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## zhbgvx()

subroutine zhbgvx | ( | character | jobz,
|  | character | range,
|  | character | uplo,
|  | integer | n,
|  | integer | ka,
|  | integer | kb,
|  | complex*16, dimension( ldab, * ) | ab,
|  | integer | ldab,
|  | complex*16, dimension( ldbb, * ) | bb,
|  | integer | ldbb,
|  | complex*16, dimension( ldq, * ) | q,
|  | integer | ldq,
|  | double precision | vl,
|  | double precision | vu,
|  | integer | il,
|  | integer | iu,
|  | double precision | abstol,
|  | integer | m,
|  | double precision, dimension( * ) | w,
|  | complex*16, dimension( ldz, * ) | z,
|  | integer | ldz,
|  | complex*16, dimension( * ) | work,
|  | double precision, dimension( * ) | rwork,
|  | integer, dimension( * ) | iwork,
|  | integer, dimension( * ) | ifail,
|  | integer | info )

ZHBGVX

Download ZHBGVX + dependencies [TGZ] [ZIP] [TXT]

Purpose:

      !>
      !> 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.
      !>

Parameters

  [in]

  JOBZ

      !>          JOBZ is CHARACTER*1
      !>          = 'N':  Compute eigenvalues only;
      !>          = 'V':  Compute eigenvalues and eigenvectors.
      !>

  [in]

  RANGE

      !>          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.
      !>

  [in]

  UPLO

      !>          UPLO is CHARACTER*1
      !>          = 'U':  Upper triangles of A and B are stored;
      !>          = 'L':  Lower triangles of A and B are stored.
      !>

  [in]

  N

      !>          N is INTEGER
      !>          The order of the matrices A and B.  N >= 0.
      !>

  [in]

  KA

      !>          KA is INTEGER
      !>          The number of superdiagonals of the matrix A if UPLO = 'U',
      !>          or the number of subdiagonals if UPLO = 'L'. KA >= 0.
      !>

  [in]

  KB

      !>          KB is INTEGER
      !>          The number of superdiagonals of the matrix B if UPLO = 'U',
      !>          or the number of subdiagonals if UPLO = 'L'. KB >= 0.
      !>

  [in,out]

  AB

      !>          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.
      !>

  [in]

  LDAB

      !>          LDAB is INTEGER
      !>          The leading dimension of the array AB.  LDAB >= KA+1.
      !>

  [in,out]

  BB

      !>          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.
      !>

  [in]

  LDBB

      !>          LDBB is INTEGER
      !>          The leading dimension of the array BB.  LDBB >= KB+1.
      !>

  [out]

  Q

      !>          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.
      !>

  [in]

  LDQ

      !>          LDQ is INTEGER
      !>          The leading dimension of the array Q.  If JOBZ = 'N',
      !>          LDQ >= 1. If JOBZ = 'V', LDQ >= max(1,N).
      !>

  [in]

  VL

      !>          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'.
      !>

  [in]

  VU

      !>          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'.
      !>

  [in]

  IL

      !>          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'.
      !>

  [in]

  IU

      !>          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'.
      !>

  [in]

  ABSTOL

      !>          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').
      !>

  [out]

  M

      !>          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.
      !>

  [out]

  W

      !>          W is DOUBLE PRECISION array, dimension (N)
      !>          If INFO = 0, the eigenvalues in ascending order.
      !>

  [out]

  Z

      !>          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.
      !>

  [in]

  LDZ

      !>          LDZ is INTEGER
      !>          The leading dimension of the array Z.  LDZ >= 1, and if
      !>          JOBZ = 'V', LDZ >= N.
      !>

  [out]

  WORK

      !>          WORK is COMPLEX*16 array, dimension (N)
      !>

  [out]

  RWORK

      !>          RWORK is DOUBLE PRECISION array, dimension (7*N)
      !>

  [out]

  IWORK

      !>          IWORK is INTEGER array, dimension (5*N)
      !>

  [out]

  IFAIL

      !>          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.
      !>

  [out]

  INFO

      !>          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.
      !>

Author
  Univ. of Tennessee
  Univ. of California Berkeley
  Univ. of Colorado Denver
  NAG Ltd.

Contributors:
  Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA