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NAME SSPEVX - compute selected eigenvalues and, optionally, eigenvectors of a real symmetric matrix A in packed storage SYNOPSIS SUBROUTINE SSPEVX( JOBZ, RANGE, UPLO, N, AP, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, IWORK, IFAIL, INFO ) CHARACTER JOBZ, RANGE, UPLO INTEGER IL, INFO, IU, LDZ, M, N REAL ABSTOL, VL, VU INTEGER IFAIL( * ), IWORK( * ) REAL AP( * ), W( * ), WORK( * ), Z( LDZ, * ) PURPOSE SSPEVX computes selected eigenvalues and, optionally, eigen- vectors of a real symmetric 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 eigen- values. 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) REAL array, dimension (N*(N+1)/2) On entry, the upper or lower triangle of the sym- metric 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 dur- ing 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 If RANGE='V', the lower bound of the interval to be searched for eigenvalues. Not referenced if RANGE = 'A' or 'I'. VU (input) REAL 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) 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 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) REAL array, dimension (N) If INFO = 0, the selected eigenvalues in ascending order. Z (output) REAL 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) REAL array, dimension (8*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.