ZHEGVD(3)      LAPACK routine of NEC Numeric Library Collection      ZHEGVD(3)



NAME
       ZHEGVD

SYNOPSIS
       SUBROUTINE ZHEGVD (ITYPE, JOBZ, UPLO, N, A, LDA, B, LDB, W, WORK,
           LWORK, RWORK, LRWORK, IWORK, LIWORK, INFO)



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.

            The divide and conquer algorithm makes very mild assumptions about
            floating point arithmetic. It will work on machines with a guard
            digit in add/subtract, or on those binary machines without guard
            digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or
            Cray-2. It could conceivably fail on hexadecimal or decimal machines
            without guard digits, but we know of none.




ARGUMENTS
           ITYPE     (input)
                     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      (input)
                     JOBZ is CHARACTER*1
                     = 'N':  Compute eigenvalues only;
                     = 'V':  Compute eigenvalues and eigenvectors.

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

           N         (input)
                     N is INTEGER
                     The order of the matrices A and B.  N >= 0.

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

           B         (input/output)
                     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       (input)
                     LDB is INTEGER
                     The leading dimension of the array B.  LDB >= max(1,N).

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

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

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

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

           LIWORK    (input)
                     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      (output)
                     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
                               minor of order i of B is not positive definite.
                               The factorization of B could not be completed and
                               no eigenvalues or eigenvectors were computed.






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.



LAPACK routine                  31 October 2017                      ZHEGVD(3)