zheevx.f

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*> \brief <b> ZHEEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for HE matrices</b>
*
*  =========== DOCUMENTATION ===========
*
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*
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*
*  Definition:
*  ===========
*
*       SUBROUTINE ZHEEVX( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU,
*                          ABSTOL, M, W, Z, LDZ, WORK, LWORK, RWORK,
*                          IWORK, IFAIL, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          JOBZ, RANGE, UPLO
*       INTEGER            IL, INFO, IU, LDA, LDZ, LWORK, M, N
*       DOUBLE PRECISION   ABSTOL, VL, VU
*       ..
*       .. Array Arguments ..
*       INTEGER            IFAIL( * ), IWORK( * )
*       DOUBLE PRECISION   RWORK( * ), W( * )
*       COMPLEX*16         A( LDA, * ), WORK( * ), Z( LDZ, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> ZHEEVX computes selected eigenvalues and, optionally, eigenvectors
*> of a complex Hermitian matrix A.  Eigenvalues and eigenvectors can
*> be selected by specifying either a range of values or a range of
*> indices for the desired eigenvalues.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] JOBZ
*> \verbatim
*>          JOBZ is CHARACTER*1
*>          = 'N':  Compute eigenvalues only;
*>          = 'V':  Compute eigenvalues and eigenvectors.
*> \endverbatim
*>
*> \param[in] RANGE
*> \verbatim
*>          RANGE is 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.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          = 'U':  Upper triangle of A is stored;
*>          = 'L':  Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is COMPLEX*16 array, dimension (LDA, N)
*>          On entry, the Hermitian matrix A.  If UPLO = 'U', the
*>          leading N-by-N upper triangular part of A contains the
*>          upper triangular part of the matrix A.  If UPLO = 'L',
*>          the leading N-by-N lower triangular part of A contains
*>          the lower triangular part of the matrix A.
*>          On exit, the lower triangle (if UPLO='L') or the upper
*>          triangle (if UPLO='U') of A, including the diagonal, is
*>          destroyed.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in] VL
*> \verbatim
*>          VL is DOUBLE PRECISION
*>          If RANGE='V', the lower bound of the interval to
*>          be searched for eigenvalues. VL < VU.
*>          Not referenced if RANGE = 'A' or 'I'.
*> \endverbatim
*>
*> \param[in] VU
*> \verbatim
*>          VU is DOUBLE PRECISION
*>          If RANGE='V', the upper bound of the interval to
*>          be searched for eigenvalues. VL < VU.
*>          Not referenced if RANGE = 'A' or 'I'.
*> \endverbatim
*>
*> \param[in] IL
*> \verbatim
*>          IL is INTEGER
*>          If RANGE='I', the index of the
*>          smallest eigenvalue 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'.
*> \endverbatim
*>
*> \param[in] IU
*> \verbatim
*>          IU is INTEGER
*>          If RANGE='I', the index of the
*>          largest eigenvalue 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'.
*> \endverbatim
*>
*> \param[in] ABSTOL
*> \verbatim
*>          ABSTOL is 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 A 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.
*> \endverbatim
*>
*> \param[out] M
*> \verbatim
*>          M is INTEGER
*>          The total number of eigenvalues found.  0 <= M <= N.
*>          If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
*> \endverbatim
*>
*> \param[out] W
*> \verbatim
*>          W is DOUBLE PRECISION array, dimension (N)
*>          On normal exit, the first M elements contain the selected
*>          eigenvalues in ascending order.
*> \endverbatim
*>
*> \param[out] Z
*> \verbatim
*>          Z is 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.
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*>          LDZ is INTEGER
*>          The leading dimension of the array Z.  LDZ >= 1, and if
*>          JOBZ = 'V', LDZ >= max(1,N).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
*>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The length of the array WORK.  LWORK >= 1, when N <= 1;
*>          otherwise 2*N.
*>          For optimal efficiency, LWORK >= (NB+1)*N,
*>          where NB is the max of the blocksize for ZHETRD and for
*>          ZUNMTR as returned by ILAENV.
*>
*>          If LWORK = -1, then a workspace query is assumed; the routine
*>          only calculates the optimal size of the WORK array, returns
*>          this value as the first entry of the WORK array, and no error
*>          message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] RWORK
*> \verbatim
*>          RWORK is DOUBLE PRECISION array, dimension (7*N)
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension (5*N)
*> \endverbatim
*>
*> \param[out] IFAIL
*> \verbatim
*>          IFAIL is 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.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is 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.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup complex16HEeigen
*
*  =====================================================================
      SUBROUTINE zheevx( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU,
     $                   ABSTOL, M, W, Z, LDZ, WORK, LWORK, RWORK,
     $                   IWORK, IFAIL, INFO )
*
*  -- LAPACK driver routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          JOBZ, RANGE, UPLO
      INTEGER            IL, INFO, IU, LDA, LDZ, LWORK, M, N
      DOUBLE PRECISION   ABSTOL, VL, VU
*     ..
*     .. Array Arguments ..
      INTEGER            IFAIL( * ), IWORK( * )
      DOUBLE PRECISION   RWORK( * ), W( * )
      COMPLEX*16         A( LDA, * ), WORK( * ), Z( LDZ, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0d+0, one = 1.0d+0 )
      COMPLEX*16         CONE
      parameter( cone = ( 1.0d+0, 0.0d+0 ) )
*     ..
*     .. Local Scalars ..
      LOGICAL            ALLEIG, INDEIG, LOWER, LQUERY, TEST, VALEIG,
     $                   WANTZ
      CHARACTER          ORDER
      INTEGER            I, IINFO, IMAX, INDD, INDE, INDEE, INDIBL,
     $                   indisp, indiwk, indrwk, indtau, indwrk, iscale,
     $                   itmp1, j, jj, llwork, lwkmin, lwkopt, nb,
     $                   nsplit
      DOUBLE PRECISION   ABSTLL, ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN,
     $                   SIGMA, SMLNUM, TMP1, VLL, VUU
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            ILAENV
      DOUBLE PRECISION   DLAMCH, ZLANHE
      EXTERNAL           lsame, ilaenv, dlamch, zlanhe
*     ..
*     .. External Subroutines ..
      EXTERNAL           dcopy, dscal, dstebz, dsterf, xerbla, zdscal,
     $                   zhetrd, zlacpy, zstein, zsteqr, zswap, zungtr,
     $                   zunmtr
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          dble, max, min, sqrt
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      lower = lsame( uplo, 'L' )
      wantz = lsame( jobz, 'V' )
      alleig = lsame( range, 'A' )
      valeig = lsame( range, 'V' )
      indeig = lsame( range, 'I' )
      lquery = ( lwork.EQ.-1 )
*
      info = 0
      IF( .NOT.( wantz .OR. lsame( jobz, 'n' ) ) ) THEN
         INFO = -1
.NOT..OR..OR.      ELSE IF( ( ALLEIG  VALEIG  INDEIG ) ) THEN
         INFO = -2
.NOT..OR.      ELSE IF( ( LOWER  LSAME( UPLO, 'u' ) ) ) THEN
         INFO = -3
.LT.      ELSE IF( N0 ) THEN
         INFO = -4
.LT.      ELSE IF( LDAMAX( 1, N ) ) THEN
         INFO = -6
      ELSE
         IF( VALEIG ) THEN
.GT..AND..LE.            IF( N0  VUVL )
     $         INFO = -8
         ELSE IF( INDEIG ) THEN
.LT..OR..GT.            IF( IL1  ILMAX( 1, N ) ) THEN
               INFO = -9
.LT..OR..GT.            ELSE IF( IUMIN( N, IL )  IUN ) THEN
               INFO = -10
            END IF
         END IF
      END IF
.EQ.      IF( INFO0 ) THEN
.LT..OR..AND..LT.         IF( LDZ1  ( WANTZ  LDZN ) ) THEN
            INFO = -15
         END IF
      END IF
*
.EQ.      IF( INFO0 ) THEN
.LE.         IF( N1 ) THEN
            LWKMIN = 1
            WORK( 1 ) = LWKMIN
         ELSE
            LWKMIN = 2*N
            NB = ILAENV( 1, 'zhetrd', UPLO, N, -1, -1, -1 )
            NB = MAX( NB, ILAENV( 1, 'zunmtr', UPLO, N, -1, -1, -1 ) )
            LWKOPT = MAX( 1, ( NB + 1 )*N )
            WORK( 1 ) = LWKOPT
         END IF
*
.LT..AND..NOT.         IF( LWORKLWKMIN  LQUERY )
     $      INFO = -17
      END IF
*
.NE.      IF( INFO0 ) THEN
         CALL XERBLA( 'zheevx', -INFO )
         RETURN
      ELSE IF( LQUERY ) THEN
         RETURN
      END IF
*
*     Quick return if possible
*
      M = 0
.EQ.      IF( N0 ) THEN
         RETURN
      END IF
*
.EQ.      IF( N1 ) THEN
.OR.         IF( ALLEIG  INDEIG ) THEN
            M = 1
            W( 1 ) = DBLE( A( 1, 1 ) )
         ELSE IF( VALEIG ) THEN
.LT..AND..GE.            IF( VLDBLE( A( 1, 1 ) )  VUDBLE( A( 1, 1 ) ) )
     $           THEN
               M = 1
               W( 1 ) = DBLE( A( 1, 1 ) )
            END IF
         END IF
         IF( WANTZ )
     $      Z( 1, 1 ) = CONE
         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
      END IF
      ANRM = ZLANHE( 'm', UPLO, N, A, LDA, RWORK )
.GT..AND..LT.      IF( ANRMZERO  ANRMRMIN ) THEN
         ISCALE = 1
         SIGMA = RMIN / ANRM
.GT.      ELSE IF( ANRMRMAX ) THEN
         ISCALE = 1
         SIGMA = RMAX / ANRM
      END IF
.EQ.      IF( ISCALE1 ) THEN
         IF( LOWER ) THEN
            DO 10 J = 1, N
               CALL ZDSCAL( N-J+1, SIGMA, A( J, J ), 1 )
   10       CONTINUE
         ELSE
            DO 20 J = 1, N
               CALL ZDSCAL( J, SIGMA, A( 1, J ), 1 )
   20       CONTINUE
         END IF
.GT.         IF( ABSTOL0 )
     $      ABSTLL = ABSTOL*SIGMA
         IF( VALEIG ) THEN
            VLL = VL*SIGMA
            VUU = VU*SIGMA
         END IF
      END IF
*
*     Call ZHETRD to reduce Hermitian matrix to tridiagonal form.
*
      INDD = 1
      INDE = INDD + N
      INDRWK = INDE + N
      INDTAU = 1
      INDWRK = INDTAU + N
      LLWORK = LWORK - INDWRK + 1
      CALL ZHETRD( UPLO, N, A, LDA, RWORK( INDD ), RWORK( INDE ),
     $             WORK( INDTAU ), WORK( INDWRK ), LLWORK, IINFO )
*
*     If all eigenvalues are desired and ABSTOL is less than or equal to
*     zero, then call DSTERF or ZUNGTR and ZSTEQR.  If this fails for
*     some eigenvalue, then try DSTEBZ.
*
      TEST = .FALSE.
      IF( INDEIG ) THEN
.EQ..AND..EQ.         IF( IL1  IUN ) THEN
            TEST = .TRUE.
         END IF
      END IF
.OR..AND..LE.      IF( ( ALLEIG  TEST )  ( ABSTOLZERO ) ) THEN
         CALL DCOPY( N, RWORK( INDD ), 1, W, 1 )
         INDEE = INDRWK + 2*N
.NOT.         IF( 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, A, LDA, Z, LDZ )
            CALL ZUNGTR( UPLO, N, Z, LDZ, WORK( INDTAU ),
     $                   WORK( INDWRK ), LLWORK, IINFO )
            CALL DCOPY( N-1, RWORK( INDE ), 1, RWORK( INDEE ), 1 )
            CALL ZSTEQR( JOBZ, N, W, RWORK( INDEE ), Z, LDZ,
     $                   RWORK( INDRWK ), INFO )
.EQ.            IF( INFO0 ) THEN
               DO 30 I = 1, N
                  IFAIL( I ) = 0
   30          CONTINUE
            END IF
         END IF
.EQ.         IF( INFO0 ) THEN
            M = N
            GO TO 40
         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.
*
         CALL ZUNMTR( 'l', UPLO, 'n', N, M, A, LDA, WORK( INDTAU ), Z,
     $                LDZ, WORK( INDWRK ), LLWORK, IINFO )
      END IF
*
*     If matrix was scaled, then rescale eigenvalues appropriately.
*
   40 CONTINUE
.EQ.      IF( ISCALE1 ) THEN
.EQ.         IF( INFO0 ) 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 60 J = 1, M - 1
            I = 0
            TMP1 = W( J )
            DO 50 JJ = J + 1, M
.LT.               IF( W( JJ )TMP1 ) THEN
                  I = JJ
                  TMP1 = W( JJ )
               END IF
   50       CONTINUE
*
.NE.            IF( I0 ) 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 )
.NE.               IF( INFO0 ) THEN
                  ITMP1 = IFAIL( I )
                  IFAIL( I ) = IFAIL( J )
                  IFAIL( J ) = ITMP1
               END IF
            END IF
   60    CONTINUE
      END IF
*
*     Set WORK(1) to optimal complex workspace size.
*
      WORK( 1 ) = LWKOPT
*
      RETURN
*
*     End of ZHEEVX
*
      END