ZHBEVX(3f)

NAME
ZHBEVX - selected eigenvalues and, optionally, eigenvectors of a complex Hermitian band matrix A

SYNOPSIS

SUBROUTINE ZHBEVX(
    JOBZ, RANGE, UPLO, N, KD, AB, LDAB, Q, LDQ, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, RWORK, IWORK, IFAIL, INFO )

CHARACTER
    JOBZ, RANGE, UPLO

INTEGER
    IL, INFO, IU, KD, LDAB, LDQ, LDZ, M, N

DOUBLE
    PRECISION ABSTOL, VL, VU

INTEGER
    IFAIL( * ), IWORK( * )

DOUBLE
    PRECISION RWORK( * ), W( * )

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

PURPOSE
ZHBEVX computes selected eigenvalues and, optionally, eigenvectors of a complex Hermitian band matrix A. Eigenvalues and eigenvectors 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.

KD      (input) INTEGER
    The number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KD >= 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 KD+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(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd). On exit, AB is overwritten by values generated during the reduction to tridiagonal form.

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

Q       (output) COMPLEX*16 array, dimension (LDQ, N)
    If JOBZ = 'V', the N-by-N unitary matrix used in the reduction to tridiagonal form. If JOBZ = 'N', the array Q is not referenced.

LDQ     (input) INTEGER
    The leading dimension of the array Q. If JOBZ = 'V', then 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 AB 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'). 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) DOUBLE PRECISION array, dimension (N)
    The first M elements contain the selected eigenvalues in ascending order.

Z       (output) COMPLEX*16 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*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, then i eigenvectors failed to converge. Their indices are stored in array IFAIL.