SPTEQR(1) LAPACK routine (version 3.2) SPTEQR(1)NAME
SPTEQR - computes 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 sym‐
metric positive definite tridiagonal matrix by first factoring the
matrix using SPTTRF, and then calling SBDSQR 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 ei‐
genvalues 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 orig‐
inal 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 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 asso‐
ciated 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 i-th 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 i-th principal
minor was not positive definite. > N the SVD algorithm
failed to converge; if INFO = N+i, i off-diagonal elements of
the bidiagonal factor did not converge to zero.
LAPACK routine (version 3.2) November 2008 SPTEQR(1)
