LAPACK: SRC/cheevx.f Source File

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00001       SUBROUTINE CHEEVX( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU,
00002      $                   ABSTOL, M, W, Z, LDZ, WORK, LWORK, RWORK,
00003      $                   IWORK, IFAIL, 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, UPLO
00012       INTEGER            IL, INFO, IU, LDA, LDZ, LWORK, M, N
00013       REAL               ABSTOL, VL, VU
00014 *     ..
00015 *     .. Array Arguments ..
00016       INTEGER            IFAIL( * ), IWORK( * )
00017       REAL               RWORK( * ), W( * )
00018       COMPLEX            A( LDA, * ), WORK( * ), Z( LDZ, * )
00019 *     ..
00020 *
00021 *  Purpose
00022 *  =======
00023 *
00024 *  CHEEVX computes selected eigenvalues and, optionally, eigenvectors
00025 *  of a complex Hermitian matrix A.  Eigenvalues and eigenvectors can
00026 *  be selected by specifying either a range of values or a range of
00027 *  indices for the desired eigenvalues.
00028 *
00029 *  Arguments
00030 *  =========
00031 *
00032 *  JOBZ    (input) CHARACTER*1
00033 *          = 'N':  Compute eigenvalues only;
00034 *          = 'V':  Compute eigenvalues and eigenvectors.
00035 *
00036 *  RANGE   (input) CHARACTER*1
00037 *          = 'A': all eigenvalues will be found.
00038 *          = 'V': all eigenvalues in the half-open interval (VL,VU]
00039 *                 will be found.
00040 *          = 'I': the IL-th through IU-th eigenvalues will be found.
00041 *
00042 *  UPLO    (input) CHARACTER*1
00043 *          = 'U':  Upper triangle of A is stored;
00044 *          = 'L':  Lower triangle of A is stored.
00045 *
00046 *  N       (input) INTEGER
00047 *          The order of the matrix A.  N >= 0.
00048 *
00049 *  A       (input/output) COMPLEX array, dimension (LDA, N)
00050 *          On entry, the Hermitian matrix A.  If UPLO = 'U', the
00051 *          leading N-by-N upper triangular part of A contains the
00052 *          upper triangular part of the matrix A.  If UPLO = 'L',
00053 *          the leading N-by-N lower triangular part of A contains
00054 *          the lower triangular part of the matrix A.
00055 *          On exit, the lower triangle (if UPLO='L') or the upper
00056 *          triangle (if UPLO='U') of A, including the diagonal, is
00057 *          destroyed.
00058 *
00059 *  LDA     (input) INTEGER
00060 *          The leading dimension of the array A.  LDA >= max(1,N).
00061 *
00062 *  VL      (input) REAL
00063 *  VU      (input) REAL
00064 *          If RANGE='V', the lower and upper bounds of the interval to
00065 *          be searched for eigenvalues. VL < VU.
00066 *          Not referenced if RANGE = 'A' or 'I'.
00067 *
00068 *  IL      (input) INTEGER
00069 *  IU      (input) INTEGER
00070 *          If RANGE='I', the indices (in ascending order) of the
00071 *          smallest and largest eigenvalues to be returned.
00072 *          1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
00073 *          Not referenced if RANGE = 'A' or 'V'.
00074 *
00075 *  ABSTOL  (input) REAL
00076 *          The absolute error tolerance for the eigenvalues.
00077 *          An approximate eigenvalue is accepted as converged
00078 *          when it is determined to lie in an interval [a,b]
00079 *          of width less than or equal to
00080 *
00081 *                  ABSTOL + EPS *   max( |a|,|b| ) ,
00082 *
00083 *          where EPS is the machine precision.  If ABSTOL is less than
00084 *          or equal to zero, then  EPS*|T|  will be used in its place,
00085 *          where |T| is the 1-norm of the tridiagonal matrix obtained
00086 *          by reducing A to tridiagonal form.
00087 *
00088 *          Eigenvalues will be computed most accurately when ABSTOL is
00089 *          set to twice the underflow threshold 2*SLAMCH('S'), not zero.
00090 *          If this routine returns with INFO>0, indicating that some
00091 *          eigenvectors did not converge, try setting ABSTOL to
00092 *          2*SLAMCH('S').
00093 *
00094 *          See "Computing Small Singular Values of Bidiagonal Matrices
00095 *          with Guaranteed High Relative Accuracy," by Demmel and
00096 *          Kahan, LAPACK Working Note #3.
00097 *
00098 *  M       (output) INTEGER
00099 *          The total number of eigenvalues found.  0 <= M <= N.
00100 *          If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
00101 *
00102 *  W       (output) REAL array, dimension (N)
00103 *          On normal exit, the first M elements contain the selected
00104 *          eigenvalues in ascending order.
00105 *
00106 *  Z       (output) COMPLEX array, dimension (LDZ, max(1,M))
00107 *          If JOBZ = 'V', then if INFO = 0, the first M columns of Z
00108 *          contain the orthonormal eigenvectors of the matrix A
00109 *          corresponding to the selected eigenvalues, with the i-th
00110 *          column of Z holding the eigenvector associated with W(i).
00111 *          If an eigenvector fails to converge, then that column of Z
00112 *          contains the latest approximation to the eigenvector, and the
00113 *          index of the eigenvector is returned in IFAIL.
00114 *          If JOBZ = 'N', then Z is not referenced.
00115 *          Note: the user must ensure that at least max(1,M) columns are
00116 *          supplied in the array Z; if RANGE = 'V', the exact value of M
00117 *          is not known in advance and an upper bound must be used.
00118 *
00119 *  LDZ     (input) INTEGER
00120 *          The leading dimension of the array Z.  LDZ >= 1, and if
00121 *          JOBZ = 'V', LDZ >= max(1,N).
00122 *
00123 *  WORK    (workspace/output) COMPLEX array, dimension (MAX(1,LWORK))
00124 *          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
00125 *
00126 *  LWORK   (input) INTEGER
00127 *          The length of the array WORK.  LWORK >= 1, when N <= 1;
00128 *          otherwise 2*N.
00129 *          For optimal efficiency, LWORK >= (NB+1)*N,
00130 *          where NB is the max of the blocksize for CHETRD and for
00131 *          CUNMTR as returned by ILAENV.
00132 *
00133 *          If LWORK = -1, then a workspace query is assumed; the routine
00134 *          only calculates the optimal size of the WORK array, returns
00135 *          this value as the first entry of the WORK array, and no error
00136 *          message related to LWORK is issued by XERBLA.
00137 *
00138 *  RWORK   (workspace) REAL array, dimension (7*N)
00139 *
00140 *  IWORK   (workspace) INTEGER array, dimension (5*N)
00141 *
00142 *  IFAIL   (output) INTEGER array, dimension (N)
00143 *          If JOBZ = 'V', then if INFO = 0, the first M elements of
00144 *          IFAIL are zero.  If INFO > 0, then IFAIL contains the
00145 *          indices of the eigenvectors that failed to converge.
00146 *          If JOBZ = 'N', then IFAIL is not referenced.
00147 *
00148 *  INFO    (output) INTEGER
00149 *          = 0:  successful exit
00150 *          < 0:  if INFO = -i, the i-th argument had an illegal value
00151 *          > 0:  if INFO = i, then i eigenvectors failed to converge.
00152 *                Their indices are stored in array IFAIL.
00153 *
00154 *  =====================================================================
00155 *
00156 *     .. Parameters ..
00157       REAL               ZERO, ONE
00158       PARAMETER          ( ZERO = 0.0E+0, ONE = 1.0E+0 )
00159       COMPLEX            CONE
00160       PARAMETER          ( CONE = ( 1.0E+0, 0.0E+0 ) )
00161 *     ..
00162 *     .. Local Scalars ..
00163       LOGICAL            ALLEIG, INDEIG, LOWER, LQUERY, TEST, VALEIG,
00164      $                   WANTZ
00165       CHARACTER          ORDER
00166       INTEGER            I, IINFO, IMAX, INDD, INDE, INDEE, INDIBL,
00167      $                   INDISP, INDIWK, INDRWK, INDTAU, INDWRK, ISCALE,
00168      $                   ITMP1, J, JJ, LLWORK, LWKMIN, LWKOPT, NB,
00169      $                   NSPLIT
00170       REAL               ABSTLL, ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN,
00171      $                   SIGMA, SMLNUM, TMP1, VLL, VUU
00172 *     ..
00173 *     .. External Functions ..
00174       LOGICAL            LSAME
00175       INTEGER            ILAENV
00176       REAL               CLANHE, SLAMCH
00177       EXTERNAL           LSAME, ILAENV, CLANHE, SLAMCH
00178 *     ..
00179 *     .. External Subroutines ..
00180       EXTERNAL           CHETRD, CLACPY, CSSCAL, CSTEIN, CSTEQR, CSWAP,
00181      $                   CUNGTR, CUNMTR, SCOPY, SSCAL, SSTEBZ, SSTERF,
00182      $                   XERBLA
00183 *     ..
00184 *     .. Intrinsic Functions ..
00185       INTRINSIC          MAX, MIN, REAL, SQRT
00186 *     ..
00187 *     .. Executable Statements ..
00188 *
00189 *     Test the input parameters.
00190 *
00191       LOWER = LSAME( UPLO, 'L' )
00192       WANTZ = LSAME( JOBZ, 'V' )
00193       ALLEIG = LSAME( RANGE, 'A' )
00194       VALEIG = LSAME( RANGE, 'V' )
00195       INDEIG = LSAME( RANGE, 'I' )
00196       LQUERY = ( LWORK.EQ.-1 )
00197 *
00198       INFO = 0
00199       IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
00200          INFO = -1
00201       ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN
00202          INFO = -2
00203       ELSE IF( .NOT.( LOWER .OR. LSAME( UPLO, 'U' ) ) ) THEN
00204          INFO = -3
00205       ELSE IF( N.LT.0 ) THEN
00206          INFO = -4
00207       ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
00208          INFO = -6
00209       ELSE
00210          IF( VALEIG ) THEN
00211             IF( N.GT.0 .AND. VU.LE.VL )
00212      $         INFO = -8
00213          ELSE IF( INDEIG ) THEN
00214             IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN
00215                INFO = -9
00216             ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN
00217                INFO = -10
00218             END IF
00219          END IF
00220       END IF
00221       IF( INFO.EQ.0 ) THEN
00222          IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
00223             INFO = -15
00224          END IF
00225       END IF
00226 *
00227       IF( INFO.EQ.0 ) THEN
00228          IF( N.LE.1 ) THEN
00229             LWKMIN = 1
00230             WORK( 1 ) = LWKMIN
00231          ELSE
00232             LWKMIN = 2*N
00233             NB = ILAENV( 1, 'CHETRD', UPLO, N, -1, -1, -1 )
00234             NB = MAX( NB, ILAENV( 1, 'CUNMTR', UPLO, N, -1, -1, -1 ) )
00235             LWKOPT = MAX( 1, ( NB + 1 )*N )
00236             WORK( 1 ) = LWKOPT
00237          END IF
00238 *
00239          IF( LWORK.LT.LWKMIN .AND. .NOT.LQUERY )
00240      $      INFO = -17
00241       END IF
00242 *
00243       IF( INFO.NE.0 ) THEN
00244          CALL XERBLA( 'CHEEVX', -INFO )
00245          RETURN
00246       ELSE IF( LQUERY ) THEN
00247          RETURN
00248       END IF
00249 *
00250 *     Quick return if possible
00251 *
00252       M = 0
00253       IF( N.EQ.0 ) THEN
00254          RETURN
00255       END IF
00256 *
00257       IF( N.EQ.1 ) THEN
00258          IF( ALLEIG .OR. INDEIG ) THEN
00259             M = 1
00260             W( 1 ) = A( 1, 1 )
00261          ELSE IF( VALEIG ) THEN
00262             IF( VL.LT.REAL( A( 1, 1 ) ) .AND. VU.GE.REAL( A( 1, 1 ) ) )
00263      $           THEN
00264                M = 1
00265                W( 1 ) = A( 1, 1 )
00266             END IF
00267          END IF
00268          IF( WANTZ )
00269      $      Z( 1, 1 ) = CONE
00270          RETURN
00271       END IF
00272 *
00273 *     Get machine constants.
00274 *
00275       SAFMIN = SLAMCH( 'Safe minimum' )
00276       EPS = SLAMCH( 'Precision' )
00277       SMLNUM = SAFMIN / EPS
00278       BIGNUM = ONE / SMLNUM
00279       RMIN = SQRT( SMLNUM )
00280       RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) )
00281 *
00282 *     Scale matrix to allowable range, if necessary.
00283 *
00284       ISCALE = 0
00285       ABSTLL = ABSTOL
00286       IF( VALEIG ) THEN
00287          VLL = VL
00288          VUU = VU
00289       END IF
00290       ANRM = CLANHE( 'M', UPLO, N, A, LDA, RWORK )
00291       IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN
00292          ISCALE = 1
00293          SIGMA = RMIN / ANRM
00294       ELSE IF( ANRM.GT.RMAX ) THEN
00295          ISCALE = 1
00296          SIGMA = RMAX / ANRM
00297       END IF
00298       IF( ISCALE.EQ.1 ) THEN
00299          IF( LOWER ) THEN
00300             DO 10 J = 1, N
00301                CALL CSSCAL( N-J+1, SIGMA, A( J, J ), 1 )
00302    10       CONTINUE
00303          ELSE
00304             DO 20 J = 1, N
00305                CALL CSSCAL( J, SIGMA, A( 1, J ), 1 )
00306    20       CONTINUE
00307          END IF
00308          IF( ABSTOL.GT.0 )
00309      $      ABSTLL = ABSTOL*SIGMA
00310          IF( VALEIG ) THEN
00311             VLL = VL*SIGMA
00312             VUU = VU*SIGMA
00313          END IF
00314       END IF
00315 *
00316 *     Call CHETRD to reduce Hermitian matrix to tridiagonal form.
00317 *
00318       INDD = 1
00319       INDE = INDD + N
00320       INDRWK = INDE + N
00321       INDTAU = 1
00322       INDWRK = INDTAU + N
00323       LLWORK = LWORK - INDWRK + 1
00324       CALL CHETRD( UPLO, N, A, LDA, RWORK( INDD ), RWORK( INDE ),
00325      $             WORK( INDTAU ), WORK( INDWRK ), LLWORK, IINFO )
00326 *
00327 *     If all eigenvalues are desired and ABSTOL is less than or equal to
00328 *     zero, then call SSTERF or CUNGTR and CSTEQR.  If this fails for
00329 *     some eigenvalue, then try SSTEBZ.
00330 *
00331       TEST = .FALSE.
00332       IF( INDEIG ) THEN
00333          IF( IL.EQ.1 .AND. IU.EQ.N ) THEN
00334             TEST = .TRUE.
00335          END IF
00336       END IF
00337       IF( ( ALLEIG .OR. TEST ) .AND. ( ABSTOL.LE.ZERO ) ) THEN
00338          CALL SCOPY( N, RWORK( INDD ), 1, W, 1 )
00339          INDEE = INDRWK + 2*N
00340          IF( .NOT.WANTZ ) THEN
00341             CALL SCOPY( N-1, RWORK( INDE ), 1, RWORK( INDEE ), 1 )
00342             CALL SSTERF( N, W, RWORK( INDEE ), INFO )
00343          ELSE
00344             CALL CLACPY( 'A', N, N, A, LDA, Z, LDZ )
00345             CALL CUNGTR( UPLO, N, Z, LDZ, WORK( INDTAU ),
00346      $                   WORK( INDWRK ), LLWORK, IINFO )
00347             CALL SCOPY( N-1, RWORK( INDE ), 1, RWORK( INDEE ), 1 )
00348             CALL CSTEQR( JOBZ, N, W, RWORK( INDEE ), Z, LDZ,
00349      $                   RWORK( INDRWK ), INFO )
00350             IF( INFO.EQ.0 ) THEN
00351                DO 30 I = 1, N
00352                   IFAIL( I ) = 0
00353    30          CONTINUE
00354             END IF
00355          END IF
00356          IF( INFO.EQ.0 ) THEN
00357             M = N
00358             GO TO 40
00359          END IF
00360          INFO = 0
00361       END IF
00362 *
00363 *     Otherwise, call SSTEBZ and, if eigenvectors are desired, CSTEIN.
00364 *
00365       IF( WANTZ ) THEN
00366          ORDER = 'B'
00367       ELSE
00368          ORDER = 'E'
00369       END IF
00370       INDIBL = 1
00371       INDISP = INDIBL + N
00372       INDIWK = INDISP + N
00373       CALL SSTEBZ( RANGE, ORDER, N, VLL, VUU, IL, IU, ABSTLL,
00374      $             RWORK( INDD ), RWORK( INDE ), M, NSPLIT, W,
00375      $             IWORK( INDIBL ), IWORK( INDISP ), RWORK( INDRWK ),
00376      $             IWORK( INDIWK ), INFO )
00377 *
00378       IF( WANTZ ) THEN
00379          CALL CSTEIN( N, RWORK( INDD ), RWORK( INDE ), M, W,
00380      $                IWORK( INDIBL ), IWORK( INDISP ), Z, LDZ,
00381      $                RWORK( INDRWK ), IWORK( INDIWK ), IFAIL, INFO )
00382 *
00383 *        Apply unitary matrix used in reduction to tridiagonal
00384 *        form to eigenvectors returned by CSTEIN.
00385 *
00386          CALL CUNMTR( 'L', UPLO, 'N', N, M, A, LDA, WORK( INDTAU ), Z,
00387      $                LDZ, WORK( INDWRK ), LLWORK, IINFO )
00388       END IF
00389 *
00390 *     If matrix was scaled, then rescale eigenvalues appropriately.
00391 *
00392    40 CONTINUE
00393       IF( ISCALE.EQ.1 ) THEN
00394          IF( INFO.EQ.0 ) THEN
00395             IMAX = M
00396          ELSE
00397             IMAX = INFO - 1
00398          END IF
00399          CALL SSCAL( IMAX, ONE / SIGMA, W, 1 )
00400       END IF
00401 *
00402 *     If eigenvalues are not in order, then sort them, along with
00403 *     eigenvectors.
00404 *
00405       IF( WANTZ ) THEN
00406          DO 60 J = 1, M - 1
00407             I = 0
00408             TMP1 = W( J )
00409             DO 50 JJ = J + 1, M
00410                IF( W( JJ ).LT.TMP1 ) THEN
00411                   I = JJ
00412                   TMP1 = W( JJ )
00413                END IF
00414    50       CONTINUE
00415 *
00416             IF( I.NE.0 ) THEN
00417                ITMP1 = IWORK( INDIBL+I-1 )
00418                W( I ) = W( J )
00419                IWORK( INDIBL+I-1 ) = IWORK( INDIBL+J-1 )
00420                W( J ) = TMP1
00421                IWORK( INDIBL+J-1 ) = ITMP1
00422                CALL CSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 )
00423                IF( INFO.NE.0 ) THEN
00424                   ITMP1 = IFAIL( I )
00425                   IFAIL( I ) = IFAIL( J )
00426                   IFAIL( J ) = ITMP1
00427                END IF
00428             END IF
00429    60    CONTINUE
00430       END IF
00431 *
00432 *     Set WORK(1) to optimal complex workspace size.
00433 *
00434       WORK( 1 ) = LWKOPT
00435 *
00436       RETURN
00437 *
00438 *     End of CHEEVX
00439 *
00440       END