SPTEQR - compute all eigenvalues and, optionally, eigen
vectors 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, eigenvec
tors of a symmetric positive definite tridiagonal matrix
by first factoring the matrix using SPTTRF, and then call
ing SBDSQR to compute the singular values of the bidiago
nal 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 eigenval
ues 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 def
inite matrix can also be found if SSYTRD, SSPTRD, or SSB
TRD has been used to reduce this matrix to tridiagonal
form. (The reduction to tridiagonal form, however, may
preclude the possibility of obtaining high relative accu
racy 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 tridiago
nal matrix. On normal exit, D contains the eigen
values, in descending order.
On entry, the (n-1) 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 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 ref
erenced.
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 i-th argument had an ille
gal value.
> 0: if INFO = i, and i is: <= N the Cholesky
factorization 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.
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