
SPTEQR(l) ) SPTEQR(l)
NAME
SPTEQR  compute all eigenvalues and, optionally, eigenvectors 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 factor
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, eigenvectors of a symmetric positive def
inite tridiagonal matrix by first factoring the matrix using SPTTRF, and then calling SBD
SQR to compute the singular values of the bidiagonal factor. This routine computes the
eigenvalues of the positive definite tridiagonal matrix to high relative accuracy. This
means that if the eigenvalues range over many orders of magnitude 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 positive definite matrix can also be found if
SSYTRD, 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 (N1)
On entry, the (n1) subdiagonal elements of the tridiagonal 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 tridiago
nal 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 (4*N)
INFO (output) INTEGER
= 0: successful exit.
< 0: if INFO = i, the ith argument had an illegal value.
> 0: if INFO = i, and i is: <= N the Cholesky factorization of the matrix could
not be performed because the ith principal minor was not positive definite. > N
the SVD algorithm failed to converge; if INFO = N+i, i offdiagonal elements of
the bidiagonal factor did not converge to zero.
LAPACK version 3.0 15 June 2000 SPTEQR(l) 
