Provided by: liblapack-doc_3.3.1-1_all bug

NAME

       LAPACK-3  -  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

SYNOPSIS

       SUBROUTINE CHPGVX( ITYPE,  JOBZ,  RANGE, UPLO, N, AP, BP, VL, VU, IL, IU, ABSTOL, M, W, Z,
                          LDZ, WORK, RWORK, IWORK, IFAIL, INFO )

           CHARACTER      JOBZ, RANGE, UPLO

           INTEGER        IL, INFO, ITYPE, IU, LDZ, M, N

           REAL           ABSTOL, VL, VU

           INTEGER        IFAIL( * ), IWORK( * )

           REAL           RWORK( * ), W( * )

           COMPLEX        AP( * ), BP( * ), WORK( * ), Z( LDZ, * )

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.

ARGUMENTS

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

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

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

        AP      (input/output) 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      (input/output) 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      (input) REAL
                VU      (input) 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) INTEGER
                IU      (input) 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) 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       (output) 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) REAL array, dimension (N)
                On normal exit, the first M elements contain the selected
                eigenvalues in ascending order.

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

        WORK    (workspace) COMPLEX array, dimension (2*N)

        RWORK   (workspace) REAL array, dimension (7*N)

        IWORK   (workspace) INTEGER array, dimension (5*N)

        IFAIL   (output) 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) 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
                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

        Based on contributions by
           Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA

 LAPACK driver routine (version 3.3.1)      April 2011                            CHPGVX(3lapack)