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

NAME

       LAPACK-3  -  computes  all  the eigenvalues, and optionally, the eigenvectors of a complex
       generalized Hermitian-definite banded eigenproblem, of the form A*x=(lambda)*B*x

SYNOPSIS

       SUBROUTINE ZHBGVX( JOBZ, RANGE, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, Q, LDQ, VL,  VU,  IL,
                          IU, ABSTOL, M, W, Z, LDZ, WORK, RWORK, IWORK, IFAIL, INFO )

           CHARACTER      JOBZ, RANGE, UPLO

           INTEGER        IL, INFO, IU, KA, KB, LDAB, LDBB, LDQ, LDZ, M, N

           DOUBLE         PRECISION ABSTOL, VL, VU

           INTEGER        IFAIL( * ), IWORK( * )

           DOUBLE         PRECISION RWORK( * ), W( * )

           COMPLEX*16     AB( LDAB, * ), BB( LDBB, * ), Q( LDQ, * ), WORK( * ), Z( LDZ, * )

PURPOSE

       ZHBGVX  computes  all  the  eigenvalues,  and  optionally,  the  eigenvectors of a complex
       generalized Hermitian-definite banded eigenproblem, of the form A*x=(lambda)*B*x.  Here  A
       and B are assumed to be Hermitian
        and banded, and B is also positive definite.  Eigenvalues and
        eigenvectors can be selected by specifying either all eigenvalues,
        a range of values or a range of indices for the desired eigenvalues.

ARGUMENTS

        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.

        KA      (input) INTEGER
                The number of superdiagonals of the matrix A if UPLO = 'U',
                or the number of subdiagonals if UPLO = 'L'. KA >= 0.

        KB      (input) INTEGER
                The number of superdiagonals of the matrix B if UPLO = 'U',
                or the number of subdiagonals if UPLO = 'L'. KB >= 0.

        AB      (input/output) COMPLEX*16 array, dimension (LDAB, N)
                On entry, the upper or lower triangle of the Hermitian band
                matrix A, stored in the first ka+1 rows of the array.  The
                j-th column of A is stored in the j-th column of the array AB
                as follows:
                if UPLO = 'U', AB(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j;
                if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+ka).
                On exit, the contents of AB are destroyed.

        LDAB    (input) INTEGER
                The leading dimension of the array AB.  LDAB >= KA+1.

        BB      (input/output) COMPLEX*16 array, dimension (LDBB, N)
                On entry, the upper or lower triangle of the Hermitian band
                matrix B, stored in the first kb+1 rows of the array.  The
                j-th column of B is stored in the j-th column of the array BB
                as follows:
                if UPLO = 'U', BB(kb+1+i-j,j) = B(i,j) for max(1,j-kb)<=i<=j;
                if UPLO = 'L', BB(1+i-j,j)    = B(i,j) for j<=i<=min(n,j+kb).
                On exit, the factor S from the split Cholesky factorization
                B = S**H*S, as returned by ZPBSTF.

        LDBB    (input) INTEGER
                The leading dimension of the array BB.  LDBB >= KB+1.

        Q       (output) COMPLEX*16 array, dimension (LDQ, N)
                If JOBZ = 'V', the n-by-n matrix used in the reduction of
                A*x = (lambda)*B*x to standard form, i.e. C*x = (lambda)*x,
                and consequently C to tridiagonal form.
                If JOBZ = 'N', the array Q is not referenced.

        LDQ     (input) INTEGER
                The leading dimension of the array Q.  If JOBZ = 'N',
                LDQ >= 1. If JOBZ = 'V', LDQ >= max(1,N).

        VL      (input) DOUBLE PRECISION
                VU      (input) DOUBLE PRECISION
                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) 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 AP to tridiagonal form.
                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       (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) DOUBLE PRECISION array, dimension (N)
                If INFO = 0, the eigenvalues in ascending order.

        Z       (output) COMPLEX*16 array, dimension (LDZ, N)
                If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of
                eigenvectors, with the i-th column of Z holding the
                eigenvector associated with W(i). The eigenvectors are
                normalized so that Z**H*B*Z = I.
                If JOBZ = 'N', then Z is not referenced.

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

        WORK    (workspace) COMPLEX*16 array, dimension (N)

        RWORK   (workspace) DOUBLE PRECISION 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:  if INFO = i, and i is:
                <= N:  then i eigenvectors failed to converge.  Their
                indices are stored in array IFAIL.
                > N:   if INFO = N + i, for 1 <= i <= N, then ZPBSTF
                returned INFO = i: 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 3.2)        April 2011                            ZHBGVX(3lapack)