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



NAME
       CHEGVX

SYNOPSIS
       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)



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.




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.

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

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

           VL        (input)
                     VL is REAL

           VU        (input)
                     VU is REAL

                     If RANGE='V', the lower and upper bounds of the interval to
                     be searched for eigenvalues. VL < VU.
                     Not referenced if RANGE = 'A' or 'I'.

           IL        (input)
                     IL is INTEGER

           IU        (input)
                     IU is INTEGER

                     If RANGE='I', the indices (in ascending order) of the
                     smallest and largest eigenvalues 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    (input)
                     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         (output)
                     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         (output)
                     W is REAL array, dimension (N)
                     The first M elements contain the selected
                     eigenvalues in ascending order.

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

           WORK      (output)
                     WORK is COMPLEX 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.  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     (output)
                     RWORK is REAL array, dimension (7*N)

           IWORK     (output)
                     IWORK is INTEGER array, dimension (5*N)

           IFAIL     (output)
                     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      (output)
                     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
                               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.



LAPACK routine                  31 October 2017                      CHEGVX(3)