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Mise à jour de lapack vers la version 3.3.0.

    1:       SUBROUTINE DSTEVX( JOBZ, RANGE, N, D, E, VL, VU, IL, IU, ABSTOL,
    2:      $                   M, W, Z, LDZ, WORK, IWORK, IFAIL, INFO )
    3: *
    4: *  -- LAPACK driver routine (version 3.2) --
    5: *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
    6: *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
    7: *     November 2006
    8: *
    9: *     .. Scalar Arguments ..
   10:       CHARACTER          JOBZ, RANGE
   11:       INTEGER            IL, INFO, IU, LDZ, M, N
   12:       DOUBLE PRECISION   ABSTOL, VL, VU
   13: *     ..
   14: *     .. Array Arguments ..
   15:       INTEGER            IFAIL( * ), IWORK( * )
   16:       DOUBLE PRECISION   D( * ), E( * ), W( * ), WORK( * ), Z( LDZ, * )
   17: *     ..
   18: *
   19: *  Purpose
   20: *  =======
   21: *
   22: *  DSTEVX computes selected eigenvalues and, optionally, eigenvectors
   23: *  of a real symmetric tridiagonal matrix A.  Eigenvalues and
   24: *  eigenvectors can be selected by specifying either a range of values
   25: *  or a range of indices for the desired eigenvalues.
   26: *
   27: *  Arguments
   28: *  =========
   29: *
   30: *  JOBZ    (input) CHARACTER*1
   31: *          = 'N':  Compute eigenvalues only;
   32: *          = 'V':  Compute eigenvalues and eigenvectors.
   33: *
   34: *  RANGE   (input) CHARACTER*1
   35: *          = 'A': all eigenvalues will be found.
   36: *          = 'V': all eigenvalues in the half-open interval (VL,VU]
   37: *                 will be found.
   38: *          = 'I': the IL-th through IU-th eigenvalues will be found.
   39: *
   40: *  N       (input) INTEGER
   41: *          The order of the matrix.  N >= 0.
   42: *
   43: *  D       (input/output) DOUBLE PRECISION array, dimension (N)
   44: *          On entry, the n diagonal elements of the tridiagonal matrix
   45: *          A.
   46: *          On exit, D may be multiplied by a constant factor chosen
   47: *          to avoid over/underflow in computing the eigenvalues.
   48: *
   49: *  E       (input/output) DOUBLE PRECISION array, dimension (max(1,N-1))
   50: *          On entry, the (n-1) subdiagonal elements of the tridiagonal
   51: *          matrix A in elements 1 to N-1 of E.
   52: *          On exit, E may be multiplied by a constant factor chosen
   53: *          to avoid over/underflow in computing the eigenvalues.
   54: *
   55: *  VL      (input) DOUBLE PRECISION
   56: *  VU      (input) DOUBLE PRECISION
   57: *          If RANGE='V', the lower and upper bounds of the interval to
   58: *          be searched for eigenvalues. VL < VU.
   59: *          Not referenced if RANGE = 'A' or 'I'.
   60: *
   61: *  IL      (input) INTEGER
   62: *  IU      (input) INTEGER
   63: *          If RANGE='I', the indices (in ascending order) of the
   64: *          smallest and largest eigenvalues to be returned.
   65: *          1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
   66: *          Not referenced if RANGE = 'A' or 'V'.
   67: *
   68: *  ABSTOL  (input) DOUBLE PRECISION
   69: *          The absolute error tolerance for the eigenvalues.
   70: *          An approximate eigenvalue is accepted as converged
   71: *          when it is determined to lie in an interval [a,b]
   72: *          of width less than or equal to
   73: *
   74: *                  ABSTOL + EPS *   max( |a|,|b| ) ,
   75: *
   76: *          where EPS is the machine precision.  If ABSTOL is less
   77: *          than or equal to zero, then  EPS*|T|  will be used in
   78: *          its place, where |T| is the 1-norm of the tridiagonal
   79: *          matrix.
   80: *
   81: *          Eigenvalues will be computed most accurately when ABSTOL is
   82: *          set to twice the underflow threshold 2*DLAMCH('S'), not zero.
   83: *          If this routine returns with INFO>0, indicating that some
   84: *          eigenvectors did not converge, try setting ABSTOL to
   85: *          2*DLAMCH('S').
   86: *
   87: *          See "Computing Small Singular Values of Bidiagonal Matrices
   88: *          with Guaranteed High Relative Accuracy," by Demmel and
   89: *          Kahan, LAPACK Working Note #3.
   90: *
   91: *  M       (output) INTEGER
   92: *          The total number of eigenvalues found.  0 <= M <= N.
   93: *          If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
   94: *
   95: *  W       (output) DOUBLE PRECISION array, dimension (N)
   96: *          The first M elements contain the selected eigenvalues in
   97: *          ascending order.
   98: *
   99: *  Z       (output) DOUBLE PRECISION array, dimension (LDZ, max(1,M) )
  100: *          If JOBZ = 'V', then if INFO = 0, the first M columns of Z
  101: *          contain the orthonormal eigenvectors of the matrix A
  102: *          corresponding to the selected eigenvalues, with the i-th
  103: *          column of Z holding the eigenvector associated with W(i).
  104: *          If an eigenvector fails to converge (INFO > 0), then that
  105: *          column of Z contains the latest approximation to the
  106: *          eigenvector, and the index of the eigenvector is returned
  107: *          in IFAIL.  If JOBZ = 'N', then Z is not referenced.
  108: *          Note: the user must ensure that at least max(1,M) columns are
  109: *          supplied in the array Z; if RANGE = 'V', the exact value of M
  110: *          is not known in advance and an upper bound must be used.
  111: *
  112: *  LDZ     (input) INTEGER
  113: *          The leading dimension of the array Z.  LDZ >= 1, and if
  114: *          JOBZ = 'V', LDZ >= max(1,N).
  115: *
  116: *  WORK    (workspace) DOUBLE PRECISION array, dimension (5*N)
  117: *
  118: *  IWORK   (workspace) INTEGER array, dimension (5*N)
  119: *
  120: *  IFAIL   (output) INTEGER array, dimension (N)
  121: *          If JOBZ = 'V', then if INFO = 0, the first M elements of
  122: *          IFAIL are zero.  If INFO > 0, then IFAIL contains the
  123: *          indices of the eigenvectors that failed to converge.
  124: *          If JOBZ = 'N', then IFAIL is not referenced.
  125: *
  126: *  INFO    (output) INTEGER
  127: *          = 0:  successful exit
  128: *          < 0:  if INFO = -i, the i-th argument had an illegal value
  129: *          > 0:  if INFO = i, then i eigenvectors failed to converge.
  130: *                Their indices are stored in array IFAIL.
  131: *
  132: *  =====================================================================
  133: *
  134: *     .. Parameters ..
  135:       DOUBLE PRECISION   ZERO, ONE
  136:       PARAMETER          ( ZERO = 0.0D0, ONE = 1.0D0 )
  137: *     ..
  138: *     .. Local Scalars ..
  139:       LOGICAL            ALLEIG, INDEIG, TEST, VALEIG, WANTZ
  140:       CHARACTER          ORDER
  141:       INTEGER            I, IMAX, INDIBL, INDISP, INDIWO, INDWRK,
  142:      $                   ISCALE, ITMP1, J, JJ, NSPLIT
  143:       DOUBLE PRECISION   BIGNUM, EPS, RMAX, RMIN, SAFMIN, SIGMA, SMLNUM,
  144:      $                   TMP1, TNRM, VLL, VUU
  145: *     ..
  146: *     .. External Functions ..
  147:       LOGICAL            LSAME
  148:       DOUBLE PRECISION   DLAMCH, DLANST
  149:       EXTERNAL           LSAME, DLAMCH, DLANST
  150: *     ..
  151: *     .. External Subroutines ..
  152:       EXTERNAL           DCOPY, DSCAL, DSTEBZ, DSTEIN, DSTEQR, DSTERF,
  153:      $                   DSWAP, XERBLA
  154: *     ..
  155: *     .. Intrinsic Functions ..
  156:       INTRINSIC          MAX, MIN, SQRT
  157: *     ..
  158: *     .. Executable Statements ..
  159: *
  160: *     Test the input parameters.
  161: *
  162:       WANTZ = LSAME( JOBZ, 'V' )
  163:       ALLEIG = LSAME( RANGE, 'A' )
  164:       VALEIG = LSAME( RANGE, 'V' )
  165:       INDEIG = LSAME( RANGE, 'I' )
  166: *
  167:       INFO = 0
  168:       IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
  169:          INFO = -1
  170:       ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN
  171:          INFO = -2
  172:       ELSE IF( N.LT.0 ) THEN
  173:          INFO = -3
  174:       ELSE
  175:          IF( VALEIG ) THEN
  176:             IF( N.GT.0 .AND. VU.LE.VL )
  177:      $         INFO = -7
  178:          ELSE IF( INDEIG ) THEN
  179:             IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN
  180:                INFO = -8
  181:             ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN
  182:                INFO = -9
  183:             END IF
  184:          END IF
  185:       END IF
  186:       IF( INFO.EQ.0 ) THEN
  187:          IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) )
  188:      $      INFO = -14
  189:       END IF
  190: *
  191:       IF( INFO.NE.0 ) THEN
  192:          CALL XERBLA( 'DSTEVX', -INFO )
  193:          RETURN
  194:       END IF
  195: *
  196: *     Quick return if possible
  197: *
  198:       M = 0
  199:       IF( N.EQ.0 )
  200:      $   RETURN
  201: *
  202:       IF( N.EQ.1 ) THEN
  203:          IF( ALLEIG .OR. INDEIG ) THEN
  204:             M = 1
  205:             W( 1 ) = D( 1 )
  206:          ELSE
  207:             IF( VL.LT.D( 1 ) .AND. VU.GE.D( 1 ) ) THEN
  208:                M = 1
  209:                W( 1 ) = D( 1 )
  210:             END IF
  211:          END IF
  212:          IF( WANTZ )
  213:      $      Z( 1, 1 ) = ONE
  214:          RETURN
  215:       END IF
  216: *
  217: *     Get machine constants.
  218: *
  219:       SAFMIN = DLAMCH( 'Safe minimum' )
  220:       EPS = DLAMCH( 'Precision' )
  221:       SMLNUM = SAFMIN / EPS
  222:       BIGNUM = ONE / SMLNUM
  223:       RMIN = SQRT( SMLNUM )
  224:       RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) )
  225: *
  226: *     Scale matrix to allowable range, if necessary.
  227: *
  228:       ISCALE = 0
  229:       IF( VALEIG ) THEN
  230:          VLL = VL
  231:          VUU = VU
  232:       ELSE
  233:          VLL = ZERO
  234:          VUU = ZERO
  235:       END IF
  236:       TNRM = DLANST( 'M', N, D, E )
  237:       IF( TNRM.GT.ZERO .AND. TNRM.LT.RMIN ) THEN
  238:          ISCALE = 1
  239:          SIGMA = RMIN / TNRM
  240:       ELSE IF( TNRM.GT.RMAX ) THEN
  241:          ISCALE = 1
  242:          SIGMA = RMAX / TNRM
  243:       END IF
  244:       IF( ISCALE.EQ.1 ) THEN
  245:          CALL DSCAL( N, SIGMA, D, 1 )
  246:          CALL DSCAL( N-1, SIGMA, E( 1 ), 1 )
  247:          IF( VALEIG ) THEN
  248:             VLL = VL*SIGMA
  249:             VUU = VU*SIGMA
  250:          END IF
  251:       END IF
  252: *
  253: *     If all eigenvalues are desired and ABSTOL is less than zero, then
  254: *     call DSTERF or SSTEQR.  If this fails for some eigenvalue, then
  255: *     try DSTEBZ.
  256: *
  257:       TEST = .FALSE.
  258:       IF( INDEIG ) THEN
  259:          IF( IL.EQ.1 .AND. IU.EQ.N ) THEN
  260:             TEST = .TRUE.
  261:          END IF
  262:       END IF
  263:       IF( ( ALLEIG .OR. TEST ) .AND. ( ABSTOL.LE.ZERO ) ) THEN
  264:          CALL DCOPY( N, D, 1, W, 1 )
  265:          CALL DCOPY( N-1, E( 1 ), 1, WORK( 1 ), 1 )
  266:          INDWRK = N + 1
  267:          IF( .NOT.WANTZ ) THEN
  268:             CALL DSTERF( N, W, WORK, INFO )
  269:          ELSE
  270:             CALL DSTEQR( 'I', N, W, WORK, Z, LDZ, WORK( INDWRK ), INFO )
  271:             IF( INFO.EQ.0 ) THEN
  272:                DO 10 I = 1, N
  273:                   IFAIL( I ) = 0
  274:    10          CONTINUE
  275:             END IF
  276:          END IF
  277:          IF( INFO.EQ.0 ) THEN
  278:             M = N
  279:             GO TO 20
  280:          END IF
  281:          INFO = 0
  282:       END IF
  283: *
  284: *     Otherwise, call DSTEBZ and, if eigenvectors are desired, SSTEIN.
  285: *
  286:       IF( WANTZ ) THEN
  287:          ORDER = 'B'
  288:       ELSE
  289:          ORDER = 'E'
  290:       END IF
  291:       INDWRK = 1
  292:       INDIBL = 1
  293:       INDISP = INDIBL + N
  294:       INDIWO = INDISP + N
  295:       CALL DSTEBZ( RANGE, ORDER, N, VLL, VUU, IL, IU, ABSTOL, D, E, M,
  296:      $             NSPLIT, W, IWORK( INDIBL ), IWORK( INDISP ),
  297:      $             WORK( INDWRK ), IWORK( INDIWO ), INFO )
  298: *
  299:       IF( WANTZ ) THEN
  300:          CALL DSTEIN( N, D, E, M, W, IWORK( INDIBL ), IWORK( INDISP ),
  301:      $                Z, LDZ, WORK( INDWRK ), IWORK( INDIWO ), IFAIL,
  302:      $                INFO )
  303:       END IF
  304: *
  305: *     If matrix was scaled, then rescale eigenvalues appropriately.
  306: *
  307:    20 CONTINUE
  308:       IF( ISCALE.EQ.1 ) THEN
  309:          IF( INFO.EQ.0 ) THEN
  310:             IMAX = M
  311:          ELSE
  312:             IMAX = INFO - 1
  313:          END IF
  314:          CALL DSCAL( IMAX, ONE / SIGMA, W, 1 )
  315:       END IF
  316: *
  317: *     If eigenvalues are not in order, then sort them, along with
  318: *     eigenvectors.
  319: *
  320:       IF( WANTZ ) THEN
  321:          DO 40 J = 1, M - 1
  322:             I = 0
  323:             TMP1 = W( J )
  324:             DO 30 JJ = J + 1, M
  325:                IF( W( JJ ).LT.TMP1 ) THEN
  326:                   I = JJ
  327:                   TMP1 = W( JJ )
  328:                END IF
  329:    30       CONTINUE
  330: *
  331:             IF( I.NE.0 ) THEN
  332:                ITMP1 = IWORK( INDIBL+I-1 )
  333:                W( I ) = W( J )
  334:                IWORK( INDIBL+I-1 ) = IWORK( INDIBL+J-1 )
  335:                W( J ) = TMP1
  336:                IWORK( INDIBL+J-1 ) = ITMP1
  337:                CALL DSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 )
  338:                IF( INFO.NE.0 ) THEN
  339:                   ITMP1 = IFAIL( I )
  340:                   IFAIL( I ) = IFAIL( J )
  341:                   IFAIL( J ) = ITMP1
  342:                END IF
  343:             END IF
  344:    40    CONTINUE
  345:       END IF
  346: *
  347:       RETURN
  348: *
  349: *     End of DSTEVX
  350: *
  351:       END