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NAME SPTEQR - compute all eigenvalues and, optionally, eigenvec- tors of a symmetric positive definite tridiagonal matrix by first factoring the matrix using SPTTRF, and then calling SBDSQR to compute the singular values of the bidiagonal fac- tor SYNOPSIS SUBROUTINE SPTEQR( COMPZ, N, D, E, Z, LDZ, WORK, INFO ) CHARACTER COMPZ INTEGER INFO, LDZ, N REAL D( * ), E( * ), WORK( * ), Z( LDZ, * ) PURPOSE SPTEQR computes all eigenvalues and, optionally, eigenvec- tors of a symmetric positive definite tridiagonal matrix by first factoring the matrix using SPTTRF, and then calling SBDSQR to compute the singular values of the bidiagonal fac- tor. This routine computes the eigenvalues of the positive defin- ite tridiagonal matrix to high relative accuracy. This means that if the eigenvalues range over many orders of mag- nitude in size, then the small eigenvalues and corresponding eigenvectors will be computed more accurately than, for example, with the standard QR method. The eigenvectors of a full or band symmetric matrix can also be found if SSYTRD or SSPTRD or SSBTRD has been used to reduce this matrix to tridiagonal form. (The reduction to tridiagonal form, however, may preclude the possibility of obtaining high relative accuracy in the small eigenvalues of the original matrix, if these eigenvalues range over many orders of magnitude.) ARGUMENTS COMPZ (input) CHARACTER*1 = 'N': Compute eigenvalues only. = 'V': Compute eigenvectors of original symmetric matrix also. Array Z contains the orthogonal matrix used to reduce the original matrix to tridiagonal form. = 'I': Compute eigenvectors of tridiagonal matrix also. N (input) INTEGER The order of the matrix. N >= 0. D (input/output) REAL array, dimension (N) On entry, the n diagonal elements of the tridiagonal matrix. On normal exit, D contains the eigenvalues, in descending order. E (input/output) REAL array, dimension (N-1) On entry, the (n-1) subdiagonal elements of the tri- diagonal matrix. On exit, E has been destroyed. Z (input/output) REAL array, dimension (LDZ, N) On entry, if COMPZ = 'V', the orthogonal matrix used in the reduction to tridiagonal form. On exit, if COMPZ = 'V', the orthonormal eigenvectors of the original symmetric matrix; if COMPZ = 'I', the orthonormal eigenvectors of the tridiagonal matrix. If INFO > 0 on exit, Z contains the eigenvectors associated with only the stored eigenvalues. If COMPZ = 'N', then Z is not referenced. LDZ (input) INTEGER The leading dimension of the array Z. LDZ >= 1, and if COMPZ = 'V' or 'I', LDZ >= max(1,N). WORK (workspace) REAL array, dimension (max(1,4*N-4)) If COMPZ = 'N', then WORK 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, and i is: <= N the Cholesky fac- torization of the matrix could not be performed because the i-th principal minor was not positive definite. > N the SVD algorithm failed to con- verge; if INFO = N+i, i off-diagonal elements of the bidiagonal factor did not converge to zero.