CHPEVX(l) ) CHPEVX(l)NAME
CHPEVX - compute 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 approxi‐
mate 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 reduc‐
ing 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 eigen‐
vectors 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 con‐
tain 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 approxi‐
mation 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.
LAPACK version 3.0 15 June 2000 CHPEVX(l)
