naginterfaces.library.lapackeig.dsbev

naginterfaces.library.lapackeig.dsbev(jobz, uplo, kd, ab)

dsbev computes all the eigenvalues and, optionally, all the eigenvectors of a real $n\times n$ symmetric band matrix $A$ of bandwidth $(2k_d+1)$.

For full information please refer to the NAG Library document for f08ha

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/f08/f08haf.html

Parameters

jobzstr, length 1

Indicates whether eigenvectors are computed.

$jobz=\text{'N'}$

Only eigenvalues are computed.

$jobz=\text{'V'}$

Eigenvalues and eigenvectors are computed.

uplostr, length 1

If $uplo=\text{'U'}$, the upper triangular part of $A$ is stored.

If $uplo=\text{'L'}$, the lower triangular part of $A$ is stored.

kdint

If $uplo=\text{'U'}$, the number of superdiagonals, $k_d$, of the matrix $A$.

If $uplo=\text{'L'}$, the number of subdiagonals, $k_d$, of the matrix $A$.

abfloat, array-like, shape $(kd+1,n)$

The upper or lower triangle of the $n\times n$ symmetric band matrix $A$.

Returns

abfloat, ndarray, shape $(kd+1,n)$

$ab$ is overwritten by values generated during the reduction to tridiagonal form.

The first superdiagonal or subdiagonal and the diagonal of the tridiagonal matrix $T$ are returned in $ab$ using the same storage format as described above.

wfloat, ndarray, shape $\left(n\right)$

The eigenvalues in ascending order.

zfloat, ndarray, shape $(:,:)$

If $jobz=\text{'V'}$, $z$ contains the orthonormal eigenvectors of the matrix $A$, with the $i$th column of $Z$ holding the eigenvector associated with $w[i-1]$.

If $jobz=\text{'N'}$, $z$ is not referenced.

Raises

NagValueError

(errno $-1$)

On entry, error in parameter $jobz$.

Constraint: $jobz=\text{'N'}$ or $\text{'V'}$.

(errno $-2$)

On entry, error in parameter $uplo$.

Constraint: $uplo=\text{'U'}$ or $\text{'L'}$.

(errno $-3$)

On entry, error in parameter $\text{n}$.

Constraint: $n\ge 0$.

(errno $-4$)

On entry, error in parameter $kd$.

Constraint: $kd\ge 0$.

(errno $i>0$)

The algorithm failed to converge; $⟨value⟩$ off-diagonal elements of an intermediate tridiagonal form did not converge to zero.

Notes

The symmetric band matrix $A$ is first reduced to tridiagonal form, using orthogonal similarity transformations, and then the $QR$ algorithm is applied to the tridiagonal matrix to compute the eigenvalues and (optionally) the eigenvectors.

References

Anderson, E, Bai, Z, Bischof, C, Blackford, S, Demmel, J, Dongarra, J J, Du Croz, J J, Greenbaum, A, Hammarling, S, McKenney, A and Sorensen, D, 1999, LAPACK Users’ Guide, (3rd Edition), SIAM, Philadelphia, https://www.netlib.org/lapack/lug

Golub, G H and Van Loan, C F, 1996, Matrix Computations, (3rd Edition), Johns Hopkins University Press, Baltimore