.TH CHPEVX 1 "November 2006" " LAPACK driver routine (version 3.1) " " LAPACK driver routine (version 3.1) " .SH NAME CHPEVX - selected eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix A in packed storage .SH SYNOPSIS .TP 19 SUBROUTINE CHPEVX( JOBZ, RANGE, UPLO, N, AP, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, RWORK, IWORK, IFAIL, INFO ) .TP 19 .ti +4 CHARACTER JOBZ, RANGE, UPLO .TP 19 .ti +4 INTEGER IL, INFO, IU, LDZ, M, N .TP 19 .ti +4 REAL ABSTOL, VL, VU .TP 19 .ti +4 INTEGER IFAIL( * ), IWORK( * ) .TP 19 .ti +4 REAL RWORK( * ), W( * ) .TP 19 .ti +4 COMPLEX AP( * ), WORK( * ), Z( LDZ, * ) .SH 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. .SH ARGUMENTS .TP 8 JOBZ (input) CHARACTER*1 = \(aqN\(aq: Compute eigenvalues only; .br = \(aqV\(aq: Compute eigenvalues and eigenvectors. .TP 8 RANGE (input) CHARACTER*1 .br = \(aqA\(aq: all eigenvalues will be found; .br = \(aqV\(aq: all eigenvalues in the half-open interval (VL,VU] will be found; = \(aqI\(aq: the IL-th through IU-th eigenvalues will be found. .TP 8 UPLO (input) CHARACTER*1 .br = \(aqU\(aq: Upper triangle of A is stored; .br = \(aqL\(aq: Lower triangle of A is stored. .TP 8 N (input) INTEGER The order of the matrix A. N >= 0. .TP 8 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 = \(aqU\(aq, AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; if UPLO = \(aqL\(aq, 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 = \(aqU\(aq, the diagonal and first superdiagonal of the tridiagonal matrix T overwrite the corresponding elements of A, and if UPLO = \(aqL\(aq, the diagonal and first subdiagonal of T overwrite the corresponding elements of A. .TP 8 VL (input) REAL VU (input) REAL If RANGE=\(aqV\(aq, the lower and upper bounds of the interval to be searched for eigenvalues. VL < VU. Not referenced if RANGE = \(aqA\(aq or \(aqI\(aq. .TP 8 IL (input) INTEGER IU (input) INTEGER If RANGE=\(aqI\(aq, 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 = \(aqA\(aq or \(aqV\(aq. .TP 8 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(\(aqS\(aq), not zero. If this routine returns with INFO>0, indicating that some eigenvectors did not converge, try setting ABSTOL to 2*SLAMCH(\(aqS\(aq). See "Computing Small Singular Values of Bidiagonal Matrices with Guaranteed High Relative Accuracy," by Demmel and Kahan, LAPACK Working Note #3. .TP 8 M (output) INTEGER The total number of eigenvalues found. 0 <= M <= N. If RANGE = \(aqA\(aq, M = N, and if RANGE = \(aqI\(aq, M = IU-IL+1. .TP 8 W (output) REAL array, dimension (N) If INFO = 0, the selected eigenvalues in ascending order. .TP 8 Z (output) COMPLEX array, dimension (LDZ, max(1,M)) If JOBZ = \(aqV\(aq, 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 = \(aqN\(aq, 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 = \(aqV\(aq, the exact value of M is not known in advance and an upper bound must be used. .TP 8 LDZ (input) INTEGER The leading dimension of the array Z. LDZ >= 1, and if JOBZ = \(aqV\(aq, LDZ >= max(1,N). .TP 8 WORK (workspace) COMPLEX array, dimension (2*N) .TP 8 RWORK (workspace) REAL array, dimension (7*N) .TP 8 IWORK (workspace) INTEGER array, dimension (5*N) .TP 8 IFAIL (output) INTEGER array, dimension (N) If JOBZ = \(aqV\(aq, 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 = \(aqN\(aq, then IFAIL is not referenced. .TP 8 INFO (output) INTEGER = 0: successful exit .br < 0: if INFO = -i, the i-th argument had an illegal value .br > 0: if INFO = i, then i eigenvectors failed to converge. Their indices are stored in array IFAIL.