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

NAME

       LAPACK-3  -  computes  selected  eigenvalues,  and  optionally,  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

SYNOPSIS

       SUBROUTINE SSYGVX( ITYPE, JOBZ, RANGE, UPLO, N, A, LDA, B, LDB, VL, VU, IL, IU, ABSTOL, M,
                          W, Z, LDZ, WORK, LWORK, IWORK, IFAIL, INFO )

           CHARACTER      JOBZ, RANGE, UPLO

           INTEGER        IL, INFO, ITYPE, IU, LDA, LDB, LDZ, LWORK, M, N

           REAL           ABSTOL, VL, VU

           INTEGER        IFAIL( * ), IWORK( * )

           REAL           A( LDA, * ), B( LDB, * ), W( * ), WORK( * ), Z( LDZ, * )

PURPOSE

       SSYGVX computes selected eigenvalues, and optionally, 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.
        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 triangle of A and B are stored;
                = 'L':  Lower triangle of A and B are stored.

        N       (input) INTEGER
                The order of the matrix pencil (A,B).  N >= 0.

        A       (input/output) 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, the lower triangle (if UPLO='L') or the upper
                triangle (if UPLO='U') of A, including the diagonal, is
                destroyed.

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

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

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

        WORK    (workspace/output) REAL array, dimension (MAX(1,LWORK))
                On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

        LWORK   (input) INTEGER
                The length of the array WORK.  LWORK >= max(1,8*N).
                For optimal efficiency, LWORK >= (NB+3)*N,
                where NB is the blocksize for SSYTRD 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.

        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:  SPOTRF or SSYEVX returned an error code:
                <= N:  if INFO = i, SSYEVX 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                            SSYGVX(3lapack)