) THEN
00230                I1 = NEXT
00231             ELSE
00232 *              Prev holds the last unconverged interval previously examined
00233                IF(PREV.GE.I1) IWORK( 2*PREV-1 ) = NEXT
00234             END IF
00235             I = NEXT
00236             GO TO 100
00237          END IF
00238          PREV = I
00239 *
00240 *        Perform one bisection step
00241 *
00242          CNT = 0
00243          S = MID
00244          DPLUS = D( 1 ) - S
00245          IF( DPLUS.LT.ZERO ) CNT = CNT + 1
00246          DO 90 J = 2, N
00247             DPLUS = D( J ) - S - E2( J-1 )/DPLUS
00248             IF( DPLUS.LT.ZERO ) CNT = CNT + 1
00249  90      CONTINUE
00250          IF( CNT.LE.I-1 ) THEN
00251             WORK( K-1 ) = MID
00252          ELSE
00253             WORK( K ) = MID
00254          END IF
00255          I = NEXT
00256
00257  100  CONTINUE
00258       ITER = ITER + 1
00259 *     do another loop if there are still unconverged intervals
00260 *     However, in the last iteration, all intervals are accepted
00261 *     since this is the best we can do.
00262       IF( ( NINT.GT.0 ).AND.(ITER.LE.MAXITR) ) GO TO 80
00263 *
00264 *
00265 *     At this point, all the intervals have converged
00266       DO 110 I = SAVI1, ILAST
00267          K = 2*I
00268          II = I - OFFSET
00269 *        All intervals marked by '0' have been refined.
00270          IF( IWORK( K-1 ).EQ.0 ) THEN
00271             W( II ) = HALF*( WORK( K-1 )+WORK( K ) )
00272             WERR( II ) = WORK( K ) - W( II )
00273          END IF
00274  110  CONTINUE
00275 *
00276
00277       RETURN
00278 *
00279 *     End of SLARRJ
00280 *
00281       END
```