LAPACK 3.3.0

dstevr.f

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00001       SUBROUTINE DSTEVR( JOBZ, RANGE, N, D, E, VL, VU, IL, IU, ABSTOL,
00002      $                   M, W, Z, LDZ, ISUPPZ, WORK, LWORK, IWORK,
00003      $                   LIWORK, INFO )
00004 *
00005 *  -- LAPACK driver routine (version 3.2) --
00006 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00007 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
00008 *     November 2006
00009 *
00010 *     .. Scalar Arguments ..
00011       CHARACTER          JOBZ, RANGE
00012       INTEGER            IL, INFO, IU, LDZ, LIWORK, LWORK, M, N
00013       DOUBLE PRECISION   ABSTOL, VL, VU
00014 *     ..
00015 *     .. Array Arguments ..
00016       INTEGER            ISUPPZ( * ), IWORK( * )
00017       DOUBLE PRECISION   D( * ), E( * ), W( * ), WORK( * ), Z( LDZ, * )
00018 *     ..
00019 *
00020 *  Purpose
00021 *  =======
00022 *
00023 *  DSTEVR computes selected eigenvalues and, optionally, eigenvectors
00024 *  of a real symmetric tridiagonal matrix T.  Eigenvalues and
00025 *  eigenvectors can be selected by specifying either a range of values
00026 *  or a range of indices for the desired eigenvalues.
00027 *
00028 *  Whenever possible, DSTEVR calls DSTEMR to compute the
00029 *  eigenspectrum using Relatively Robust Representations.  DSTEMR
00030 *  computes eigenvalues by the dqds algorithm, while orthogonal
00031 *  eigenvectors are computed from various "good" L D L^T representations
00032 *  (also known as Relatively Robust Representations). Gram-Schmidt
00033 *  orthogonalization is avoided as far as possible. More specifically,
00034 *  the various steps of the algorithm are as follows. For the i-th
00035 *  unreduced block of T,
00036 *     (a) Compute T - sigma_i = L_i D_i L_i^T, such that L_i D_i L_i^T
00037 *          is a relatively robust representation,
00038 *     (b) Compute the eigenvalues, lambda_j, of L_i D_i L_i^T to high
00039 *         relative accuracy by the dqds algorithm,
00040 *     (c) If there is a cluster of close eigenvalues, "choose" sigma_i
00041 *         close to the cluster, and go to step (a),
00042 *     (d) Given the approximate eigenvalue lambda_j of L_i D_i L_i^T,
00043 *         compute the corresponding eigenvector by forming a
00044 *         rank-revealing twisted factorization.
00045 *  The desired accuracy of the output can be specified by the input
00046 *  parameter ABSTOL.
00047 *
00048 *  For more details, see "A new O(n^2) algorithm for the symmetric
00049 *  tridiagonal eigenvalue/eigenvector problem", by Inderjit Dhillon,
00050 *  Computer Science Division Technical Report No. UCB//CSD-97-971,
00051 *  UC Berkeley, May 1997.
00052 *
00053 *
00054 *  Note 1 : DSTEVR calls DSTEMR when the full spectrum is requested
00055 *  on machines which conform to the ieee-754 floating point standard.
00056 *  DSTEVR calls DSTEBZ and DSTEIN on non-ieee machines and
00057 *  when partial spectrum requests are made.
00058 *
00059 *  Normal execution of DSTEMR may create NaNs and infinities and
00060 *  hence may abort due to a floating point exception in environments
00061 *  which do not handle NaNs and infinities in the ieee standard default
00062 *  manner.
00063 *
00064 *  Arguments
00065 *  =========
00066 *
00067 *  JOBZ    (input) CHARACTER*1
00068 *          = 'N':  Compute eigenvalues only;
00069 *          = 'V':  Compute eigenvalues and eigenvectors.
00070 *
00071 *  RANGE   (input) CHARACTER*1
00072 *          = 'A': all eigenvalues will be found.
00073 *          = 'V': all eigenvalues in the half-open interval (VL,VU]
00074 *                 will be found.
00075 *          = 'I': the IL-th through IU-th eigenvalues will be found.
00076 ********** For RANGE = 'V' or 'I' and IU - IL < N - 1, DSTEBZ and
00077 ********** DSTEIN are called
00078 *
00079 *  N       (input) INTEGER
00080 *          The order of the matrix.  N >= 0.
00081 *
00082 *  D       (input/output) DOUBLE PRECISION array, dimension (N)
00083 *          On entry, the n diagonal elements of the tridiagonal matrix
00084 *          A.
00085 *          On exit, D may be multiplied by a constant factor chosen
00086 *          to avoid over/underflow in computing the eigenvalues.
00087 *
00088 *  E       (input/output) DOUBLE PRECISION array, dimension (max(1,N-1))
00089 *          On entry, the (n-1) subdiagonal elements of the tridiagonal
00090 *          matrix A in elements 1 to N-1 of E.
00091 *          On exit, E may be multiplied by a constant factor chosen
00092 *          to avoid over/underflow in computing the eigenvalues.
00093 *
00094 *  VL      (input) DOUBLE PRECISION
00095 *  VU      (input) DOUBLE PRECISION
00096 *          If RANGE='V', the lower and upper bounds of the interval to
00097 *          be searched for eigenvalues. VL < VU.
00098 *          Not referenced if RANGE = 'A' or 'I'.
00099 *
00100 *  IL      (input) INTEGER
00101 *  IU      (input) INTEGER
00102 *          If RANGE='I', the indices (in ascending order) of the
00103 *          smallest and largest eigenvalues to be returned.
00104 *          1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
00105 *          Not referenced if RANGE = 'A' or 'V'.
00106 *
00107 *  ABSTOL  (input) DOUBLE PRECISION
00108 *          The absolute error tolerance for the eigenvalues.
00109 *          An approximate eigenvalue is accepted as converged
00110 *          when it is determined to lie in an interval [a,b]
00111 *          of width less than or equal to
00112 *
00113 *                  ABSTOL + EPS *   max( |a|,|b| ) ,
00114 *
00115 *          where EPS is the machine precision.  If ABSTOL is less than
00116 *          or equal to zero, then  EPS*|T|  will be used in its place,
00117 *          where |T| is the 1-norm of the tridiagonal matrix obtained
00118 *          by reducing A to tridiagonal form.
00119 *
00120 *          See "Computing Small Singular Values of Bidiagonal Matrices
00121 *          with Guaranteed High Relative Accuracy," by Demmel and
00122 *          Kahan, LAPACK Working Note #3.
00123 *
00124 *          If high relative accuracy is important, set ABSTOL to
00125 *          DLAMCH( 'Safe minimum' ).  Doing so will guarantee that
00126 *          eigenvalues are computed to high relative accuracy when
00127 *          possible in future releases.  The current code does not
00128 *          make any guarantees about high relative accuracy, but
00129 *          future releases will. See J. Barlow and J. Demmel,
00130 *          "Computing Accurate Eigensystems of Scaled Diagonally
00131 *          Dominant Matrices", LAPACK Working Note #7, for a discussion
00132 *          of which matrices define their eigenvalues to high relative
00133 *          accuracy.
00134 *
00135 *  M       (output) INTEGER
00136 *          The total number of eigenvalues found.  0 <= M <= N.
00137 *          If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
00138 *
00139 *  W       (output) DOUBLE PRECISION array, dimension (N)
00140 *          The first M elements contain the selected eigenvalues in
00141 *          ascending order.
00142 *
00143 *  Z       (output) DOUBLE PRECISION array, dimension (LDZ, max(1,M) )
00144 *          If JOBZ = 'V', then if INFO = 0, the first M columns of Z
00145 *          contain the orthonormal eigenvectors of the matrix A
00146 *          corresponding to the selected eigenvalues, with the i-th
00147 *          column of Z holding the eigenvector associated with W(i).
00148 *          Note: the user must ensure that at least max(1,M) columns are
00149 *          supplied in the array Z; if RANGE = 'V', the exact value of M
00150 *          is not known in advance and an upper bound must be used.
00151 *
00152 *  LDZ     (input) INTEGER
00153 *          The leading dimension of the array Z.  LDZ >= 1, and if
00154 *          JOBZ = 'V', LDZ >= max(1,N).
00155 *
00156 *  ISUPPZ  (output) INTEGER array, dimension ( 2*max(1,M) )
00157 *          The support of the eigenvectors in Z, i.e., the indices
00158 *          indicating the nonzero elements in Z. The i-th eigenvector
00159 *          is nonzero only in elements ISUPPZ( 2*i-1 ) through
00160 *          ISUPPZ( 2*i ).
00161 ********** Implemented only for RANGE = 'A' or 'I' and IU - IL = N - 1
00162 *
00163 *  WORK    (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK))
00164 *          On exit, if INFO = 0, WORK(1) returns the optimal (and
00165 *          minimal) LWORK.
00166 *
00167 *  LWORK   (input) INTEGER
00168 *          The dimension of the array WORK.  LWORK >= max(1,20*N).
00169 *
00170 *          If LWORK = -1, then a workspace query is assumed; the routine
00171 *          only calculates the optimal sizes of the WORK and IWORK
00172 *          arrays, returns these values as the first entries of the WORK
00173 *          and IWORK arrays, and no error message related to LWORK or
00174 *          LIWORK is issued by XERBLA.
00175 *
00176 *  IWORK   (workspace/output) INTEGER array, dimension (MAX(1,LIWORK))
00177 *          On exit, if INFO = 0, IWORK(1) returns the optimal (and
00178 *          minimal) LIWORK.
00179 *
00180 *  LIWORK  (input) INTEGER
00181 *          The dimension of the array IWORK.  LIWORK >= max(1,10*N).
00182 *
00183 *          If LIWORK = -1, then a workspace query is assumed; the
00184 *          routine only calculates the optimal sizes of the WORK and
00185 *          IWORK arrays, returns these values as the first entries of
00186 *          the WORK and IWORK arrays, and no error message related to
00187 *          LWORK or LIWORK is issued by XERBLA.
00188 *
00189 *  INFO    (output) INTEGER
00190 *          = 0:  successful exit
00191 *          < 0:  if INFO = -i, the i-th argument had an illegal value
00192 *          > 0:  Internal error
00193 *
00194 *  Further Details
00195 *  ===============
00196 *
00197 *  Based on contributions by
00198 *     Inderjit Dhillon, IBM Almaden, USA
00199 *     Osni Marques, LBNL/NERSC, USA
00200 *     Ken Stanley, Computer Science Division, University of
00201 *       California at Berkeley, USA
00202 *
00203 *  =====================================================================
00204 *
00205 *     .. Parameters ..
00206       DOUBLE PRECISION   ZERO, ONE, TWO
00207       PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0, TWO = 2.0D+0 )
00208 *     ..
00209 *     .. Local Scalars ..
00210       LOGICAL            ALLEIG, INDEIG, TEST, LQUERY, VALEIG, WANTZ,
00211      $                   TRYRAC
00212       CHARACTER          ORDER
00213       INTEGER            I, IEEEOK, IMAX, INDIBL, INDIFL, INDISP,
00214      $                   INDIWO, ISCALE, ITMP1, J, JJ, LIWMIN, LWMIN,
00215      $                   NSPLIT
00216       DOUBLE PRECISION   BIGNUM, EPS, RMAX, RMIN, SAFMIN, SIGMA, SMLNUM,
00217      $                   TMP1, TNRM, VLL, VUU
00218 *     ..
00219 *     .. External Functions ..
00220       LOGICAL            LSAME
00221       INTEGER            ILAENV
00222       DOUBLE PRECISION   DLAMCH, DLANST
00223       EXTERNAL           LSAME, ILAENV, DLAMCH, DLANST
00224 *     ..
00225 *     .. External Subroutines ..
00226       EXTERNAL           DCOPY, DSCAL, DSTEBZ, DSTEMR, DSTEIN, DSTERF,
00227      $                   DSWAP, XERBLA
00228 *     ..
00229 *     .. Intrinsic Functions ..
00230       INTRINSIC          MAX, MIN, SQRT
00231 *     ..
00232 *     .. Executable Statements ..
00233 *
00234 *
00235 *     Test the input parameters.
00236 *
00237       IEEEOK = ILAENV( 10, 'DSTEVR', 'N', 1, 2, 3, 4 )
00238 *
00239       WANTZ = LSAME( JOBZ, 'V' )
00240       ALLEIG = LSAME( RANGE, 'A' )
00241       VALEIG = LSAME( RANGE, 'V' )
00242       INDEIG = LSAME( RANGE, 'I' )
00243 *
00244       LQUERY = ( ( LWORK.EQ.-1 ) .OR. ( LIWORK.EQ.-1 ) )
00245       LWMIN = MAX( 1, 20*N )
00246       LIWMIN = MAX( 1, 10*N )
00247 *
00248 *
00249       INFO = 0
00250       IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
00251          INFO = -1
00252       ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN
00253          INFO = -2
00254       ELSE IF( N.LT.0 ) THEN
00255          INFO = -3
00256       ELSE
00257          IF( VALEIG ) THEN
00258             IF( N.GT.0 .AND. VU.LE.VL )
00259      $         INFO = -7
00260          ELSE IF( INDEIG ) THEN
00261             IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN
00262                INFO = -8
00263             ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN
00264                INFO = -9
00265             END IF
00266          END IF
00267       END IF
00268       IF( INFO.EQ.0 ) THEN
00269          IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
00270             INFO = -14
00271          END IF
00272       END IF
00273 *
00274       IF( INFO.EQ.0 ) THEN
00275          WORK( 1 ) = LWMIN
00276          IWORK( 1 ) = LIWMIN
00277 *
00278          IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
00279             INFO = -17
00280          ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN
00281             INFO = -19
00282          END IF
00283       END IF
00284 *
00285       IF( INFO.NE.0 ) THEN
00286          CALL XERBLA( 'DSTEVR', -INFO )
00287          RETURN
00288       ELSE IF( LQUERY ) THEN
00289          RETURN
00290       END IF
00291 *
00292 *     Quick return if possible
00293 *
00294       M = 0
00295       IF( N.EQ.0 )
00296      $   RETURN
00297 *
00298       IF( N.EQ.1 ) THEN
00299          IF( ALLEIG .OR. INDEIG ) THEN
00300             M = 1
00301             W( 1 ) = D( 1 )
00302          ELSE
00303             IF( VL.LT.D( 1 ) .AND. VU.GE.D( 1 ) ) THEN
00304                M = 1
00305                W( 1 ) = D( 1 )
00306             END IF
00307          END IF
00308          IF( WANTZ )
00309      $      Z( 1, 1 ) = ONE
00310          RETURN
00311       END IF
00312 *
00313 *     Get machine constants.
00314 *
00315       SAFMIN = DLAMCH( 'Safe minimum' )
00316       EPS = DLAMCH( 'Precision' )
00317       SMLNUM = SAFMIN / EPS
00318       BIGNUM = ONE / SMLNUM
00319       RMIN = SQRT( SMLNUM )
00320       RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) )
00321 *
00322 *
00323 *     Scale matrix to allowable range, if necessary.
00324 *
00325       ISCALE = 0
00326       VLL = VL
00327       VUU = VU
00328 *
00329       TNRM = DLANST( 'M', N, D, E )
00330       IF( TNRM.GT.ZERO .AND. TNRM.LT.RMIN ) THEN
00331          ISCALE = 1
00332          SIGMA = RMIN / TNRM
00333       ELSE IF( TNRM.GT.RMAX ) THEN
00334          ISCALE = 1
00335          SIGMA = RMAX / TNRM
00336       END IF
00337       IF( ISCALE.EQ.1 ) THEN
00338          CALL DSCAL( N, SIGMA, D, 1 )
00339          CALL DSCAL( N-1, SIGMA, E( 1 ), 1 )
00340          IF( VALEIG ) THEN
00341             VLL = VL*SIGMA
00342             VUU = VU*SIGMA
00343          END IF
00344       END IF
00345 
00346 *     Initialize indices into workspaces.  Note: These indices are used only
00347 *     if DSTERF or DSTEMR fail.
00348 
00349 *     IWORK(INDIBL:INDIBL+M-1) corresponds to IBLOCK in DSTEBZ and
00350 *     stores the block indices of each of the M<=N eigenvalues.
00351       INDIBL = 1
00352 *     IWORK(INDISP:INDISP+NSPLIT-1) corresponds to ISPLIT in DSTEBZ and
00353 *     stores the starting and finishing indices of each block.
00354       INDISP = INDIBL + N
00355 *     IWORK(INDIFL:INDIFL+N-1) stores the indices of eigenvectors
00356 *     that corresponding to eigenvectors that fail to converge in
00357 *     DSTEIN.  This information is discarded; if any fail, the driver
00358 *     returns INFO > 0.
00359       INDIFL = INDISP + N
00360 *     INDIWO is the offset of the remaining integer workspace.
00361       INDIWO = INDISP + N
00362 *
00363 *     If all eigenvalues are desired, then
00364 *     call DSTERF or DSTEMR.  If this fails for some eigenvalue, then
00365 *     try DSTEBZ.
00366 *
00367 *
00368       TEST = .FALSE.
00369       IF( INDEIG ) THEN
00370          IF( IL.EQ.1 .AND. IU.EQ.N ) THEN
00371             TEST = .TRUE.
00372          END IF
00373       END IF
00374       IF( ( ALLEIG .OR. TEST ) .AND. IEEEOK.EQ.1 ) THEN
00375          CALL DCOPY( N-1, E( 1 ), 1, WORK( 1 ), 1 )
00376          IF( .NOT.WANTZ ) THEN
00377             CALL DCOPY( N, D, 1, W, 1 )
00378             CALL DSTERF( N, W, WORK, INFO )
00379          ELSE
00380             CALL DCOPY( N, D, 1, WORK( N+1 ), 1 )
00381             IF (ABSTOL .LE. TWO*N*EPS) THEN
00382                TRYRAC = .TRUE.
00383             ELSE
00384                TRYRAC = .FALSE.
00385             END IF
00386             CALL DSTEMR( JOBZ, 'A', N, WORK( N+1 ), WORK, VL, VU, IL,
00387      $                   IU, M, W, Z, LDZ, N, ISUPPZ, TRYRAC,
00388      $                   WORK( 2*N+1 ), LWORK-2*N, IWORK, LIWORK, INFO )
00389 *
00390          END IF
00391          IF( INFO.EQ.0 ) THEN
00392             M = N
00393             GO TO 10
00394          END IF
00395          INFO = 0
00396       END IF
00397 *
00398 *     Otherwise, call DSTEBZ and, if eigenvectors are desired, DSTEIN.
00399 *
00400       IF( WANTZ ) THEN
00401          ORDER = 'B'
00402       ELSE
00403          ORDER = 'E'
00404       END IF
00405 
00406       CALL DSTEBZ( RANGE, ORDER, N, VLL, VUU, IL, IU, ABSTOL, D, E, M,
00407      $             NSPLIT, W, IWORK( INDIBL ), IWORK( INDISP ), WORK,
00408      $             IWORK( INDIWO ), INFO )
00409 *
00410       IF( WANTZ ) THEN
00411          CALL DSTEIN( N, D, E, M, W, IWORK( INDIBL ), IWORK( INDISP ),
00412      $                Z, LDZ, WORK, IWORK( INDIWO ), IWORK( INDIFL ),
00413      $                INFO )
00414       END IF
00415 *
00416 *     If matrix was scaled, then rescale eigenvalues appropriately.
00417 *
00418    10 CONTINUE
00419       IF( ISCALE.EQ.1 ) THEN
00420          IF( INFO.EQ.0 ) THEN
00421             IMAX = M
00422          ELSE
00423             IMAX = INFO - 1
00424          END IF
00425          CALL DSCAL( IMAX, ONE / SIGMA, W, 1 )
00426       END IF
00427 *
00428 *     If eigenvalues are not in order, then sort them, along with
00429 *     eigenvectors.
00430 *
00431       IF( WANTZ ) THEN
00432          DO 30 J = 1, M - 1
00433             I = 0
00434             TMP1 = W( J )
00435             DO 20 JJ = J + 1, M
00436                IF( W( JJ ).LT.TMP1 ) THEN
00437                   I = JJ
00438                   TMP1 = W( JJ )
00439                END IF
00440    20       CONTINUE
00441 *
00442             IF( I.NE.0 ) THEN
00443                ITMP1 = IWORK( I )
00444                W( I ) = W( J )
00445                IWORK( I ) = IWORK( J )
00446                W( J ) = TMP1
00447                IWORK( J ) = ITMP1
00448                CALL DSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 )
00449             END IF
00450    30    CONTINUE
00451       END IF
00452 *
00453 *      Causes problems with tests 19 & 20:
00454 *      IF (wantz .and. INDEIG ) Z( 1,1) = Z(1,1) / 1.002 + .002
00455 *
00456 *
00457       WORK( 1 ) = LWMIN
00458       IWORK( 1 ) = LIWMIN
00459       RETURN
00460 *
00461 *     End of DSTEVR
00462 *
00463       END
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