zhbevx.f

          SUBROUTINE ZHBEVX( JOBZ, RANGE, UPLO, N, KD, AB, LDAB, Q, LDQ, VL,
         $                   VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, RWORK,
         $                   IWORK, IFAIL, INFO )
    *
    *  -- LAPACK driver routine (version 3.1) --
    *     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
    *     November 2006
    *
    *     .. Scalar Arguments ..
          CHARACTER          JOBZ, RANGE, UPLO
          INTEGER            IL, INFO, IU, KD, LDAB, LDQ, LDZ, M, N
          DOUBLE PRECISION   ABSTOL, VL, VU
    *     ..
    *     .. Array Arguments ..
          INTEGER            IFAIL( * ), IWORK( * )
          DOUBLE PRECISION   RWORK( * ), W( * )
          COMPLEX*16         AB( LDAB, * ), Q( LDQ, * ), WORK( * ),
         $                   Z( LDZ, * )
    *     ..
    *
    *  Purpose
    *  =======
    *
    *  ZHBEVX computes selected eigenvalues and, optionally, eigenvectors
    *  of a complex Hermitian band matrix A.  Eigenvalues and eigenvectors
    *  can be selected by specifying either a range of values or a range of
    *  indices for the desired eigenvalues.
    *
    *  Arguments
    *  =========
    *
    *  JOBZ    (input) CHARACTER*1
    *          = 'N':  Compute eigenvalues only;
    *          = 'V':  Compute eigenvalues and eigenvectors.
    *
    *  RANGE   (input) CHARACTER*1
    *          = 'A': all eigenvalues will be found;
    *          = 'V': all eigenvalues in the half-open interval (VL,VU]
    *                 will be found;
    *          = 'I': the IL-th through IU-th eigenvalues will be found.
    *
    *  UPLO    (input) CHARACTER*1
    *          = 'U':  Upper triangle of A is stored;
    *          = 'L':  Lower triangle of A is stored.
    *
    *  N       (input) INTEGER
    *          The order of the matrix A.  N >= 0.
    *
    *  KD      (input) INTEGER
    *          The number of superdiagonals of the matrix A if UPLO = 'U',
    *          or the number of subdiagonals if UPLO = 'L'.  KD >= 0.
    *
    *  AB      (input/output) COMPLEX*16 array, dimension (LDAB, N)
    *          On entry, the upper or lower triangle of the Hermitian band
    *          matrix A, stored in the first KD+1 rows of the array.  The
    *          j-th column of A is stored in the j-th column of the array AB
    *          as follows:
    *          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
    *          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
    *
    *          On exit, AB is overwritten by values generated during the
    *          reduction to tridiagonal form.
    *
    *  LDAB    (input) INTEGER
    *          The leading dimension of the array AB.  LDAB >= KD + 1.
    *
    *  Q       (output) COMPLEX*16 array, dimension (LDQ, N)
    *          If JOBZ = 'V', the N-by-N unitary matrix used in the
    *                          reduction to tridiagonal form.
    *          If JOBZ = 'N', the array Q is not referenced.
    *
    *  LDQ     (input) INTEGER
    *          The leading dimension of the array Q.  If JOBZ = 'V', then
    *          LDQ >= max(1,N).
    *
    *  VL      (input) DOUBLE PRECISION
    *  VU      (input) DOUBLE PRECISION
    *          If RANGE='V', the lower and upper bounds of the interval to
    *          be searched for eigenvalues. VL < VU.
    *          Not referenced if RANGE = 'A' or 'I'.
    *
    *  IL      (input) INTEGER
    *  IU      (input) INTEGER
    *          If RANGE='I', the indices (in ascending order) of the
    *          smallest and largest eigenvalues to be returned.
    *          1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
    *          Not referenced if RANGE = 'A' or 'V'.
    *
    *  ABSTOL  (input) DOUBLE PRECISION
    *          The absolute error tolerance for the eigenvalues.
    *          An approximate eigenvalue is accepted as converged
    *          when it is determined to lie in an interval [a,b]
    *          of width less than or equal to
    *
    *                  ABSTOL + EPS *   max( |a|,|b| ) ,
    *
    *          where EPS is the machine precision.  If ABSTOL is less than
    *          or equal to zero, then  EPS*|T|  will be used in its place,
    *          where |T| is the 1-norm of the tridiagonal matrix obtained
    *          by reducing AB to tridiagonal form.
    *
    *          Eigenvalues will be computed most accurately when ABSTOL is
    *          set to twice the underflow threshold 2*DLAMCH('S'), not zero.
    *          If this routine returns with INFO>0, indicating that some
    *          eigenvectors did not converge, try setting ABSTOL to
    *          2*DLAMCH('S').
    *
    *          See "Computing Small Singular Values of Bidiagonal Matrices
    *          with Guaranteed High Relative Accuracy," by Demmel and
    *          Kahan, LAPACK Working Note #3.
    *
    *  M       (output) INTEGER
    *          The total number of eigenvalues found.  0 <= M <= N.
    *          If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
    *
    *  W       (output) DOUBLE PRECISION array, dimension (N)
    *          The first M elements contain the selected eigenvalues in
    *          ascending order.
    *
    *  Z       (output) COMPLEX*16 array, dimension (LDZ, max(1,M))
    *          If JOBZ = 'V', then if INFO = 0, the first M columns of Z
    *          contain the orthonormal eigenvectors of the matrix A
    *          corresponding to the selected eigenvalues, with the i-th
    *          column of Z holding the eigenvector associated with W(i).
    *          If an eigenvector fails to converge, then that column of Z
    *          contains the latest approximation to the eigenvector, and the
    *          index of the eigenvector is returned in IFAIL.
    *          If JOBZ = 'N', then Z is not referenced.
    *          Note: the user must ensure that at least max(1,M) columns are
    *          supplied in the array Z; if RANGE = 'V', the exact value of M
    *          is not known in advance and an upper bound must be used.
    *
    *  LDZ     (input) INTEGER
    *          The leading dimension of the array Z.  LDZ >= 1, and if
    *          JOBZ = 'V', LDZ >= max(1,N).
    *
    *  WORK    (workspace) COMPLEX*16 array, dimension (N)
    *
    *  RWORK   (workspace) DOUBLE PRECISION array, dimension (7*N)
    *
    *  IWORK   (workspace) INTEGER array, dimension (5*N)
    *
    *  IFAIL   (output) INTEGER array, dimension (N)
    *          If JOBZ = 'V', then if INFO = 0, the first M elements of
    *          IFAIL are zero.  If INFO > 0, then IFAIL contains the
    *          indices of the eigenvectors that failed to converge.
    *          If JOBZ = 'N', then IFAIL is not referenced.
    *
    *  INFO    (output) INTEGER
    *          = 0:  successful exit
    *          < 0:  if INFO = -i, the i-th argument had an illegal value
    *          > 0:  if INFO = i, then i eigenvectors failed to converge.
    *                Their indices are stored in array IFAIL.
    *
    *  =====================================================================
    *
    *     .. Parameters ..
          DOUBLE PRECISION   ZERO, ONE
          PARAMETER          ( ZERO = 0.0D0, ONE = 1.0D0 )
          COMPLEX*16         CZERO, CONE
          PARAMETER          ( CZERO = ( 0.0D0, 0.0D0 ),
         $                   CONE = ( 1.0D0, 0.0D0 ) )
    *     ..
    *     .. Local Scalars ..
          LOGICAL            ALLEIG, INDEIG, LOWER, TEST, VALEIG, WANTZ
          CHARACTER          ORDER
          INTEGER            I, IINFO, IMAX, INDD, INDE, INDEE, INDIBL,
         $                   INDISP, INDIWK, INDRWK, INDWRK, ISCALE, ITMP1,
         $                   J, JJ, NSPLIT
          DOUBLE PRECISION   ABSTLL, ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN,
         $                   SIGMA, SMLNUM, TMP1, VLL, VUU
          COMPLEX*16         CTMP1
    *     ..
    *     .. External Functions ..
          LOGICAL            LSAME
          DOUBLE PRECISION   DLAMCH, ZLANHB
          EXTERNAL           LSAME, DLAMCH, ZLANHB
    *     ..
    *     .. External Subroutines ..
          EXTERNAL           DCOPY, DSCAL, DSTEBZ, DSTERF, XERBLA, ZCOPY,
         $                   ZGEMV, ZHBTRD, ZLACPY, ZLASCL, ZSTEIN, ZSTEQR,
         $                   ZSWAP
    *     ..
    *     .. Intrinsic Functions ..
          INTRINSIC          DBLE, MAX, MIN, SQRT
    *     ..
    *     .. Executable Statements ..
    *
    *     Test the input parameters.
    *
          WANTZ = LSAME( JOBZ, 'V' )
          ALLEIG = LSAME( RANGE, 'A' )
          VALEIG = LSAME( RANGE, 'V' )
          INDEIG = LSAME( RANGE, 'I' )
          LOWER = LSAME( UPLO, 'L' )
    *
          INFO = 0
          IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
             INFO = -1
          ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN
             INFO = -2
          ELSE IF( .NOT.( LOWER .OR. LSAME( UPLO, 'U' ) ) ) THEN
             INFO = -3
          ELSE IF( N.LT.0 ) THEN
             INFO = -4
          ELSE IF( KD.LT.0 ) THEN
             INFO = -5
          ELSE IF( LDAB.LT.KD+1 ) THEN
             INFO = -7
          ELSE IF( WANTZ .AND. LDQ.LT.MAX( 1, N ) ) THEN
             INFO = -9
          ELSE
             IF( VALEIG ) THEN
                IF( N.GT.0 .AND. VU.LE.VL )
         $         INFO = -11
             ELSE IF( INDEIG ) THEN
                IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN
                   INFO = -12
                ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN
                   INFO = -13
                END IF
             END IF
          END IF
          IF( INFO.EQ.0 ) THEN
             IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) )
         $      INFO = -18
          END IF
    *
          IF( INFO.NE.0 ) THEN
             CALL XERBLA( 'ZHBEVX', -INFO )
             RETURN
          END IF
    *
    *     Quick return if possible
    *
          M = 0
          IF( N.EQ.0 )
         $   RETURN
    *
          IF( N.EQ.1 ) THEN
             M = 1
             IF( LOWER ) THEN
                CTMP1 = AB( 1, 1 )
             ELSE
                CTMP1 = AB( KD+1, 1 )
             END IF
             TMP1 = DBLE( CTMP1 )
             IF( VALEIG ) THEN
                IF( .NOT.( VL.LT.TMP1 .AND. VU.GE.TMP1 ) )
         $         M = 0
             END IF
             IF( M.EQ.1 ) THEN
                W( 1 ) = CTMP1
                IF( WANTZ )
         $         Z( 1, 1 ) = CONE
             END IF
             RETURN
          END IF
    *
    *     Get machine constants.
    *
          SAFMIN = DLAMCH( 'Safe minimum' )
          EPS = DLAMCH( 'Precision' )
          SMLNUM = SAFMIN / EPS
          BIGNUM = ONE / SMLNUM
          RMIN = SQRT( SMLNUM )
          RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) )
    *
    *     Scale matrix to allowable range, if necessary.
    *
          ISCALE = 0
          ABSTLL = ABSTOL
          IF( VALEIG ) THEN
             VLL = VL
             VUU = VU
          ELSE
             VLL = ZERO
             VUU = ZERO
          END IF
          ANRM = ZLANHB( 'M', UPLO, N, KD, AB, LDAB, RWORK )
          IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN
             ISCALE = 1
             SIGMA = RMIN / ANRM
          ELSE IF( ANRM.GT.RMAX ) THEN
             ISCALE = 1
             SIGMA = RMAX / ANRM
          END IF
          IF( ISCALE.EQ.1 ) THEN
             IF( LOWER ) THEN
                CALL ZLASCL( 'B', KD, KD, ONE, SIGMA, N, N, AB, LDAB, INFO )
             ELSE
                CALL ZLASCL( 'Q', KD, KD, ONE, SIGMA, N, N, AB, LDAB, INFO )
             END IF
             IF( ABSTOL.GT.0 )
         $      ABSTLL = ABSTOL*SIGMA
             IF( VALEIG ) THEN
                VLL = VL*SIGMA
                VUU = VU*SIGMA
             END IF
          END IF
    *
    *     Call ZHBTRD to reduce Hermitian band matrix to tridiagonal form.
    *
          INDD = 1
          INDE = INDD + N
          INDRWK = INDE + N
          INDWRK = 1
          CALL ZHBTRD( JOBZ, UPLO, N, KD, AB, LDAB, RWORK( INDD ),
         $             RWORK( INDE ), Q, LDQ, WORK( INDWRK ), IINFO )
    *
    *     If all eigenvalues are desired and ABSTOL is less than or equal
    *     to zero, then call DSTERF or ZSTEQR.  If this fails for some
    *     eigenvalue, then try DSTEBZ.
    *
          TEST = .FALSE.
          IF (INDEIG) THEN
             IF (IL.EQ.1 .AND. IU.EQ.N) THEN
                TEST = .TRUE.
             END IF
          END IF
          IF ((ALLEIG .OR. TEST) .AND. (ABSTOL.LE.ZERO)) THEN
             CALL DCOPY( N, RWORK( INDD ), 1, W, 1 )
             INDEE = INDRWK + 2*N
             IF( .NOT.WANTZ ) THEN
                CALL DCOPY( N-1, RWORK( INDE ), 1, RWORK( INDEE ), 1 )
                CALL DSTERF( N, W, RWORK( INDEE ), INFO )
             ELSE
                CALL ZLACPY( 'A', N, N, Q, LDQ, Z, LDZ )
                CALL DCOPY( N-1, RWORK( INDE ), 1, RWORK( INDEE ), 1 )
                CALL ZSTEQR( JOBZ, N, W, RWORK( INDEE ), Z, LDZ,
         $                   RWORK( INDRWK ), INFO )
                IF( INFO.EQ.0 ) THEN
                   DO 10 I = 1, N
                      IFAIL( I ) = 0
       10          CONTINUE
                END IF
             END IF
             IF( INFO.EQ.0 ) THEN
                M = N
                GO TO 30
             END IF
             INFO = 0
          END IF
    *
    *     Otherwise, call DSTEBZ and, if eigenvectors are desired, ZSTEIN.
    *
          IF( WANTZ ) THEN
             ORDER = 'B'
          ELSE
             ORDER = 'E'
          END IF
          INDIBL = 1
          INDISP = INDIBL + N
          INDIWK = INDISP + N
          CALL DSTEBZ( RANGE, ORDER, N, VLL, VUU, IL, IU, ABSTLL,
         $             RWORK( INDD ), RWORK( INDE ), M, NSPLIT, W,
         $             IWORK( INDIBL ), IWORK( INDISP ), RWORK( INDRWK ),
         $             IWORK( INDIWK ), INFO )
    *
          IF( WANTZ ) THEN
             CALL ZSTEIN( N, RWORK( INDD ), RWORK( INDE ), M, W,
         $                IWORK( INDIBL ), IWORK( INDISP ), Z, LDZ,
         $                RWORK( INDRWK ), IWORK( INDIWK ), IFAIL, INFO )
    *
    *        Apply unitary matrix used in reduction to tridiagonal
    *        form to eigenvectors returned by ZSTEIN.
    *
             DO 20 J = 1, M
                CALL ZCOPY( N, Z( 1, J ), 1, WORK( 1 ), 1 )
                CALL ZGEMV( 'N', N, N, CONE, Q, LDQ, WORK, 1, CZERO,
         $                  Z( 1, J ), 1 )
       20    CONTINUE
          END IF
    *
    *     If matrix was scaled, then rescale eigenvalues appropriately.
    *
       30 CONTINUE
          IF( ISCALE.EQ.1 ) THEN
             IF( INFO.EQ.0 ) THEN
                IMAX = M
             ELSE
                IMAX = INFO - 1
             END IF
             CALL DSCAL( IMAX, ONE / SIGMA, W, 1 )
          END IF
    *
    *     If eigenvalues are not in order, then sort them, along with
    *     eigenvectors.
    *
          IF( WANTZ ) THEN
             DO 50 J = 1, M - 1
                I = 0
                TMP1 = W( J )
                DO 40 JJ = J + 1, M
                   IF( W( JJ ).LT.TMP1 ) THEN
                      I = JJ
                      TMP1 = W( JJ )
                   END IF
       40       CONTINUE
    *
                IF( I.NE.0 ) THEN
                   ITMP1 = IWORK( INDIBL+I-1 )
                   W( I ) = W( J )
                   IWORK( INDIBL+I-1 ) = IWORK( INDIBL+J-1 )
                   W( J ) = TMP1
                   IWORK( INDIBL+J-1 ) = ITMP1
                   CALL ZSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 )
                   IF( INFO.NE.0 ) THEN
                      ITMP1 = IFAIL( I )
                      IFAIL( I ) = IFAIL( J )
                      IFAIL( J ) = ITMP1
                   END IF
                END IF
       50    CONTINUE
          END IF
    *
          RETURN
    *
    *     End of ZHBEVX
    *
          END