chbevx.f

Go to the documentation of this file.

*> \brief <b> CHBEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices</b>
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download CHBEVX + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/chbevx.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/chbevx.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/chbevx.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE CHBEVX( JOBZ, RANGE, UPLO, N, KD, AB, LDAB, Q, LDQ, VL,
*                          VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, RWORK,
*                          IWORK, IFAIL, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          JOBZ, RANGE, UPLO
*       INTEGER            IL, INFO, IU, KD, LDAB, LDQ, LDZ, M, N
*       REAL               ABSTOL, VL, VU
*       ..
*       .. Array Arguments ..
*       INTEGER            IFAIL( * ), IWORK( * )
*       REAL               RWORK( * ), W( * )
*       COMPLEX            AB( LDAB, * ), Q( LDQ, * ), WORK( * ),
*      $                   Z( LDZ, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> CHBEVX 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.
*> \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] KD
*> \verbatim
*>          KD is INTEGER
*>          The number of superdiagonals of the matrix A if UPLO = 'U',
*>          or the number of subdiagonals if UPLO = 'L'.  KD >= 0.
*> \endverbatim
*>
*> \param[in,out] AB
*> \verbatim
*>          AB is COMPLEX 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.
*> \endverbatim
*>
*> \param[in] LDAB
*> \verbatim
*>          LDAB is INTEGER
*>          The leading dimension of the array AB.  LDAB >= KD + 1.
*> \endverbatim
*>
*> \param[out] Q
*> \verbatim
*>          Q is COMPLEX 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.
*> \endverbatim
*>
*> \param[in] LDQ
*> \verbatim
*>          LDQ is INTEGER
*>          The leading dimension of the array Q.  If JOBZ = 'V', then
*>          LDQ >= max(1,N).
*> \endverbatim
*>
*> \param[in] VL
*> \verbatim
*>          VL is REAL
*>          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 REAL
*>          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 REAL
*>          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*SLAMCH('S'), not zero.
*>          If this routine returns with INFO>0, indicating that some
*>          eigenvectors did not converge, try setting ABSTOL to
*>          2*SLAMCH('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 REAL array, dimension (N)
*>          The first M elements contain the selected eigenvalues in
*>          ascending order.
*> \endverbatim
*>
*> \param[out] Z
*> \verbatim
*>          Z is COMPLEX 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 array, dimension (N)
*> \endverbatim
*>
*> \param[out] RWORK
*> \verbatim
*>          RWORK is REAL 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 complexOTHEReigen
*
*  =====================================================================
      SUBROUTINE chbevx( 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 --
*  -- 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, KD, LDAB, LDQ, LDZ, M, N
      REAL               ABSTOL, VL, VU
*     ..
*     .. Array Arguments ..
      INTEGER            IFAIL( * ), IWORK( * )
      REAL               RWORK( * ), W( * )
      COMPLEX            AB( LDAB, * ), Q( LDQ, * ), WORK( * ),
     $                   z( ldz, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      REAL               ZERO, ONE
      PARAMETER          ( ZERO = 0.0e0, one = 1.0e0 )
      COMPLEX            CZERO, CONE
      parameter( czero = ( 0.0e0, 0.0e0 ),
     $                   cone = ( 1.0e0, 0.0e0 ) )
*     ..
*     .. 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
      REAL               ABSTLL, ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN,
     $                   SIGMA, SMLNUM, TMP1, VLL, VUU
      COMPLEX            CTMP1
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      REAL               CLANHB, SLAMCH
      EXTERNAL           lsame, clanhb, slamch
*     ..
*     .. External Subroutines ..
      EXTERNAL           ccopy, cgemv, chbtrd, clacpy, clascl, cstein,
     $                   csteqr, cswap, scopy, sscal, sstebz, ssterf,
     $                   xerbla
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          max, min, real, 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( 'chbevx', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      M = 0
.EQ.      IF( N0 )
     $   RETURN
*
.EQ.      IF( N1 ) THEN
         M = 1
         IF( LOWER ) THEN
            CTMP1 = AB( 1, 1 )
         ELSE
            CTMP1 = AB( KD+1, 1 )
         END IF
         TMP1 = REAL( CTMP1 )
         IF( VALEIG ) THEN
.NOT..LT..AND..GE.            IF( ( VLTMP1  VUTMP1 ) )
     $         M = 0
         END IF
.EQ.         IF( M1 ) THEN
            W( 1 ) = REAL( CTMP1 )
            IF( WANTZ )
     $         Z( 1, 1 ) = CONE
         END IF
         RETURN
      END IF
*
*     Get machine constants.
*
      SAFMIN = SLAMCH( 'safe minimum' )
      EPS = SLAMCH( '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
      ENDIF
      ANRM = CLANHB( 'm', UPLO, N, KD, AB, LDAB, 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
            CALL CLASCL( 'b', KD, KD, ONE, SIGMA, N, N, AB, LDAB, INFO )
         ELSE
            CALL CLASCL( 'q', KD, KD, ONE, SIGMA, N, N, AB, LDAB, INFO )
         END IF
.GT.         IF( ABSTOL0 )
     $      ABSTLL = ABSTOL*SIGMA
         IF( VALEIG ) THEN
            VLL = VL*SIGMA
            VUU = VU*SIGMA
         END IF
      END IF
*
*     Call CHBTRD to reduce Hermitian band matrix to tridiagonal form.
*
      INDD = 1
      INDE = INDD + N
      INDRWK = INDE + N
      INDWRK = 1
      CALL CHBTRD( 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 SSTERF or CSTEQR.  If this fails for some
*     eigenvalue, then try SSTEBZ.
*
      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 SCOPY( N, RWORK( INDD ), 1, W, 1 )
         INDEE = INDRWK + 2*N
.NOT.         IF( WANTZ ) THEN
            CALL SCOPY( N-1, RWORK( INDE ), 1, RWORK( INDEE ), 1 )
            CALL SSTERF( N, W, RWORK( INDEE ), INFO )
         ELSE
            CALL CLACPY( 'a', N, N, Q, LDQ, Z, LDZ )
            CALL SCOPY( N-1, RWORK( INDE ), 1, RWORK( INDEE ), 1 )
            CALL CSTEQR( JOBZ, N, W, RWORK( INDEE ), Z, LDZ,
     $                   RWORK( INDRWK ), INFO )
.EQ.            IF( INFO0 ) THEN
               DO 10 I = 1, N
                  IFAIL( I ) = 0
   10          CONTINUE
            END IF
         END IF
.EQ.         IF( INFO0 ) THEN
            M = N
            GO TO 30
         END IF
         INFO = 0
      END IF
*
*     Otherwise, call SSTEBZ and, if eigenvectors are desired, CSTEIN.
*
      IF( WANTZ ) THEN
         ORDER = 'b'
      ELSE
         ORDER = 'e'
      END IF
      INDIBL = 1
      INDISP = INDIBL + N
      INDIWK = INDISP + N
      CALL SSTEBZ( 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 CSTEIN( 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 CSTEIN.
*
         DO 20 J = 1, M
            CALL CCOPY( N, Z( 1, J ), 1, WORK( 1 ), 1 )
            CALL CGEMV( '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
.EQ.      IF( ISCALE1 ) THEN
.EQ.         IF( INFO0 ) THEN
            IMAX = M
         ELSE
            IMAX = INFO - 1
         END IF
         CALL SSCAL( 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
.LT.               IF( W( JJ )TMP1 ) THEN
                  I = JJ
                  TMP1 = W( JJ )
               END IF
   40       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 CSWAP( 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
   50    CONTINUE
      END IF
*
      RETURN
*
*     End of CHBEVX
*
      END