naginterfaces.library.lapackeig.dstevd

naginterfaces.library.lapackeig.dstevd(job, d, e)

dstevd computes all the eigenvalues and, optionally, all the eigenvectors of a real symmetric tridiagonal matrix. If the eigenvectors are requested, then it uses a divide-and-conquer algorithm to compute eigenvalues and eigenvectors. However, if only eigenvalues are required, then it uses the Pal–Walker–Kahan variant of the $QL$ or $QR$ algorithm.

For full information please refer to the NAG Library document for f08jc

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/f08/f08jcf.html

Parameters

jobstr, length 1

Indicates whether eigenvectors are computed.

$job=\text{'N'}$

Only eigenvalues are computed.

$job=\text{'V'}$

Eigenvalues and eigenvectors are computed.

dfloat, array-like, shape $\left(n\right)$

The $n$ diagonal elements of the tridiagonal matrix $T$.

efloat, array-like, shape $\left(n\right)$

The $n-1$ off-diagonal elements of the tridiagonal matrix $T$. The $n$th element of this array is used as workspace.

Returns

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

The eigenvalues of the matrix $T$ in ascending order.

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

$e$ is overwritten with intermediate results.

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

If $job=\text{'V'}$, $z$ is overwritten by the orthogonal matrix $Z$ which contains the eigenvectors of $T$.

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

Raises

NagValueError

(errno $-1$)

On entry, error in parameter $job$.

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

(errno $-2$)

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

Constraint: $n\ge 0$.

(errno $i>0$)

The algorithm failed to converge; $⟨value⟩$ off-diagonal elements of $e$ did not converge to zero.

Notes

dstevd computes all the eigenvalues and, optionally, all the eigenvectors of a real symmetric tridiagonal matrix $T$. In other words, it can compute the spectral factorization of $T$ as

$$T=Z\Lambda Z^T\text{,}$$

where $\Lambda$ is a diagonal matrix whose diagonal elements are the eigenvalues ${\lambda }_i$, and $Z$ is the orthogonal matrix whose columns are the eigenvectors $z_i$. Thus

$$T{z}_i={\lambda }_i{z}_i\text{,}i=1,2,\dots ,n\text{.}$$

References

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