CHPEVX(3f)

NAME
CHPEVX - selected eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix A in packed storage

SYNOPSIS

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

CHARACTER
    JOBZ, RANGE, UPLO

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

REAL
    ABSTOL, VL, VU

INTEGER
    IFAIL( * ), IWORK( * )

REAL
    RWORK( * ), W( * )

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

PURPOSE
CHPEVX computes selected eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix A in packed storage. Eigenvalues/vectors can be selected by specifying either 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 triangle of A is stored;
    = 'L': Lower triangle of A is stored.

N       (input) INTEGER
    The order of the matrix A. 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, AP is overwritten by values generated during the reduction to tridiagonal form. If UPLO = 'U', the diagonal and first superdiagonal of the tridiagonal matrix T overwrite the corresponding elements of A, and if UPLO = 'L', the diagonal and first subdiagonal of T overwrite the corresponding elements of A.

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'). See "Computing Small Singular Values of Bidiagonal Matrices with Guaranteed High Relative Accuracy," by Demmel and Kahan, LAPACK Working Note #3.

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)
    If INFO = 0, the selected eigenvalues in ascending order.

Z       (output) COMPLEX array, dimension (LDZ, max(1,M))
    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). 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. If JOBZ = 'N', then Z is not referenced. 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: if INFO = i, then i eigenvectors failed to converge. Their indices are stored in array IFAIL.