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NAME ZHEEVX - compute selected eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix A SYNOPSIS SUBROUTINE ZHEEVX( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, LWORK, RWORK, IWORK, IFAIL, INFO ) CHARACTER JOBZ, RANGE, UPLO INTEGER IL, INFO, IU, LDA, LDZ, LWORK, M, N DOUBLE PRECISION ABSTOL, VL, VU INTEGER IFAIL( * ), IWORK( * ) DOUBLE PRECISION RWORK( * ), W( * ) COMPLEX*16 A( LDA, * ), WORK( * ), Z( LDZ, * ) PURPOSE ZHEEVX computes selected eigenvalues and, optionally, eigen- vectors of a complex Hermitian 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. A (input/workspace) COMPLEX*16 array, dimension (LDA, N) On entry, the Hermitian matrix A. If UPLO = 'U', the leading N-by-N upper triangular part of A con- tains 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). VL (input) DOUBLE PRECISION If RANGE='V', the lower bound of the interval to be searched for eigenvalues. Not referenced if RANGE = 'A' or 'I'. VU (input) DOUBLE PRECISION If RANGE='V', the upper bound of the interval to be searched for eigenvalues. Not referenced if RANGE = 'A' or 'I'. IL (input) INTEGER If RANGE='I', the index (from smallest to largest) of the smallest eigenvalue to be returned. IL >= 1. Not referenced if RANGE = 'A' or 'V'. IU (input) INTEGER If RANGE='I', the index (from smallest to largest) of the largest eigenvalue to be returned. min(IL,N) <= IU <= N. 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 A to tridi- agonal form. 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) On normal exit, the first M entries 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 corresponding to the selected eigenvalues. 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 (LWORK) On exit, if INFO = 0, WORK(1) returns the optimal LWORK. LWORK (input) INTEGER The length of the array WORK. LWORK >= max(1,2*N- 1). For optimal efficiency, LWORK >= (NB+1)*N, where NB is the blocksize for ZHETRD returned by ILAENV. 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 ele- ments 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.