naginterfaces.library.lapackeig.dspev

naginterfaces.library.lapackeig.dspev(jobz, uplo, n, ap)

dspev computes all the eigenvalues and, optionally, all the eigenvectors of a real $n\times n$ symmetric matrix $A$ in packed storage.

For full information please refer to the NAG Library document for f08ga

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/f08/f08gaf.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.

nint

$n$, the order of the matrix $A$.

apfloat, array-like, shape $(n\times (n+1)/2)$

The upper or lower triangle of the $n\times n$ symmetric matrix $A$, packed by columns.

Returns

apfloat, ndarray, shape $(n\times (n+1)/2)$

$ap$ is overwritten by the values generated during the reduction to tridiagonal form. The elements of the diagonal and the off-diagonal of the tridiagonal matrix overwrite the corresponding elements of $A$.

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 $n$.

Constraint: $n\ge 0$.

(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 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