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spteqr

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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.
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