hpgvx(3)

NAME
hpgvx - {hp,sp}gvx: eig, bisection

SYNOPSIS

  Functions

subroutine chpgvx (itype, jobz, range, uplo, n, ap, bp, vl, vu, il, iu, abstol, m, w, z, ldz, work, rwork, iwork, ifail, info)
CHPGVX
subroutine dspgvx (itype, jobz, range, uplo, n, ap, bp, vl, vu, il, iu, abstol, m, w, z, ldz, work, iwork, ifail, info)
DSPGVX
subroutine sspgvx (itype, jobz, range, uplo, n, ap, bp, vl, vu, il, iu, abstol, m, w, z, ldz, work, iwork, ifail, info)
SSPGVX
subroutine zhpgvx (itype, jobz, range, uplo, n, ap, bp, vl, vu, il, iu, abstol, m, w, z, ldz, work, rwork, iwork, ifail, info)
ZHPGVX

Detailed Description

Function Documentation

  subroutine chpgvx (integer itype, character jobz, character range, character uplo, integer n, complex, dimension( * ) ap, complex, dimension( * ) bp, 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)

CHPGVX

Purpose:

    !>
    !> CHPGVX computes selected eigenvalues and, optionally, eigenvectors
    !> of a complex generalized Hermitian-definite eigenproblem, of the form
    !> A*x=(lambda)*B*x,  A*Bx=(lambda)*x,  or B*A*x=(lambda)*x.  Here A and
    !> B are assumed to be Hermitian, stored in packed format, and B is also
    !> positive definite.  Eigenvalues and eigenvectors can be selected by
    !> specifying either a range of values or a range of indices for the
    !> desired eigenvalues.
    !>

Parameters
    ITYPE

    !>          ITYPE is INTEGER
    !>          Specifies the problem type to be solved:
    !>          = 1:  A*x = (lambda)*B*x
    !>          = 2:  A*B*x = (lambda)*x
    !>          = 3:  B*A*x = (lambda)*x
    !>

    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 triangles of A and B are stored;
    !>          = 'L':  Lower triangles of A and B are stored.
    !>

    N

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

    AP

    !>          AP is COMPLEX array, dimension (N*(N+1)/2)
    !>          On entry, the upper or lower triangle of the Hermitian matrix
    !>          A, packed columnwise in a linear array.  The j-th column of A
    !>          is stored in the array AP as follows:
    !>          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
    !>          if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
    !>
    !>          On exit, the contents of AP are destroyed.
    !>

    BP

    !>          BP is COMPLEX array, dimension (N*(N+1)/2)
    !>          On entry, the upper or lower triangle of the Hermitian matrix
    !>          B, packed columnwise in a linear array.  The j-th column of B
    !>          is stored in the array BP as follows:
    !>          if UPLO = 'U', BP(i + (j-1)*j/2) = B(i,j) for 1<=i<=j;
    !>          if UPLO = 'L', BP(i + (j-1)*(2*n-j)/2) = B(i,j) for j<=i<=n.
    !>
    !>          On exit, the triangular factor U or L from the Cholesky
    !>          factorization B = U**H*U or B = L*L**H, in the same storage
    !>          format as B.
    !>

    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 AP 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').
    !>

    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)
    !>          On normal exit, the first M elements contain the selected
    !>          eigenvalues in ascending order.
    !>

    Z

    !>          Z is COMPLEX array, dimension (LDZ, N)
    !>          If JOBZ = 'N', then Z is not referenced.
    !>          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).
    !>          The eigenvectors are normalized as follows:
    !>          if ITYPE = 1 or 2, Z**H*B*Z = I;
    !>          if ITYPE = 3, Z**H*inv(B)*Z = 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.
    !>          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 (2*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:  CPPTRF or CHPEVX returned an error code:
    !>             <= N:  if INFO = i, CHPEVX failed to converge;
    !>                    i eigenvectors failed to converge.  Their indices
    !>                    are stored in array IFAIL.
    !>             > N:   if INFO = N + i, for 1 <= i <= n, then the leading
    !>                    principal minor of order i of B is not positive.
    !>                    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

  subroutine dspgvx (integer itype, character jobz, character range, character uplo, integer n, double precision, dimension( * ) ap, double precision, dimension( * ) bp, 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)

DSPGVX

Purpose:

    !>
    !> DSPGVX computes selected eigenvalues, and optionally, eigenvectors
    !> of a real generalized symmetric-definite eigenproblem, of the form
    !> A*x=(lambda)*B*x,  A*Bx=(lambda)*x,  or B*A*x=(lambda)*x.  Here A
    !> and B are assumed to be symmetric, stored in packed storage, and B
    !> is also positive definite.  Eigenvalues and eigenvectors can be
    !> selected by specifying either a range of values or a range of indices
    !> for the desired eigenvalues.
    !>

Parameters
    ITYPE

    !>          ITYPE is INTEGER
    !>          Specifies the problem type to be solved:
    !>          = 1:  A*x = (lambda)*B*x
    !>          = 2:  A*B*x = (lambda)*x
    !>          = 3:  B*A*x = (lambda)*x
    !>

    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 and B are stored;
    !>          = 'L':  Lower triangle of A and B are stored.
    !>

    N

    !>          N is INTEGER
    !>          The order of the matrix pencil (A,B).  N >= 0.
    !>

    AP

    !>          AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
    !>          On entry, the upper or lower triangle of the symmetric matrix
    !>          A, packed columnwise in a linear array.  The j-th column of A
    !>          is stored in the array AP as follows:
    !>          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
    !>          if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
    !>
    !>          On exit, the contents of AP are destroyed.
    !>

    BP

    !>          BP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
    !>          On entry, the upper or lower triangle of the symmetric matrix
    !>          B, packed columnwise in a linear array.  The j-th column of B
    !>          is stored in the array BP as follows:
    !>          if UPLO = 'U', BP(i + (j-1)*j/2) = B(i,j) for 1<=i<=j;
    !>          if UPLO = 'L', BP(i + (j-1)*(2*n-j)/2) = B(i,j) for j<=i<=n.
    !>
    !>          On exit, the triangular factor U or L from the Cholesky
    !>          factorization B = U**T*U or B = L*L**T, in the same storage
    !>          format as B.
    !>

    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 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').
    !>

    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)
    !>          On normal exit, the first M elements contain the selected
    !>          eigenvalues in ascending order.
    !>

    Z

    !>          Z is DOUBLE PRECISION array, dimension (LDZ, max(1,M))
    !>          If JOBZ = 'N', then Z is not referenced.
    !>          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).
    !>          The eigenvectors are normalized as follows:
    !>          if ITYPE = 1 or 2, Z**T*B*Z = I;
    !>          if ITYPE = 3, Z**T*inv(B)*Z = 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.
    !>          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 (8*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:  DPPTRF or DSPEVX returned an error code:
    !>             <= N:  if INFO = i, DSPEVX failed to converge;
    !>                    i eigenvectors failed to converge.  Their indices
    !>                    are stored in array IFAIL.
    !>             > N:   if INFO = N + i, for 1 <= i <= N, then the leading
    !>                    principal minor of order i of B is not positive.
    !>                    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

  subroutine sspgvx (integer itype, character jobz, character range, character uplo, integer n, real, dimension( * ) ap, real, dimension( * ) bp, 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)

SSPGVX

Purpose:

    !>
    !> SSPGVX computes selected eigenvalues, and optionally, eigenvectors
    !> of a real generalized symmetric-definite eigenproblem, of the form
    !> A*x=(lambda)*B*x,  A*Bx=(lambda)*x,  or B*A*x=(lambda)*x.  Here A
    !> and B are assumed to be symmetric, stored in packed storage, and B
    !> is also positive definite.  Eigenvalues and eigenvectors can be
    !> selected by specifying either a range of values or a range of indices
    !> for the desired eigenvalues.
    !>

Parameters
    ITYPE

    !>          ITYPE is INTEGER
    !>          Specifies the problem type to be solved:
    !>          = 1:  A*x = (lambda)*B*x
    !>          = 2:  A*B*x = (lambda)*x
    !>          = 3:  B*A*x = (lambda)*x
    !>

    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 and B are stored;
    !>          = 'L':  Lower triangle of A and B are stored.
    !>

    N

    !>          N is INTEGER
    !>          The order of the matrix pencil (A,B).  N >= 0.
    !>

    AP

    !>          AP is REAL array, dimension (N*(N+1)/2)
    !>          On entry, the upper or lower triangle of the symmetric matrix
    !>          A, packed columnwise in a linear array.  The j-th column of A
    !>          is stored in the array AP as follows:
    !>          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
    !>          if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
    !>
    !>          On exit, the contents of AP are destroyed.
    !>

    BP

    !>          BP is REAL array, dimension (N*(N+1)/2)
    !>          On entry, the upper or lower triangle of the symmetric matrix
    !>          B, packed columnwise in a linear array.  The j-th column of B
    !>          is stored in the array BP as follows:
    !>          if UPLO = 'U', BP(i + (j-1)*j/2) = B(i,j) for 1<=i<=j;
    !>          if UPLO = 'L', BP(i + (j-1)*(2*n-j)/2) = B(i,j) for j<=i<=n.
    !>
    !>          On exit, the triangular factor U or L from the Cholesky
    !>          factorization B = U**T*U or B = L*L**T, in the same storage
    !>          format as B.
    !>

    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 A 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').
    !>

    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)
    !>          On normal exit, the first M elements contain the selected
    !>          eigenvalues in ascending order.
    !>

    Z

    !>          Z is REAL array, dimension (LDZ, max(1,M))
    !>          If JOBZ = 'N', then Z is not referenced.
    !>          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).
    !>          The eigenvectors are normalized as follows:
    !>          if ITYPE = 1 or 2, Z**T*B*Z = I;
    !>          if ITYPE = 3, Z**T*inv(B)*Z = 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.
    !>          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 (8*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:  SPPTRF or SSPEVX returned an error code:
    !>             <= N:  if INFO = i, SSPEVX failed to converge;
    !>                    i eigenvectors failed to converge.  Their indices
    !>                    are stored in array IFAIL.
    !>             > N:   if INFO = N + i, for 1 <= i <= N, then the leading
    !>                    principal minor of order i of B is not positive.
    !>                    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

  subroutine zhpgvx (integer itype, character jobz, character range, character uplo, integer n, complex*16, dimension( * ) ap, complex*16, dimension( * ) bp, 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)

ZHPGVX

Purpose:

    !>
    !> ZHPGVX computes selected eigenvalues and, optionally, eigenvectors
    !> of a complex generalized Hermitian-definite eigenproblem, of the form
    !> A*x=(lambda)*B*x,  A*Bx=(lambda)*x,  or B*A*x=(lambda)*x.  Here A and
    !> B are assumed to be Hermitian, stored in packed format, and B is also
    !> positive definite.  Eigenvalues and eigenvectors can be selected by
    !> specifying either a range of values or a range of indices for the
    !> desired eigenvalues.
    !>

Parameters
    ITYPE

    !>          ITYPE is INTEGER
    !>          Specifies the problem type to be solved:
    !>          = 1:  A*x = (lambda)*B*x
    !>          = 2:  A*B*x = (lambda)*x
    !>          = 3:  B*A*x = (lambda)*x
    !>

    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 triangles of A and B are stored;
    !>          = 'L':  Lower triangles of A and B are stored.
    !>

    N

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

    AP

    !>          AP is COMPLEX*16 array, dimension (N*(N+1)/2)
    !>          On entry, the upper or lower triangle of the Hermitian matrix
    !>          A, packed columnwise in a linear array.  The j-th column of A
    !>          is stored in the array AP as follows:
    !>          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
    !>          if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
    !>
    !>          On exit, the contents of AP are destroyed.
    !>

    BP

    !>          BP is COMPLEX*16 array, dimension (N*(N+1)/2)
    !>          On entry, the upper or lower triangle of the Hermitian matrix
    !>          B, packed columnwise in a linear array.  The j-th column of B
    !>          is stored in the array BP as follows:
    !>          if UPLO = 'U', BP(i + (j-1)*j/2) = B(i,j) for 1<=i<=j;
    !>          if UPLO = 'L', BP(i + (j-1)*(2*n-j)/2) = B(i,j) for j<=i<=n.
    !>
    !>          On exit, the triangular factor U or L from the Cholesky
    !>          factorization B = U**H*U or B = L*L**H, in the same storage
    !>          format as B.
    !>

    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 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').
    !>

    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)
    !>          On normal exit, the first M elements contain the selected
    !>          eigenvalues in ascending order.
    !>

    Z

    !>          Z is COMPLEX*16 array, dimension (LDZ, N)
    !>          If JOBZ = 'N', then Z is not referenced.
    !>          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).
    !>          The eigenvectors are normalized as follows:
    !>          if ITYPE = 1 or 2, Z**H*B*Z = I;
    !>          if ITYPE = 3, Z**H*inv(B)*Z = 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.
    !>          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 (2*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:  ZPPTRF or ZHPEVX returned an error code:
    !>             <= N:  if INFO = i, ZHPEVX failed to converge;
    !>                    i eigenvectors failed to converge.  Their indices
    !>                    are stored in array IFAIL.
    !>             > N:   if INFO = N + i, for 1 <= i <= n, then the leading
    !>                    principal minor of order i of B is not positive.
    !>                    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

Author

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