zhbgvx.f

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*> \brief \b ZHBGVX
*
*  =========== DOCUMENTATION ===========
*
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*
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*
*  Definition:
*  ===========
*
*       SUBROUTINE ZHBGVX( JOBZ, RANGE, UPLO, N, KA, KB, AB, LDAB, BB,
*                          LDBB, 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, KA, KB, LDAB, LDBB, LDQ, LDZ, M,
*      $                   N
*       DOUBLE PRECISION   ABSTOL, VL, VU
*       ..
*       .. Array Arguments ..
*       INTEGER            IFAIL( * ), IWORK( * )
*       DOUBLE PRECISION   RWORK( * ), W( * )
*       COMPLEX*16         AB( LDAB, * ), BB( LDBB, * ), Q( LDQ, * ),
*      $                   WORK( * ), Z( LDZ, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> ZHBGVX computes all the eigenvalues, and optionally, the eigenvectors
*> of a complex generalized Hermitian-definite banded eigenproblem, of
*> the form A*x=(lambda)*B*x. Here A and B are assumed to be Hermitian
*> and banded, and B is also positive definite.  Eigenvalues and
*> eigenvectors can be selected by specifying either all eigenvalues,
*> 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 triangles of A and B are stored;
*>          = 'L':  Lower triangles of A and B are stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrices A and B.  N >= 0.
*> \endverbatim
*>
*> \param[in] KA
*> \verbatim
*>          KA is INTEGER
*>          The number of superdiagonals of the matrix A if UPLO = 'U',
*>          or the number of subdiagonals if UPLO = 'L'. KA >= 0.
*> \endverbatim
*>
*> \param[in] KB
*> \verbatim
*>          KB is INTEGER
*>          The number of superdiagonals of the matrix B if UPLO = 'U',
*>          or the number of subdiagonals if UPLO = 'L'. KB >= 0.
*> \endverbatim
*>
*> \param[in,out] AB
*> \verbatim
*>          AB is COMPLEX*16 array, dimension (LDAB, N)
*>          On entry, the upper or lower triangle of the Hermitian band
*>          matrix A, stored in the first ka+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(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j;
*>          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+ka).
*>
*>          On exit, the contents of AB are destroyed.
*> \endverbatim
*>
*> \param[in] LDAB
*> \verbatim
*>          LDAB is INTEGER
*>          The leading dimension of the array AB.  LDAB >= KA+1.
*> \endverbatim
*>
*> \param[in,out] BB
*> \verbatim
*>          BB is COMPLEX*16 array, dimension (LDBB, N)
*>          On entry, the upper or lower triangle of the Hermitian band
*>          matrix B, stored in the first kb+1 rows of the array.  The
*>          j-th column of B is stored in the j-th column of the array BB
*>          as follows:
*>          if UPLO = 'U', BB(kb+1+i-j,j) = B(i,j) for max(1,j-kb)<=i<=j;
*>          if UPLO = 'L', BB(1+i-j,j)    = B(i,j) for j<=i<=min(n,j+kb).
*>
*>          On exit, the factor S from the split Cholesky factorization
*>          B = S**H*S, as returned by ZPBSTF.
*> \endverbatim
*>
*> \param[in] LDBB
*> \verbatim
*>          LDBB is INTEGER
*>          The leading dimension of the array BB.  LDBB >= KB+1.
*> \endverbatim
*>
*> \param[out] Q
*> \verbatim
*>          Q is COMPLEX*16 array, dimension (LDQ, N)
*>          If JOBZ = 'V', the n-by-n matrix used in the reduction of
*>          A*x = (lambda)*B*x to standard form, i.e. C*x = (lambda)*x,
*>          and consequently C 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 = 'N',
*>          LDQ >= 1. If JOBZ = 'V', LDQ >= 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 AP 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').
*> \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)
*>          If INFO = 0, the eigenvalues in ascending order.
*> \endverbatim
*>
*> \param[out] Z
*> \verbatim
*>          Z is COMPLEX*16 array, dimension (LDZ, N)
*>          If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of
*>          eigenvectors, with the i-th column of Z holding the
*>          eigenvector associated with W(i). The eigenvectors are
*>          normalized so that Z**H*B*Z = I.
*>          If JOBZ = 'N', then Z is not referenced.
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*>          LDZ is INTEGER
*>          The leading dimension of the array Z.  LDZ >= 1, and if
*>          JOBZ = 'V', LDZ >= N.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is COMPLEX*16 array, dimension (N)
*> \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, and i is:
*>             <= N:  then i eigenvectors failed to converge.  Their
*>                    indices are stored in array IFAIL.
*>             > N:   if INFO = N + i, for 1 <= i <= N, then ZPBSTF
*>                    returned INFO = i: B is not positive definite.
*>                    The factorization of B could not be completed and
*>                    no eigenvalues or eigenvectors were computed.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date June 2016
*
*> \ingroup complex16OTHEReigen
*
*> \par Contributors:
*  ==================
*>
*>     Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA
*
*  =====================================================================
      SUBROUTINE zhbgvx( JOBZ, RANGE, UPLO, N, KA, KB, AB, LDAB, BB,
     $                   LDBB, Q, LDQ, VL, VU, IL, IU, ABSTOL, M, W, Z,
     $                   LDZ, WORK, RWORK, IWORK, IFAIL, INFO )
*
*  -- LAPACK driver routine (version 3.7.0) --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*     June 2016
*
*     .. Scalar Arguments ..
      CHARACTER          JOBZ, RANGE, UPLO
      INTEGER            IL, INFO, IU, KA, KB, LDAB, LDBB, LDQ, LDZ, M,
     $                   n
      DOUBLE PRECISION   ABSTOL, VL, VU
*     ..
*     .. Array Arguments ..
      INTEGER            IFAIL( * ), IWORK( * )
      DOUBLE PRECISION   RWORK( * ), W( * )
      COMPLEX*16         AB( ldab, * ), BB( ldbb, * ), Q( ldq, * ),
     $                   work( * ), z( ldz, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO
      parameter( zero = 0.0d+0 )
      COMPLEX*16         CZERO, CONE
      parameter( czero = ( 0.0d+0, 0.0d+0 ),
     $                   cone = ( 1.0d+0, 0.0d+0 ) )
*     ..
*     .. Local Scalars ..
      LOGICAL            ALLEIG, INDEIG, TEST, UPPER, VALEIG, WANTZ
      CHARACTER          ORDER, VECT
      INTEGER            I, IINFO, INDD, INDE, INDEE, INDIBL, INDISP,
     $                   indiwk, indrwk, indwrk, itmp1, j, jj, nsplit
      DOUBLE PRECISION   TMP1
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           lsame
*     ..
*     .. External Subroutines ..
      EXTERNAL           dcopy, dstebz, dsterf, xerbla, zcopy, zgemv,
     $                   zhbgst, zhbtrd, zlacpy, zpbstf, zstein, zsteqr,
     $                   zswap
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          min
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      wantz = lsame( jobz, 'V' )
      upper = lsame( uplo, 'U' )
      alleig = lsame( range, 'A' )
      valeig = lsame( range, 'V' )
      indeig = lsame( range, 'I' )
*
      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.( upper .OR. lsame( uplo, 'L' ) ) ) THEN
         info = -3
      ELSE IF( n.LT.0 ) THEN
         info = -4
      ELSE IF( ka.LT.0 ) THEN
         info = -5
      ELSE IF( kb.LT.0 .OR. kb.GT.ka ) THEN
         info = -6
      ELSE IF( ldab.LT.ka+1 ) THEN
         info = -8
      ELSE IF( ldbb.LT.kb+1 ) THEN
         info = -10
      ELSE IF( ldq.LT.1 .OR. ( wantz .AND. ldq.LT.n ) ) THEN
         info = -12
      ELSE
         IF( valeig ) THEN
            IF( n.GT.0 .AND. vu.LE.vl )
     $         info = -14
         ELSE IF( indeig ) THEN
            IF( il.LT.1 .OR. il.GT.max( 1, n ) ) THEN
               info = -15
            ELSE IF ( iu.LT.min( n, il ) .OR. iu.GT.n ) THEN
               info = -16
            END IF
         END IF
      END IF
      IF( info.EQ.0) THEN
         IF( ldz.LT.1 .OR. ( wantz .AND. ldz.LT.n ) ) THEN
            info = -21
         END IF
      END IF
*
      IF( info.NE.0 ) THEN
         CALL xerbla( 'ZHBGVX', -info )
         RETURN
      END IF
*
*     Quick return if possible
*
      m = 0
      IF( n.EQ.0 )
     $   RETURN
*
*     Form a split Cholesky factorization of B.
*
      CALL zpbstf( uplo, n, kb, bb, ldbb, info )
      IF( info.NE.0 ) THEN
         info = n + info
         RETURN
      END IF
*
*     Transform problem to standard eigenvalue problem.
*
      CALL zhbgst( jobz, uplo, n, ka, kb, ab, ldab, bb, ldbb, q, ldq,
     $             work, rwork, iinfo )
*
*     Solve the standard eigenvalue problem.
*     Reduce Hermitian band matrix to tridiagonal form.
*
      indd = 1
      inde = indd + n
      indrwk = inde + n
      indwrk = 1
      IF( wantz ) THEN
         vect = 'U'
      ELSE
         vect = 'N'
      END IF
      CALL zhbtrd( vect, uplo, n, ka, 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
         CALL dcopy( n-1, rwork( inde ), 1, rwork( indee ), 1 )
         IF( .NOT.wantz ) THEN
            CALL dsterf( n, w, rwork( indee ), info )
         ELSE
            CALL zlacpy( 'A', n, n, q, ldq, z, ldz )
            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,
*     call ZSTEIN.
*
      IF( wantz ) THEN
         order = 'B'
      ELSE
         order = 'E'
      END IF
      indibl = 1
      indisp = indibl + n
      indiwk = indisp + n
      CALL dstebz( range, order, n, vl, vu, il, iu, abstol,
     $             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
*
   30 CONTINUE
*
*     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 ZHBGVX
*
      END