hegvx(3)

NAME
hegvx - {he,sy}gvx: eig, bisection

SYNOPSIS

  Functions

subroutine chegvx (itype, jobz, range, uplo, n, a, lda, b, ldb, vl, vu, il, iu, abstol, m, w, z, ldz, work, lwork, rwork, iwork, ifail, info)
CHEGVX
subroutine dsygvx (itype, jobz, range, uplo, n, a, lda, b, ldb, vl, vu, il, iu, abstol, m, w, z, ldz, work, lwork, iwork, ifail, info)
DSYGVX
subroutine ssygvx (itype, jobz, range, uplo, n, a, lda, b, ldb, vl, vu, il, iu, abstol, m, w, z, ldz, work, lwork, iwork, ifail, info)
SSYGVX
subroutine zhegvx (itype, jobz, range, uplo, n, a, lda, b, ldb, vl, vu, il, iu, abstol, m, w, z, ldz, work, lwork, rwork, iwork, ifail, info)
ZHEGVX

Detailed Description

Function Documentation

  subroutine chegvx (integer itype, character jobz, character range, character uplo, integer n, complex, dimension( lda, * ) a, integer lda, complex, dimension( ldb, * ) b, integer ldb, real vl, real vu, integer il, integer iu, real abstol, integer m, real, dimension( * ) w, complex, dimension( ldz, * ) z, integer ldz, complex, dimension( * ) work, integer lwork, real, dimension( * ) rwork, integer, dimension( * ) iwork, integer, dimension( * ) ifail, integer info)

CHEGVX

Purpose:

    !>
    !> CHEGVX 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 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.
    !>

    A

    !>          A is COMPLEX array, dimension (LDA, N)
    !>          On entry, the Hermitian matrix A.  If UPLO = 'U', the
    !>          leading N-by-N upper triangular part of A contains the
    !>          upper triangular part of the matrix A.  If UPLO = 'L',
    !>          the leading N-by-N lower triangular part of A contains
    !>          the lower triangular part of the matrix A.
    !>
    !>          On exit,  the lower triangle (if UPLO='L') or the upper
    !>          triangle (if UPLO='U') of A, including the diagonal, is
    !>          destroyed.
    !>

    LDA

    !>          LDA is INTEGER
    !>          The leading dimension of the array A.  LDA >= max(1,N).
    !>

    B

    !>          B is COMPLEX array, dimension (LDB, N)
    !>          On entry, the Hermitian matrix B.  If UPLO = 'U', the
    !>          leading N-by-N upper triangular part of B contains the
    !>          upper triangular part of the matrix B.  If UPLO = 'L',
    !>          the leading N-by-N lower triangular part of B contains
    !>          the lower triangular part of the matrix B.
    !>
    !>          On exit, if INFO <= N, the part of B containing the matrix is
    !>          overwritten by the triangular factor U or L from the Cholesky
    !>          factorization B = U**H*U or B = L*L**H.
    !>

    LDB

    !>          LDB is INTEGER
    !>          The leading dimension of the array B.  LDB >= 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 C to tridiagonal form, where C is the symmetric
    !>          matrix of the standard symmetric problem to which the
    !>          generalized problem is transformed.
    !>
    !>          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)
    !>          The first M elements contain the selected
    !>          eigenvalues in ascending order.
    !>

    Z

    !>          Z is COMPLEX 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 COMPLEX array, dimension (MAX(1,LWORK))
    !>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
    !>

    LWORK

    !>          LWORK is INTEGER
    !>          The length of the array WORK.  LWORK >= max(1,2*N).
    !>          For optimal efficiency, LWORK >= (NB+1)*N,
    !>          where NB is the blocksize for CHETRD returned by ILAENV.
    !>
    !>          If LWORK = -1, then a workspace query is assumed; the routine
    !>          only calculates the optimal size of the WORK array, returns
    !>          this value as the first entry of the WORK array, and no error
    !>          message related to LWORK is issued by XERBLA.
    !>

    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:  CPOTRF or CHEEVX returned an error code:
    !>             <= N:  if INFO = i, CHEEVX 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 dsygvx (integer itype, character jobz, character range, character uplo, integer n, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( ldb, * ) b, integer ldb, 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 lwork, integer, dimension( * ) iwork, integer, dimension( * ) ifail, integer info)

DSYGVX

Purpose:

    !>
    !> DSYGVX 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 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.
    !>

    A

    !>          A is DOUBLE PRECISION array, dimension (LDA, N)
    !>          On entry, the symmetric matrix A.  If UPLO = 'U', the
    !>          leading N-by-N upper triangular part of A contains the
    !>          upper triangular part of the matrix A.  If UPLO = 'L',
    !>          the leading N-by-N lower triangular part of A contains
    !>          the lower triangular part of the matrix A.
    !>
    !>          On exit, the lower triangle (if UPLO='L') or the upper
    !>          triangle (if UPLO='U') of A, including the diagonal, is
    !>          destroyed.
    !>

    LDA

    !>          LDA is INTEGER
    !>          The leading dimension of the array A.  LDA >= max(1,N).
    !>

    B

    !>          B is DOUBLE PRECISION array, dimension (LDB, N)
    !>          On entry, the symmetric matrix B.  If UPLO = 'U', the
    !>          leading N-by-N upper triangular part of B contains the
    !>          upper triangular part of the matrix B.  If UPLO = 'L',
    !>          the leading N-by-N lower triangular part of B contains
    !>          the lower triangular part of the matrix B.
    !>
    !>          On exit, if INFO <= N, the part of B containing the matrix is
    !>          overwritten by the triangular factor U or L from the Cholesky
    !>          factorization B = U**T*U or B = L*L**T.
    !>

    LDB

    !>          LDB is INTEGER
    !>          The leading dimension of the array B.  LDB >= 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 C to tridiagonal form, where C is the symmetric
    !>          matrix of the standard symmetric problem to which the
    !>          generalized problem is transformed.
    !>
    !>          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 (MAX(1,LWORK))
    !>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
    !>

    LWORK

    !>          LWORK is INTEGER
    !>          The length of the array WORK.  LWORK >= max(1,8*N).
    !>          For optimal efficiency, LWORK >= (NB+3)*N,
    !>          where NB is the blocksize for DSYTRD returned by ILAENV.
    !>
    !>          If LWORK = -1, then a workspace query is assumed; the routine
    !>          only calculates the optimal size of the WORK array, returns
    !>          this value as the first entry of the WORK array, and no error
    !>          message related to LWORK is issued by XERBLA.
    !>

    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:  DPOTRF or DSYEVX returned an error code:
    !>             <= N:  if INFO = i, DSYEVX 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 ssygvx (integer itype, character jobz, character range, character uplo, integer n, real, dimension( lda, * ) a, integer lda, real, dimension( ldb, * ) b, integer ldb, 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 lwork, integer, dimension( * ) iwork, integer, dimension( * ) ifail, integer info)

SSYGVX

Purpose:

    !>
    !> SSYGVX 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 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.
    !>

    A

    !>          A is REAL array, dimension (LDA, N)
    !>          On entry, the symmetric matrix A.  If UPLO = 'U', the
    !>          leading N-by-N upper triangular part of A contains the
    !>          upper triangular part of the matrix A.  If UPLO = 'L',
    !>          the leading N-by-N lower triangular part of A contains
    !>          the lower triangular part of the matrix A.
    !>
    !>          On exit, the lower triangle (if UPLO='L') or the upper
    !>          triangle (if UPLO='U') of A, including the diagonal, is
    !>          destroyed.
    !>

    LDA

    !>          LDA is INTEGER
    !>          The leading dimension of the array A.  LDA >= max(1,N).
    !>

    B

    !>          B is REAL array, dimension (LDB, N)
    !>          On entry, the symmetric matrix B.  If UPLO = 'U', the
    !>          leading N-by-N upper triangular part of B contains the
    !>          upper triangular part of the matrix B.  If UPLO = 'L',
    !>          the leading N-by-N lower triangular part of B contains
    !>          the lower triangular part of the matrix B.
    !>
    !>          On exit, if INFO <= N, the part of B containing the matrix is
    !>          overwritten by the triangular factor U or L from the Cholesky
    !>          factorization B = U**T*U or B = L*L**T.
    !>

    LDB

    !>          LDB is INTEGER
    !>          The leading dimension of the array B.  LDB >= 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 C to tridiagonal form, where C is the symmetric
    !>          matrix of the standard symmetric problem to which the
    !>          generalized problem is transformed.
    !>
    !>          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*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 (MAX(1,LWORK))
    !>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
    !>

    LWORK

    !>          LWORK is INTEGER
    !>          The length of the array WORK.  LWORK >= max(1,8*N).
    !>          For optimal efficiency, LWORK >= (NB+3)*N,
    !>          where NB is the blocksize for SSYTRD returned by ILAENV.
    !>
    !>          If LWORK = -1, then a workspace query is assumed; the routine
    !>          only calculates the optimal size of the WORK array, returns
    !>          this value as the first entry of the WORK array, and no error
    !>          message related to LWORK is issued by XERBLA.
    !>

    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:  SPOTRF or SSYEVX returned an error code:
    !>             <= N:  if INFO = i, SSYEVX 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 zhegvx (integer itype, character jobz, character range, character uplo, integer n, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( ldb, * ) b, integer ldb, 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, integer lwork, double precision, dimension( * ) rwork, integer, dimension( * ) iwork, integer, dimension( * ) ifail, integer info)

ZHEGVX

Purpose:

    !>
    !> ZHEGVX 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 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.
    !>

    A

    !>          A is COMPLEX*16 array, dimension (LDA, N)
    !>          On entry, the Hermitian matrix A.  If UPLO = 'U', the
    !>          leading N-by-N upper triangular part of A contains the
    !>          upper triangular part of the matrix A.  If UPLO = 'L',
    !>          the leading N-by-N lower triangular part of A contains
    !>          the lower triangular part of the matrix A.
    !>
    !>          On exit,  the lower triangle (if UPLO='L') or the upper
    !>          triangle (if UPLO='U') of A, including the diagonal, is
    !>          destroyed.
    !>

    LDA

    !>          LDA is INTEGER
    !>          The leading dimension of the array A.  LDA >= max(1,N).
    !>

    B

    !>          B is COMPLEX*16 array, dimension (LDB, N)
    !>          On entry, the Hermitian matrix B.  If UPLO = 'U', the
    !>          leading N-by-N upper triangular part of B contains the
    !>          upper triangular part of the matrix B.  If UPLO = 'L',
    !>          the leading N-by-N lower triangular part of B contains
    !>          the lower triangular part of the matrix B.
    !>
    !>          On exit, if INFO <= N, the part of B containing the matrix is
    !>          overwritten by the triangular factor U or L from the Cholesky
    !>          factorization B = U**H*U or B = L*L**H.
    !>

    LDB

    !>          LDB is INTEGER
    !>          The leading dimension of the array B.  LDB >= 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 C to tridiagonal form, where C is the symmetric
    !>          matrix of the standard symmetric problem to which the
    !>          generalized problem is transformed.
    !>
    !>          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)
    !>          The first M elements contain the selected
    !>          eigenvalues in ascending order.
    !>

    Z

    !>          Z is COMPLEX*16 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 COMPLEX*16 array, dimension (MAX(1,LWORK))
    !>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
    !>

    LWORK

    !>          LWORK is INTEGER
    !>          The length of the array WORK.  LWORK >= max(1,2*N).
    !>          For optimal efficiency, LWORK >= (NB+1)*N,
    !>          where NB is the blocksize for ZHETRD returned by ILAENV.
    !>
    !>          If LWORK = -1, then a workspace query is assumed; the routine
    !>          only calculates the optimal size of the WORK array, returns
    !>          this value as the first entry of the WORK array, and no error
    !>          message related to LWORK is issued by XERBLA.
    !>

    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:  ZPOTRF or ZHEEVX returned an error code:
    !>             <= N:  if INFO = i, ZHEEVX 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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