LAPACK: SRC/zhpevx.f Source File

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00001       SUBROUTINE ZHPEVX( JOBZ, RANGE, UPLO, N, AP, VL, VU, IL, IU,
00002      $                   ABSTOL, M, W, Z, LDZ, WORK, RWORK, IWORK,
00003      $                   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, LDZ, M, N
00013       DOUBLE PRECISION   ABSTOL, VL, VU
00014 *     ..
00015 *     .. Array Arguments ..
00016       INTEGER            IFAIL( * ), IWORK( * )
00017       DOUBLE PRECISION   RWORK( * ), W( * )
00018       COMPLEX*16         AP( * ), WORK( * ), Z( LDZ, * )
00019 *     ..
00020 *
00021 *  Purpose
00022 *  =======
00023 *
00024 *  ZHPEVX computes selected eigenvalues and, optionally, eigenvectors
00025 *  of a complex Hermitian matrix A in packed storage.
00026 *  Eigenvalues/vectors can be selected by specifying either a range of
00027 *  values or a range of 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 *  AP      (input/output) COMPLEX*16 array, dimension (N*(N+1)/2)
00050 *          On entry, the upper or lower triangle of the Hermitian matrix
00051 *          A, packed columnwise in a linear array.  The j-th column of A
00052 *          is stored in the array AP as follows:
00053 *          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
00054 *          if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
00055 *
00056 *          On exit, AP is overwritten by values generated during the
00057 *          reduction to tridiagonal form.  If UPLO = 'U', the diagonal
00058 *          and first superdiagonal of the tridiagonal matrix T overwrite
00059 *          the corresponding elements of A, and if UPLO = 'L', the
00060 *          diagonal and first subdiagonal of T overwrite the
00061 *          corresponding elements of A.
00062 *
00063 *  VL      (input) DOUBLE PRECISION
00064 *  VU      (input) DOUBLE PRECISION
00065 *          If RANGE='V', the lower and upper bounds of the interval to
00066 *          be searched for eigenvalues. VL < VU.
00067 *          Not referenced if RANGE = 'A' or 'I'.
00068 *
00069 *  IL      (input) INTEGER
00070 *  IU      (input) INTEGER
00071 *          If RANGE='I', the indices (in ascending order) of the
00072 *          smallest and largest eigenvalues to be returned.
00073 *          1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
00074 *          Not referenced if RANGE = 'A' or 'V'.
00075 *
00076 *  ABSTOL  (input) DOUBLE PRECISION
00077 *          The absolute error tolerance for the eigenvalues.
00078 *          An approximate eigenvalue is accepted as converged
00079 *          when it is determined to lie in an interval [a,b]
00080 *          of width less than or equal to
00081 *
00082 *                  ABSTOL + EPS *   max( |a|,|b| ) ,
00083 *
00084 *          where EPS is the machine precision.  If ABSTOL is less than
00085 *          or equal to zero, then  EPS*|T|  will be used in its place,
00086 *          where |T| is the 1-norm of the tridiagonal matrix obtained
00087 *          by reducing AP to tridiagonal form.
00088 *
00089 *          Eigenvalues will be computed most accurately when ABSTOL is
00090 *          set to twice the underflow threshold 2*DLAMCH('S'), not zero.
00091 *          If this routine returns with INFO>0, indicating that some
00092 *          eigenvectors did not converge, try setting ABSTOL to
00093 *          2*DLAMCH('S').
00094 *
00095 *          See "Computing Small Singular Values of Bidiagonal Matrices
00096 *          with Guaranteed High Relative Accuracy," by Demmel and
00097 *          Kahan, LAPACK Working Note #3.
00098 *
00099 *  M       (output) INTEGER
00100 *          The total number of eigenvalues found.  0 <= M <= N.
00101 *          If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
00102 *
00103 *  W       (output) DOUBLE PRECISION array, dimension (N)
00104 *          If INFO = 0, the selected eigenvalues in ascending order.
00105 *
00106 *  Z       (output) COMPLEX*16 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
00113 *          the 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) COMPLEX*16 array, dimension (2*N)
00124 *
00125 *  RWORK   (workspace) DOUBLE PRECISION array, dimension (7*N)
00126 *
00127 *  IWORK   (workspace) INTEGER array, dimension (5*N)
00128 *
00129 *  IFAIL   (output) INTEGER array, dimension (N)
00130 *          If JOBZ = 'V', then if INFO = 0, the first M elements of
00131 *          IFAIL are zero.  If INFO > 0, then IFAIL contains the
00132 *          indices of the eigenvectors that failed to converge.
00133 *          If JOBZ = 'N', then IFAIL is not referenced.
00134 *
00135 *  INFO    (output) INTEGER
00136 *          = 0:  successful exit
00137 *          < 0:  if INFO = -i, the i-th argument had an illegal value
00138 *          > 0:  if INFO = i, then i eigenvectors failed to converge.
00139 *                Their indices are stored in array IFAIL.
00140 *
00141 *  =====================================================================
00142 *
00143 *     .. Parameters ..
00144       DOUBLE PRECISION   ZERO, ONE
00145       PARAMETER          ( ZERO = 0.0D0, ONE = 1.0D0 )
00146       COMPLEX*16         CONE
00147       PARAMETER          ( CONE = ( 1.0D0, 0.0D0 ) )
00148 *     ..
00149 *     .. Local Scalars ..
00150       LOGICAL            ALLEIG, INDEIG, TEST, VALEIG, WANTZ
00151       CHARACTER          ORDER
00152       INTEGER            I, IINFO, IMAX, INDD, INDE, INDEE, INDIBL,
00153      $                   INDISP, INDIWK, INDRWK, INDTAU, INDWRK, ISCALE,
00154      $                   ITMP1, J, JJ, NSPLIT
00155       DOUBLE PRECISION   ABSTLL, ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN,
00156      $                   SIGMA, SMLNUM, TMP1, VLL, VUU
00157 *     ..
00158 *     .. External Functions ..
00159       LOGICAL            LSAME
00160       DOUBLE PRECISION   DLAMCH, ZLANHP
00161       EXTERNAL           LSAME, DLAMCH, ZLANHP
00162 *     ..
00163 *     .. External Subroutines ..
00164       EXTERNAL           DCOPY, DSCAL, DSTEBZ, DSTERF, XERBLA, ZDSCAL,
00165      $                   ZHPTRD, ZSTEIN, ZSTEQR, ZSWAP, ZUPGTR, ZUPMTR
00166 *     ..
00167 *     .. Intrinsic Functions ..
00168       INTRINSIC          DBLE, MAX, MIN, SQRT
00169 *     ..
00170 *     .. Executable Statements ..
00171 *
00172 *     Test the input parameters.
00173 *
00174       WANTZ = LSAME( JOBZ, 'V' )
00175       ALLEIG = LSAME( RANGE, 'A' )
00176       VALEIG = LSAME( RANGE, 'V' )
00177       INDEIG = LSAME( RANGE, 'I' )
00178 *
00179       INFO = 0
00180       IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
00181          INFO = -1
00182       ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN
00183          INFO = -2
00184       ELSE IF( .NOT.( LSAME( UPLO, 'L' ) .OR. LSAME( UPLO, 'U' ) ) )
00185      $          THEN
00186          INFO = -3
00187       ELSE IF( N.LT.0 ) THEN
00188          INFO = -4
00189       ELSE
00190          IF( VALEIG ) THEN
00191             IF( N.GT.0 .AND. VU.LE.VL )
00192      $         INFO = -7
00193          ELSE IF( INDEIG ) THEN
00194             IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN
00195                INFO = -8
00196             ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN
00197                INFO = -9
00198             END IF
00199          END IF
00200       END IF
00201       IF( INFO.EQ.0 ) THEN
00202          IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) )
00203      $      INFO = -14
00204       END IF
00205 *
00206       IF( INFO.NE.0 ) THEN
00207          CALL XERBLA( 'ZHPEVX', -INFO )
00208          RETURN
00209       END IF
00210 *
00211 *     Quick return if possible
00212 *
00213       M = 0
00214       IF( N.EQ.0 )
00215      $   RETURN
00216 *
00217       IF( N.EQ.1 ) THEN
00218          IF( ALLEIG .OR. INDEIG ) THEN
00219             M = 1
00220             W( 1 ) = AP( 1 )
00221          ELSE
00222             IF( VL.LT.DBLE( AP( 1 ) ) .AND. VU.GE.DBLE( AP( 1 ) ) ) THEN
00223                M = 1
00224                W( 1 ) = AP( 1 )
00225             END IF
00226          END IF
00227          IF( WANTZ )
00228      $      Z( 1, 1 ) = CONE
00229          RETURN
00230       END IF
00231 *
00232 *     Get machine constants.
00233 *
00234       SAFMIN = DLAMCH( 'Safe minimum' )
00235       EPS = DLAMCH( 'Precision' )
00236       SMLNUM = SAFMIN / EPS
00237       BIGNUM = ONE / SMLNUM
00238       RMIN = SQRT( SMLNUM )
00239       RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) )
00240 *
00241 *     Scale matrix to allowable range, if necessary.
00242 *
00243       ISCALE = 0
00244       ABSTLL = ABSTOL
00245       IF( VALEIG ) THEN
00246          VLL = VL
00247          VUU = VU
00248       ELSE
00249          VLL = ZERO
00250          VUU = ZERO
00251       END IF
00252       ANRM = ZLANHP( 'M', UPLO, N, AP, RWORK )
00253       IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN
00254          ISCALE = 1
00255          SIGMA = RMIN / ANRM
00256       ELSE IF( ANRM.GT.RMAX ) THEN
00257          ISCALE = 1
00258          SIGMA = RMAX / ANRM
00259       END IF
00260       IF( ISCALE.EQ.1 ) THEN
00261          CALL ZDSCAL( ( N*( N+1 ) ) / 2, SIGMA, AP, 1 )
00262          IF( ABSTOL.GT.0 )
00263      $      ABSTLL = ABSTOL*SIGMA
00264          IF( VALEIG ) THEN
00265             VLL = VL*SIGMA
00266             VUU = VU*SIGMA
00267          END IF
00268       END IF
00269 *
00270 *     Call ZHPTRD to reduce Hermitian packed matrix to tridiagonal form.
00271 *
00272       INDD = 1
00273       INDE = INDD + N
00274       INDRWK = INDE + N
00275       INDTAU = 1
00276       INDWRK = INDTAU + N
00277       CALL ZHPTRD( UPLO, N, AP, RWORK( INDD ), RWORK( INDE ),
00278      $             WORK( INDTAU ), IINFO )
00279 *
00280 *     If all eigenvalues are desired and ABSTOL is less than or equal
00281 *     to zero, then call DSTERF or ZUPGTR and ZSTEQR.  If this fails
00282 *     for some eigenvalue, then try DSTEBZ.
00283 *
00284       TEST = .FALSE.
00285       IF (INDEIG) THEN
00286          IF (IL.EQ.1 .AND. IU.EQ.N) THEN
00287             TEST = .TRUE.
00288          END IF
00289       END IF
00290       IF ((ALLEIG .OR. TEST) .AND. (ABSTOL.LE.ZERO)) THEN
00291          CALL DCOPY( N, RWORK( INDD ), 1, W, 1 )
00292          INDEE = INDRWK + 2*N
00293          IF( .NOT.WANTZ ) THEN
00294             CALL DCOPY( N-1, RWORK( INDE ), 1, RWORK( INDEE ), 1 )
00295             CALL DSTERF( N, W, RWORK( INDEE ), INFO )
00296          ELSE
00297             CALL ZUPGTR( UPLO, N, AP, WORK( INDTAU ), Z, LDZ,
00298      $                   WORK( INDWRK ), IINFO )
00299             CALL DCOPY( N-1, RWORK( INDE ), 1, RWORK( INDEE ), 1 )
00300             CALL ZSTEQR( JOBZ, N, W, RWORK( INDEE ), Z, LDZ,
00301      $                   RWORK( INDRWK ), INFO )
00302             IF( INFO.EQ.0 ) THEN
00303                DO 10 I = 1, N
00304                   IFAIL( I ) = 0
00305    10          CONTINUE
00306             END IF
00307          END IF
00308          IF( INFO.EQ.0 ) THEN
00309             M = N
00310             GO TO 20
00311          END IF
00312          INFO = 0
00313       END IF
00314 *
00315 *     Otherwise, call DSTEBZ and, if eigenvectors are desired, ZSTEIN.
00316 *
00317       IF( WANTZ ) THEN
00318          ORDER = 'B'
00319       ELSE
00320          ORDER = 'E'
00321       END IF
00322       INDIBL = 1
00323       INDISP = INDIBL + N
00324       INDIWK = INDISP + N
00325       CALL DSTEBZ( RANGE, ORDER, N, VLL, VUU, IL, IU, ABSTLL,
00326      $             RWORK( INDD ), RWORK( INDE ), M, NSPLIT, W,
00327      $             IWORK( INDIBL ), IWORK( INDISP ), RWORK( INDRWK ),
00328      $             IWORK( INDIWK ), INFO )
00329 *
00330       IF( WANTZ ) THEN
00331          CALL ZSTEIN( N, RWORK( INDD ), RWORK( INDE ), M, W,
00332      $                IWORK( INDIBL ), IWORK( INDISP ), Z, LDZ,
00333      $                RWORK( INDRWK ), IWORK( INDIWK ), IFAIL, INFO )
00334 *
00335 *        Apply unitary matrix used in reduction to tridiagonal
00336 *        form to eigenvectors returned by ZSTEIN.
00337 *
00338          INDWRK = INDTAU + N
00339          CALL ZUPMTR( 'L', UPLO, 'N', N, M, AP, WORK( INDTAU ), Z, LDZ,
00340      $                WORK( INDWRK ), IINFO )
00341       END IF
00342 *
00343 *     If matrix was scaled, then rescale eigenvalues appropriately.
00344 *
00345    20 CONTINUE
00346       IF( ISCALE.EQ.1 ) THEN
00347          IF( INFO.EQ.0 ) THEN
00348             IMAX = M
00349          ELSE
00350             IMAX = INFO - 1
00351          END IF
00352          CALL DSCAL( IMAX, ONE / SIGMA, W, 1 )
00353       END IF
00354 *
00355 *     If eigenvalues are not in order, then sort them, along with
00356 *     eigenvectors.
00357 *
00358       IF( WANTZ ) THEN
00359          DO 40 J = 1, M - 1
00360             I = 0
00361             TMP1 = W( J )
00362             DO 30 JJ = J + 1, M
00363                IF( W( JJ ).LT.TMP1 ) THEN
00364                   I = JJ
00365                   TMP1 = W( JJ )
00366                END IF
00367    30       CONTINUE
00368 *
00369             IF( I.NE.0 ) THEN
00370                ITMP1 = IWORK( INDIBL+I-1 )
00371                W( I ) = W( J )
00372                IWORK( INDIBL+I-1 ) = IWORK( INDIBL+J-1 )
00373                W( J ) = TMP1
00374                IWORK( INDIBL+J-1 ) = ITMP1
00375                CALL ZSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 )
00376                IF( INFO.NE.0 ) THEN
00377                   ITMP1 = IFAIL( I )
00378                   IFAIL( I ) = IFAIL( J )
00379                   IFAIL( J ) = ITMP1
00380                END IF
00381             END IF
00382    40    CONTINUE
00383       END IF
00384 *
00385       RETURN
00386 *
00387 *     End of ZHPEVX
00388 *
00389       END