LAPACK 3.3.1
Linear Algebra PACKage

sstevr.f

Go to the documentation of this file.
00001       SUBROUTINE SSTEVR( 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       REAL               ABSTOL, VL, VU
00014 *     ..
00015 *     .. Array Arguments ..
00016       INTEGER            ISUPPZ( * ), IWORK( * )
00017       REAL               D( * ), E( * ), W( * ), WORK( * ), Z( LDZ, * )
00018 *     ..
00019 *
00020 *  Purpose
00021 *  =======
00022 *
00023 *  SSTEVR 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, SSTEVR calls SSTEMR to compute the
00029 *  eigenspectrum using Relatively Robust Representations.  SSTEMR
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 : SSTEVR calls SSTEMR when the full spectrum is requested
00055 *  on machines which conform to the ieee-754 floating point standard.
00056 *  SSTEVR calls SSTEBZ and SSTEIN on non-ieee machines and
00057 *  when partial spectrum requests are made.
00058 *
00059 *  Normal execution of SSTEMR 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, SSTEBZ and
00077 ********** SSTEIN are called
00078 *
00079 *  N       (input) INTEGER
00080 *          The order of the matrix.  N >= 0.
00081 *
00082 *  D       (input/output) REAL 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) REAL 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) REAL
00095 *  VU      (input) REAL
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) REAL
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 *          SLAMCH( '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) REAL array, dimension (N)
00140 *          The first M elements contain the selected eigenvalues in
00141 *          ascending order.
00142 *
00143 *  Z       (output) REAL 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) REAL 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 >= 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 >= 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 *     Jason Riedy, Computer Science Division, University of
00203 *       California at Berkeley, USA
00204 *
00205 *  =====================================================================
00206 *
00207 *     .. Parameters ..
00208       REAL               ZERO, ONE, TWO
00209       PARAMETER          ( ZERO = 0.0E+0, ONE = 1.0E+0, TWO = 2.0E+0 )
00210 *     ..
00211 *     .. Local Scalars ..
00212       LOGICAL            ALLEIG, INDEIG, TEST, LQUERY, VALEIG, WANTZ,
00213      $                   TRYRAC
00214       CHARACTER          ORDER
00215       INTEGER            I, IEEEOK, IMAX, INDIBL, INDIFL, INDISP,
00216      $                   INDIWO, ISCALE, J, JJ, LIWMIN, LWMIN, NSPLIT
00217       REAL               BIGNUM, EPS, RMAX, RMIN, SAFMIN, SIGMA, SMLNUM,
00218      $                   TMP1, TNRM, VLL, VUU
00219 *     ..
00220 *     .. External Functions ..
00221       LOGICAL            LSAME
00222       INTEGER            ILAENV
00223       REAL               SLAMCH, SLANST
00224       EXTERNAL           LSAME, ILAENV, SLAMCH, SLANST
00225 *     ..
00226 *     .. External Subroutines ..
00227       EXTERNAL           SCOPY, SSCAL, SSTEBZ, SSTEMR, SSTEIN, SSTERF,
00228      $                   SSWAP, XERBLA
00229 *     ..
00230 *     .. Intrinsic Functions ..
00231       INTRINSIC          MAX, MIN, SQRT
00232 *     ..
00233 *     .. Executable Statements ..
00234 *
00235 *
00236 *     Test the input parameters.
00237 *
00238       IEEEOK = ILAENV( 10, 'SSTEVR', 'N', 1, 2, 3, 4 )
00239 *
00240       WANTZ = LSAME( JOBZ, 'V' )
00241       ALLEIG = LSAME( RANGE, 'A' )
00242       VALEIG = LSAME( RANGE, 'V' )
00243       INDEIG = LSAME( RANGE, 'I' )
00244 *
00245       LQUERY = ( ( LWORK.EQ.-1 ) .OR. ( LIWORK.EQ.-1 ) )
00246       LWMIN = MAX( 1, 20*N )
00247       LIWMIN = MAX(1, 10*N )
00248 *
00249 *
00250       INFO = 0
00251       IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
00252          INFO = -1
00253       ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN
00254          INFO = -2
00255       ELSE IF( N.LT.0 ) THEN
00256          INFO = -3
00257       ELSE
00258          IF( VALEIG ) THEN
00259             IF( N.GT.0 .AND. VU.LE.VL )
00260      $         INFO = -7
00261          ELSE IF( INDEIG ) THEN
00262             IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN
00263                INFO = -8
00264             ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN
00265                INFO = -9
00266             END IF
00267          END IF
00268       END IF
00269       IF( INFO.EQ.0 ) THEN
00270          IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
00271             INFO = -14
00272          END IF
00273       END IF
00274 *
00275       IF( INFO.EQ.0 ) THEN
00276          WORK( 1 ) = LWMIN
00277          IWORK( 1 ) = LIWMIN
00278 *
00279          IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
00280             INFO = -17
00281          ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN
00282             INFO = -19
00283          END IF
00284       END IF
00285 *
00286       IF( INFO.NE.0 ) THEN
00287          CALL XERBLA( 'SSTEVR', -INFO )
00288          RETURN
00289       ELSE IF( LQUERY ) THEN
00290          RETURN
00291       END IF
00292 *
00293 *     Quick return if possible
00294 *
00295       M = 0
00296       IF( N.EQ.0 )
00297      $   RETURN
00298 *
00299       IF( N.EQ.1 ) THEN
00300          IF( ALLEIG .OR. INDEIG ) THEN
00301             M = 1
00302             W( 1 ) = D( 1 )
00303          ELSE
00304             IF( VL.LT.D( 1 ) .AND. VU.GE.D( 1 ) ) THEN
00305                M = 1
00306                W( 1 ) = D( 1 )
00307             END IF
00308          END IF
00309          IF( WANTZ )
00310      $      Z( 1, 1 ) = ONE
00311          RETURN
00312       END IF
00313 *
00314 *     Get machine constants.
00315 *
00316       SAFMIN = SLAMCH( 'Safe minimum' )
00317       EPS = SLAMCH( 'Precision' )
00318       SMLNUM = SAFMIN / EPS
00319       BIGNUM = ONE / SMLNUM
00320       RMIN = SQRT( SMLNUM )
00321       RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) )
00322 *
00323 *
00324 *     Scale matrix to allowable range, if necessary.
00325 *
00326       ISCALE = 0
00327       VLL = VL
00328       VUU = VU
00329 *
00330       TNRM = SLANST( 'M', N, D, E )
00331       IF( TNRM.GT.ZERO .AND. TNRM.LT.RMIN ) THEN
00332          ISCALE = 1
00333          SIGMA = RMIN / TNRM
00334       ELSE IF( TNRM.GT.RMAX ) THEN
00335          ISCALE = 1
00336          SIGMA = RMAX / TNRM
00337       END IF
00338       IF( ISCALE.EQ.1 ) THEN
00339          CALL SSCAL( N, SIGMA, D, 1 )
00340          CALL SSCAL( N-1, SIGMA, E( 1 ), 1 )
00341          IF( VALEIG ) THEN
00342             VLL = VL*SIGMA
00343             VUU = VU*SIGMA
00344          END IF
00345       END IF
00346 
00347 *     Initialize indices into workspaces.  Note: These indices are used only
00348 *     if SSTERF or SSTEMR fail.
00349 
00350 *     IWORK(INDIBL:INDIBL+M-1) corresponds to IBLOCK in SSTEBZ and
00351 *     stores the block indices of each of the M<=N eigenvalues.
00352       INDIBL = 1
00353 *     IWORK(INDISP:INDISP+NSPLIT-1) corresponds to ISPLIT in SSTEBZ and
00354 *     stores the starting and finishing indices of each block.
00355       INDISP = INDIBL + N
00356 *     IWORK(INDIFL:INDIFL+N-1) stores the indices of eigenvectors
00357 *     that corresponding to eigenvectors that fail to converge in
00358 *     SSTEIN.  This information is discarded; if any fail, the driver
00359 *     returns INFO > 0.
00360       INDIFL = INDISP + N
00361 *     INDIWO is the offset of the remaining integer workspace.
00362       INDIWO = INDISP + N
00363 *
00364 *     If all eigenvalues are desired, then
00365 *     call SSTERF or SSTEMR.  If this fails for some eigenvalue, then
00366 *     try SSTEBZ.
00367 *
00368 *
00369       TEST = .FALSE.
00370       IF( INDEIG ) THEN
00371          IF( IL.EQ.1 .AND. IU.EQ.N ) THEN
00372             TEST = .TRUE.
00373          END IF
00374       END IF
00375       IF( ( ALLEIG .OR. TEST ) .AND. IEEEOK.EQ.1 ) THEN
00376          CALL SCOPY( N-1, E( 1 ), 1, WORK( 1 ), 1 )
00377          IF( .NOT.WANTZ ) THEN
00378             CALL SCOPY( N, D, 1, W, 1 )
00379             CALL SSTERF( N, W, WORK, INFO )
00380          ELSE
00381             CALL SCOPY( N, D, 1, WORK( N+1 ), 1 )
00382             IF (ABSTOL .LE. TWO*N*EPS) THEN
00383                TRYRAC = .TRUE.
00384             ELSE
00385                TRYRAC = .FALSE.
00386             END IF
00387             CALL SSTEMR( JOBZ, 'A', N, WORK( N+1 ), WORK, VL, VU, IL,
00388      $                   IU, M, W, Z, LDZ, N, ISUPPZ, TRYRAC,
00389      $                   WORK( 2*N+1 ), LWORK-2*N, IWORK, LIWORK, INFO )
00390 *
00391          END IF
00392          IF( INFO.EQ.0 ) THEN
00393             M = N
00394             GO TO 10
00395          END IF
00396          INFO = 0
00397       END IF
00398 *
00399 *     Otherwise, call SSTEBZ and, if eigenvectors are desired, SSTEIN.
00400 *
00401       IF( WANTZ ) THEN
00402          ORDER = 'B'
00403       ELSE
00404          ORDER = 'E'
00405       END IF
00406 
00407       CALL SSTEBZ( RANGE, ORDER, N, VLL, VUU, IL, IU, ABSTOL, D, E, M,
00408      $             NSPLIT, W, IWORK( INDIBL ), IWORK( INDISP ), WORK,
00409      $             IWORK( INDIWO ), INFO )
00410 *
00411       IF( WANTZ ) THEN
00412          CALL SSTEIN( N, D, E, M, W, IWORK( INDIBL ), IWORK( INDISP ),
00413      $                Z, LDZ, WORK, IWORK( INDIWO ), IWORK( INDIFL ),
00414      $                INFO )
00415       END IF
00416 *
00417 *     If matrix was scaled, then rescale eigenvalues appropriately.
00418 *
00419    10 CONTINUE
00420       IF( ISCALE.EQ.1 ) THEN
00421          IF( INFO.EQ.0 ) THEN
00422             IMAX = M
00423          ELSE
00424             IMAX = INFO - 1
00425          END IF
00426          CALL SSCAL( IMAX, ONE / SIGMA, W, 1 )
00427       END IF
00428 *
00429 *     If eigenvalues are not in order, then sort them, along with
00430 *     eigenvectors.
00431 *
00432       IF( WANTZ ) THEN
00433          DO 30 J = 1, M - 1
00434             I = 0
00435             TMP1 = W( J )
00436             DO 20 JJ = J + 1, M
00437                IF( W( JJ ).LT.TMP1 ) THEN
00438                   I = JJ
00439                   TMP1 = W( JJ )
00440                END IF
00441    20       CONTINUE
00442 *
00443             IF( I.NE.0 ) THEN
00444                W( I ) = W( J )
00445                W( J ) = TMP1
00446                CALL SSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 )
00447             END IF
00448    30    CONTINUE
00449       END IF
00450 *
00451 *      Causes problems with tests 19 & 20:
00452 *      IF (wantz .and. INDEIG ) Z( 1,1) = Z(1,1) / 1.002 + .002
00453 *
00454 *
00455       WORK( 1 ) = LWMIN
00456       IWORK( 1 ) = LIWMIN
00457       RETURN
00458 *
00459 *     End of SSTEVR
00460 *
00461       END
 All Files Functions