hegvd(3)

NAME
hegvd - {he,sy}gvd: eig, divide and conquer

SYNOPSIS

  Functions

subroutine chegvd (itype, jobz, uplo, n, a, lda, b, ldb, w, work, lwork, rwork, lrwork, iwork, liwork, info)
CHEGVD
subroutine dsygvd (itype, jobz, uplo, n, a, lda, b, ldb, w, work, lwork, iwork, liwork, info)
DSYGVD
subroutine ssygvd (itype, jobz, uplo, n, a, lda, b, ldb, w, work, lwork, iwork, liwork, info)
SSYGVD
subroutine zhegvd (itype, jobz, uplo, n, a, lda, b, ldb, w, work, lwork, rwork, lrwork, iwork, liwork, info)
ZHEGVD

Detailed Description

Function Documentation

  subroutine chegvd (integer itype, character jobz, character uplo, integer n, complex, dimension( lda, * ) a, integer lda, complex, dimension( ldb, * ) b, integer ldb, real, dimension( * ) w, complex, dimension( * ) work, integer lwork, real, dimension( * ) rwork, integer lrwork, integer, dimension( * ) iwork, integer liwork, integer info)

CHEGVD

Purpose:

    !>
    !> CHEGVD computes all the eigenvalues, and optionally, the 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.
    !> If eigenvectors are desired, it uses a divide and conquer algorithm.
    !>
    !>

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.
    !>

    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, if JOBZ = 'V', then if INFO = 0, A contains the
    !>          matrix Z of eigenvectors.  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 JOBZ = 'N', then on exit the upper triangle (if UPLO='U')
    !>          or the lower triangle (if UPLO='L') 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).
    !>

    W

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

    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.
    !>          If N <= 1,                LWORK >= 1.
    !>          If JOBZ  = 'N' and N > 1, LWORK >= N + 1.
    !>          If JOBZ  = 'V' and N > 1, LWORK >= 2*N + N**2.
    !>
    !>          If LWORK = -1, then a workspace query is assumed; the routine
    !>          only calculates the optimal sizes of the WORK, RWORK and
    !>          IWORK arrays, returns these values as the first entries of
    !>          the WORK, RWORK and IWORK arrays, and no error message
    !>          related to LWORK or LRWORK or LIWORK is issued by XERBLA.
    !>

    RWORK

    !>          RWORK is REAL array, dimension (MAX(1,LRWORK))
    !>          On exit, if INFO = 0, RWORK(1) returns the optimal LRWORK.
    !>

    LRWORK

    !>          LRWORK is INTEGER
    !>          The dimension of the array RWORK.
    !>          If N <= 1,                LRWORK >= 1.
    !>          If JOBZ  = 'N' and N > 1, LRWORK >= N.
    !>          If JOBZ  = 'V' and N > 1, LRWORK >= 1 + 5*N + 2*N**2.
    !>
    !>          If LRWORK = -1, then a workspace query is assumed; the
    !>          routine only calculates the optimal sizes of the WORK, RWORK
    !>          and IWORK arrays, returns these values as the first entries
    !>          of the WORK, RWORK and IWORK arrays, and no error message
    !>          related to LWORK or LRWORK or LIWORK is issued by XERBLA.
    !>

    IWORK

    !>          IWORK is INTEGER array, dimension (MAX(1,LIWORK))
    !>          On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
    !>

    LIWORK

    !>          LIWORK is INTEGER
    !>          The dimension of the array IWORK.
    !>          If N <= 1,                LIWORK >= 1.
    !>          If JOBZ  = 'N' and N > 1, LIWORK >= 1.
    !>          If JOBZ  = 'V' and N > 1, LIWORK >= 3 + 5*N.
    !>
    !>          If LIWORK = -1, then a workspace query is assumed; the
    !>          routine only calculates the optimal sizes of the WORK, RWORK
    !>          and IWORK arrays, returns these values as the first entries
    !>          of the WORK, RWORK and IWORK arrays, and no error message
    !>          related to LWORK or LRWORK or LIWORK is issued by XERBLA.
    !>

    INFO

    !>          INFO is INTEGER
    !>          = 0:  successful exit
    !>          < 0:  if INFO = -i, the i-th argument had an illegal value
    !>          > 0:  CPOTRF or CHEEVD returned an error code:
    !>             <= N:  if INFO = i and JOBZ = 'N', then the algorithm
    !>                    failed to converge; i off-diagonal elements of an
    !>                    intermediate tridiagonal form did not converge to
    !>                    zero;
    !>                    if INFO = i and JOBZ = 'V', then the algorithm
    !>                    failed to compute an eigenvalue while working on
    !>                    the submatrix lying in rows and columns INFO/(N+1)
    !>                    through mod(INFO,N+1);
    !>             > 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.

Further Details:

    !>
    !>  Modified so that no backsubstitution is performed if CHEEVD fails to
    !>  converge (NEIG in old code could be greater than N causing out of
    !>  bounds reference to A - reported by Ralf Meyer).  Also corrected the
    !>  description of INFO and the test on ITYPE. Sven, 16 Feb 05.
    !>

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

  subroutine dsygvd (integer itype, character jobz, character uplo, integer n, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( ldb, * ) b, integer ldb, double precision, dimension( * ) w, double precision, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer liwork, integer info)

DSYGVD

Purpose:

    !>
    !> DSYGVD computes all the eigenvalues, and optionally, the 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.
    !> If eigenvectors are desired, it uses a divide and conquer algorithm.
    !>
    !>

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.
    !>

    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 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, if JOBZ = 'V', then if INFO = 0, A contains the
    !>          matrix Z of eigenvectors.  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 JOBZ = 'N', then on exit the upper triangle (if UPLO='U')
    !>          or the lower triangle (if UPLO='L') 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).
    !>

    W

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

    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 dimension of the array WORK.
    !>          If N <= 1,               LWORK >= 1.
    !>          If JOBZ = 'N' and N > 1, LWORK >= 2*N+1.
    !>          If JOBZ = 'V' and N > 1, LWORK >= 1 + 6*N + 2*N**2.
    !>
    !>          If LWORK = -1, then a workspace query is assumed; the routine
    !>          only calculates the optimal sizes of the WORK and IWORK
    !>          arrays, returns these values as the first entries of the WORK
    !>          and IWORK arrays, and no error message related to LWORK or
    !>          LIWORK is issued by XERBLA.
    !>

    IWORK

    !>          IWORK is INTEGER array, dimension (MAX(1,LIWORK))
    !>          On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
    !>

    LIWORK

    !>          LIWORK is INTEGER
    !>          The dimension of the array IWORK.
    !>          If N <= 1,                LIWORK >= 1.
    !>          If JOBZ  = 'N' and N > 1, LIWORK >= 1.
    !>          If JOBZ  = 'V' and N > 1, LIWORK >= 3 + 5*N.
    !>
    !>          If LIWORK = -1, then a workspace query is assumed; the
    !>          routine only calculates the optimal sizes of the WORK and
    !>          IWORK arrays, returns these values as the first entries of
    !>          the WORK and IWORK arrays, and no error message related to
    !>          LWORK or LIWORK is issued by XERBLA.
    !>

    INFO

    !>          INFO is INTEGER
    !>          = 0:  successful exit
    !>          < 0:  if INFO = -i, the i-th argument had an illegal value
    !>          > 0:  DPOTRF or DSYEVD returned an error code:
    !>             <= N:  if INFO = i and JOBZ = 'N', then the algorithm
    !>                    failed to converge; i off-diagonal elements of an
    !>                    intermediate tridiagonal form did not converge to
    !>                    zero;
    !>                    if INFO = i and JOBZ = 'V', then the algorithm
    !>                    failed to compute an eigenvalue while working on
    !>                    the submatrix lying in rows and columns INFO/(N+1)
    !>                    through mod(INFO,N+1);
    !>             > 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.

Further Details:

    !>
    !>  Modified so that no backsubstitution is performed if DSYEVD fails to
    !>  converge (NEIG in old code could be greater than N causing out of
    !>  bounds reference to A - reported by Ralf Meyer).  Also corrected the
    !>  description of INFO and the test on ITYPE. Sven, 16 Feb 05.
    !>

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

  subroutine ssygvd (integer itype, character jobz, character uplo, integer n, real, dimension( lda, * ) a, integer lda, real, dimension( ldb, * ) b, integer ldb, real, dimension( * ) w, real, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer liwork, integer info)

SSYGVD

Purpose:

    !>
    !> SSYGVD computes all the eigenvalues, and optionally, the 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.
    !> If eigenvectors are desired, it uses a divide and conquer algorithm.
    !>
    !>

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.
    !>

    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 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, if JOBZ = 'V', then if INFO = 0, A contains the
    !>          matrix Z of eigenvectors.  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 JOBZ = 'N', then on exit the upper triangle (if UPLO='U')
    !>          or the lower triangle (if UPLO='L') 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).
    !>

    W

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

    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 dimension of the array WORK.
    !>          If N <= 1,               LWORK >= 1.
    !>          If JOBZ = 'N' and N > 1, LWORK >= 2*N+1.
    !>          If JOBZ = 'V' and N > 1, LWORK >= 1 + 6*N + 2*N**2.
    !>
    !>          If LWORK = -1, then a workspace query is assumed; the routine
    !>          only calculates the optimal sizes of the WORK and IWORK
    !>          arrays, returns these values as the first entries of the WORK
    !>          and IWORK arrays, and no error message related to LWORK or
    !>          LIWORK is issued by XERBLA.
    !>

    IWORK

    !>          IWORK is INTEGER array, dimension (MAX(1,LIWORK))
    !>          On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
    !>

    LIWORK

    !>          LIWORK is INTEGER
    !>          The dimension of the array IWORK.
    !>          If N <= 1,                LIWORK >= 1.
    !>          If JOBZ  = 'N' and N > 1, LIWORK >= 1.
    !>          If JOBZ  = 'V' and N > 1, LIWORK >= 3 + 5*N.
    !>
    !>          If LIWORK = -1, then a workspace query is assumed; the
    !>          routine only calculates the optimal sizes of the WORK and
    !>          IWORK arrays, returns these values as the first entries of
    !>          the WORK and IWORK arrays, and no error message related to
    !>          LWORK or LIWORK is issued by XERBLA.
    !>

    INFO

    !>          INFO is INTEGER
    !>          = 0:  successful exit
    !>          < 0:  if INFO = -i, the i-th argument had an illegal value
    !>          > 0:  SPOTRF or SSYEVD returned an error code:
    !>             <= N:  if INFO = i and JOBZ = 'N', then the algorithm
    !>                    failed to converge; i off-diagonal elements of an
    !>                    intermediate tridiagonal form did not converge to
    !>                    zero;
    !>                    if INFO = i and JOBZ = 'V', then the algorithm
    !>                    failed to compute an eigenvalue while working on
    !>                    the submatrix lying in rows and columns INFO/(N+1)
    !>                    through mod(INFO,N+1);
    !>             > 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.

Further Details:

    !>
    !>  Modified so that no backsubstitution is performed if SSYEVD fails to
    !>  converge (NEIG in old code could be greater than N causing out of
    !>  bounds reference to A - reported by Ralf Meyer).  Also corrected the
    !>  description of INFO and the test on ITYPE. Sven, 16 Feb 05.
    !>

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

  subroutine zhegvd (integer itype, character jobz, character uplo, integer n, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( ldb, * ) b, integer ldb, double precision, dimension( * ) w, complex*16, dimension( * ) work, integer lwork, double precision, dimension( * ) rwork, integer lrwork, integer, dimension( * ) iwork, integer liwork, integer info)

ZHEGVD

Purpose:

    !>
    !> ZHEGVD computes all the eigenvalues, and optionally, the 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.
    !> If eigenvectors are desired, it uses a divide and conquer algorithm.
    !>
    !>

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.
    !>

    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, if JOBZ = 'V', then if INFO = 0, A contains the
    !>          matrix Z of eigenvectors.  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 JOBZ = 'N', then on exit the upper triangle (if UPLO='U')
    !>          or the lower triangle (if UPLO='L') 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).
    !>

    W

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

    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.
    !>          If N <= 1,                LWORK >= 1.
    !>          If JOBZ  = 'N' and N > 1, LWORK >= N + 1.
    !>          If JOBZ  = 'V' and N > 1, LWORK >= 2*N + N**2.
    !>
    !>          If LWORK = -1, then a workspace query is assumed; the routine
    !>          only calculates the optimal sizes of the WORK, RWORK and
    !>          IWORK arrays, returns these values as the first entries of
    !>          the WORK, RWORK and IWORK arrays, and no error message
    !>          related to LWORK or LRWORK or LIWORK is issued by XERBLA.
    !>

    RWORK

    !>          RWORK is DOUBLE PRECISION array, dimension (MAX(1,LRWORK))
    !>          On exit, if INFO = 0, RWORK(1) returns the optimal LRWORK.
    !>

    LRWORK

    !>          LRWORK is INTEGER
    !>          The dimension of the array RWORK.
    !>          If N <= 1,                LRWORK >= 1.
    !>          If JOBZ  = 'N' and N > 1, LRWORK >= N.
    !>          If JOBZ  = 'V' and N > 1, LRWORK >= 1 + 5*N + 2*N**2.
    !>
    !>          If LRWORK = -1, then a workspace query is assumed; the
    !>          routine only calculates the optimal sizes of the WORK, RWORK
    !>          and IWORK arrays, returns these values as the first entries
    !>          of the WORK, RWORK and IWORK arrays, and no error message
    !>          related to LWORK or LRWORK or LIWORK is issued by XERBLA.
    !>

    IWORK

    !>          IWORK is INTEGER array, dimension (MAX(1,LIWORK))
    !>          On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
    !>

    LIWORK

    !>          LIWORK is INTEGER
    !>          The dimension of the array IWORK.
    !>          If N <= 1,                LIWORK >= 1.
    !>          If JOBZ  = 'N' and N > 1, LIWORK >= 1.
    !>          If JOBZ  = 'V' and N > 1, LIWORK >= 3 + 5*N.
    !>
    !>          If LIWORK = -1, then a workspace query is assumed; the
    !>          routine only calculates the optimal sizes of the WORK, RWORK
    !>          and IWORK arrays, returns these values as the first entries
    !>          of the WORK, RWORK and IWORK arrays, and no error message
    !>          related to LWORK or LRWORK or LIWORK is issued by XERBLA.
    !>

    INFO

    !>          INFO is INTEGER
    !>          = 0:  successful exit
    !>          < 0:  if INFO = -i, the i-th argument had an illegal value
    !>          > 0:  ZPOTRF or ZHEEVD returned an error code:
    !>             <= N:  if INFO = i and JOBZ = 'N', then the algorithm
    !>                    failed to converge; i off-diagonal elements of an
    !>                    intermediate tridiagonal form did not converge to
    !>                    zero;
    !>                    if INFO = i and JOBZ = 'V', then the algorithm
    !>                    failed to compute an eigenvalue while working on
    !>                    the submatrix lying in rows and columns INFO/(N+1)
    !>                    through mod(INFO,N+1);
    !>             > 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.

Further Details:

    !>
    !>  Modified so that no backsubstitution is performed if ZHEEVD fails to
    !>  converge (NEIG in old code could be greater than N causing out of
    !>  bounds reference to A - reported by Ralf Meyer).  Also corrected the
    !>  description of INFO and the test on ITYPE. Sven, 16 Feb 05.
    !>

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

Author

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