SPTEQR(3F) SPTEQR(3F)
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
SUBROUTINE SPTEQR( COMPZ, N, D, E, Z, LDZ, WORK, INFO )
CHARACTER COMPZ
INTEGER INFO, LDZ, N
REAL D( * ), E( * ), WORK( * ), Z( LDZ, * )
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.
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.)
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.
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SPTEQR(3F) SPTEQR(3F)
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
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 (LWORK)
If COMPZ = 'N', then LWORK = 2*N If COMPZ = 'V' or 'I', then
LWORK = MAX(1,4*N-4)
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.
SPTEQR(3F) SPTEQR(3F)
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
SUBROUTINE SPTEQR( COMPZ, N, D, E, Z, LDZ, WORK, INFO )
CHARACTER COMPZ
INTEGER INFO, LDZ, N
REAL D( * ), E( * ), WORK( * ), Z( LDZ, * )
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.
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.)
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.
Page 1
SPTEQR(3F) SPTEQR(3F)
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
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 (LWORK)
If COMPZ = 'N', then LWORK = 2*N If COMPZ = 'V' or 'I', then
LWORK = MAX(1,4*N-4)
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.
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