hbevx(3)

NAME
hbevx - {hb,sb}evx: eig, bisection

SYNOPSIS

  Functions

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)
CHBEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices
subroutine dsbevx (jobz, range, uplo, n, kd, ab, ldab, q, ldq, vl, vu, il, iu, abstol, m, w, z, ldz, work, iwork, ifail, info)
DSBEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices
subroutine ssbevx (jobz, range, uplo, n, kd, ab, ldab, q, ldq, vl, vu, il, iu, abstol, m, w, z, ldz, work, iwork, ifail, info)
SSBEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices
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)
ZHBEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices

Detailed Description

Function Documentation

  subroutine chbevx (character jobz, character range, character uplo, integer n, integer kd, complex, dimension( ldab, * ) ab, integer ldab, complex, dimension( ldq, * ) q, integer ldq, real vl, real vu, integer il, integer iu, real abstol, integer m, real, dimension( * ) w, complex, dimension( ldz, * ) z, integer ldz, complex, dimension( * ) work, real, dimension( * ) rwork, integer, dimension( * ) iwork, integer, dimension( * ) ifail, integer info)

CHBEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices

Purpose:

    !>
    !> 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.
    !>

Parameters
    JOBZ

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

    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.
    !>

    UPLO

    !>          UPLO is CHARACTER*1
    !>          = 'U':  Upper triangle of A is stored;
    !>          = 'L':  Lower triangle of A is stored.
    !>

    N

    !>          N is INTEGER
    !>          The order of the matrix A.  N >= 0.
    !>

    KD

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

    AB

    !>          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.
    !>

    LDAB

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

    Q

    !>          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.
    !>

    LDQ

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

    VL

    !>          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'.
    !>

    VU

    !>          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'.
    !>

    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'.
    !>

    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'.
    !>

    ABSTOL

    !>          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  by Demmel and
    !>          Kahan, LAPACK Working Note #3.
    !>

    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.
    !>

    W

    !>          W is REAL array, dimension (N)
    !>          The first M elements contain the selected eigenvalues in
    !>          ascending order.
    !>

    Z

    !>          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.
    !>

    LDZ

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

    WORK

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

    RWORK

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

    IWORK

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

    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.
    !>

    INFO

    !>          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.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

  subroutine dsbevx (character jobz, character range, character uplo, integer n, integer kd, double precision, dimension( ldab, * ) ab, integer ldab, double precision, dimension( ldq, * ) q, integer ldq, double precision vl, double precision vu, integer il, integer iu, double precision abstol, integer m, double precision, dimension( * ) w, double precision, dimension( ldz, * ) z, integer ldz, double precision, dimension( * ) work, integer, dimension( * ) iwork, integer, dimension( * ) ifail, integer info)

DSBEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices

Purpose:

    !>
    !> DSBEVX computes selected eigenvalues and, optionally, eigenvectors
    !> of a real symmetric 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.
    !>

Parameters
    JOBZ

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

    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.
    !>

    UPLO

    !>          UPLO is CHARACTER*1
    !>          = 'U':  Upper triangle of A is stored;
    !>          = 'L':  Lower triangle of A is stored.
    !>

    N

    !>          N is INTEGER
    !>          The order of the matrix A.  N >= 0.
    !>

    KD

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

    AB

    !>          AB is DOUBLE PRECISION array, dimension (LDAB, N)
    !>          On entry, the upper or lower triangle of the symmetric 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.  If UPLO = 'U', the first
    !>          superdiagonal and the diagonal of the tridiagonal matrix T
    !>          are returned in rows KD and KD+1 of AB, and if UPLO = 'L',
    !>          the diagonal and first subdiagonal of T are returned in the
    !>          first two rows of AB.
    !>

    LDAB

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

    Q

    !>          Q is DOUBLE PRECISION array, dimension (LDQ, N)
    !>          If JOBZ = 'V', the N-by-N orthogonal matrix used in the
    !>                         reduction to tridiagonal form.
    !>          If JOBZ = 'N', the array Q is not referenced.
    !>

    LDQ

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

    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'.
    !>

    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'.
    !>

    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'.
    !>

    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'.
    !>

    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 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  by Demmel and
    !>          Kahan, LAPACK Working Note #3.
    !>

    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.
    !>

    W

    !>          W is DOUBLE PRECISION array, dimension (N)
    !>          The first M elements contain the selected eigenvalues in
    !>          ascending order.
    !>

    Z

    !>          Z is DOUBLE PRECISION 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

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

    WORK

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

    IWORK

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

    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.
    !>

    INFO

    !>          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.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

  subroutine ssbevx (character jobz, character range, character uplo, integer n, integer kd, real, dimension( ldab, * ) ab, integer ldab, real, dimension( ldq, * ) q, integer ldq, real vl, real vu, integer il, integer iu, real abstol, integer m, real, dimension( * ) w, real, dimension( ldz, * ) z, integer ldz, real, dimension( * ) work, integer, dimension( * ) iwork, integer, dimension( * ) ifail, integer info)

SSBEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices

Purpose:

    !>
    !> SSBEVX computes selected eigenvalues and, optionally, eigenvectors
    !> of a real symmetric 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.
    !>

Parameters
    JOBZ

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

    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.
    !>

    UPLO

    !>          UPLO is CHARACTER*1
    !>          = 'U':  Upper triangle of A is stored;
    !>          = 'L':  Lower triangle of A is stored.
    !>

    N

    !>          N is INTEGER
    !>          The order of the matrix A.  N >= 0.
    !>

    KD

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

    AB

    !>          AB is REAL array, dimension (LDAB, N)
    !>          On entry, the upper or lower triangle of the symmetric 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.  If UPLO = 'U', the first
    !>          superdiagonal and the diagonal of the tridiagonal matrix T
    !>          are returned in rows KD and KD+1 of AB, and if UPLO = 'L',
    !>          the diagonal and first subdiagonal of T are returned in the
    !>          first two rows of AB.
    !>

    LDAB

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

    Q

    !>          Q is REAL array, dimension (LDQ, N)
    !>          If JOBZ = 'V', the N-by-N orthogonal matrix used in the
    !>                         reduction to tridiagonal form.
    !>          If JOBZ = 'N', the array Q is not referenced.
    !>

    LDQ

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

    VL

    !>          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'.
    !>

    VU

    !>          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'.
    !>

    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'.
    !>

    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'.
    !>

    ABSTOL

    !>          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  by Demmel and
    !>          Kahan, LAPACK Working Note #3.
    !>

    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.
    !>

    W

    !>          W is REAL array, dimension (N)
    !>          The first M elements contain the selected eigenvalues in
    !>          ascending order.
    !>

    Z

    !>          Z is REAL 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

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

    WORK

    !>          WORK is REAL array, dimension (7*N)
    !>

    IWORK

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

    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.
    !>

    INFO

    !>          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.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

  subroutine zhbevx (character jobz, character range, character uplo, integer n, integer kd, complex*16, dimension( ldab, * ) ab, integer ldab, 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)

ZHBEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices

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.
    !>

Parameters
    JOBZ

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

    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.
    !>

    UPLO

    !>          UPLO is CHARACTER*1
    !>          = 'U':  Upper triangle of A is stored;
    !>          = 'L':  Lower triangle of A is stored.
    !>

    N

    !>          N is INTEGER
    !>          The order of the matrix A.  N >= 0.
    !>

    KD

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

    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 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

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

    Q

    !>          Q is 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

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

    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'.
    !>

    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'.
    !>

    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'.
    !>

    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'.
    !>

    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 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  by Demmel and
    !>          Kahan, LAPACK Working Note #3.
    !>

    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.
    !>

    W

    !>          W is DOUBLE PRECISION array, dimension (N)
    !>          The first M elements contain the selected eigenvalues in
    !>          ascending order.
    !>

    Z

    !>          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.
    !>

    LDZ

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

    WORK

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

    RWORK

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

    IWORK

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

    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.
    !>

    INFO

    !>          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.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Author

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