naginterfaces.library.lapackeig.dspevd

naginterfaces.library.lapackeig.dspevd(job, uplo, n, ap)

dspevd computes all the eigenvalues and, optionally, all the eigenvectors of a real symmetric matrix held in packed storage. 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 f08gc

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/f08/f08gcf.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.

uplostr, length 1

Indicates whether the upper or lower triangular part of $A$ is stored.

$uplo=\text{'U'}$

The upper triangular part of $A$ is stored.

$uplo=\text{'L'}$

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 of the matrix $A$ in ascending order.

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

If $job=\text{'V'}$, $z$ is overwritten by the orthogonal matrix $Z$ which contains the eigenvectors of $A$.

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 $uplo$.

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

(errno $-3$)

On entry, error in parameter $n$.

Constraint: $n\ge 0$.

(errno $i>0$)

If $\text{errno}=⟨value⟩$ and $job=\text{'N'}$, the algorithm failed to converge; $⟨value⟩$ elements of an intermediate tridiagonal form did not converge to zero; if $\text{errno}=⟨value⟩$ and $job=\text{'V'}$, then the algorithm failed to compute an eigenvalue while working on the submatrix lying in rows and column $⟨value⟩/(n+1)$ through $mod(⟨value⟩,n+1)$.

Notes

dspevd computes all the eigenvalues and, optionally, all the eigenvectors of a real symmetric matrix $A$ (held in packed storage). In other words, it can compute the spectral factorization of $A$ as

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

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

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

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