naginterfaces.library.lapackeig.zstedc

naginterfaces.library.lapackeig.zstedc(compz, d, e, z)

zstedc computes all the eigenvalues and, optionally, all the eigenvectors of a real $n\times n$ symmetric tridiagonal matrix, or of a complex full or banded Hermitian matrix which has been reduced to tridiagonal form.

For full information please refer to the NAG Library document for f08jv

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/f08/f08jvf.html

Parameters

compzstr, length 1

Indicates whether the eigenvectors are to be computed.

$compz=\text{'N'}$

Only the eigenvalues are computed (and the array $z$ is not referenced).

$compz=\text{'V'}$

The eigenvalues and eigenvectors of $A$ are computed (and the array $z$ must contain the matrix $Q$ on entry).

$compz=\text{'I'}$

The eigenvalues and eigenvectors of $T$ are computed (and the array $z$ is initialized by the function).

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

The diagonal elements of the tridiagonal matrix.

efloat, array-like, shape $(max(1,n-1))$

The subdiagonal elements of the tridiagonal matrix.

zcomplex, array-like, shape $(:,:)$

Note: the required extent for this argument in dimension 1 is determined as follows: if $compz\text{in}(\text{'V'},\text{'I'})$: $n$; otherwise: $1$.

Note: the required extent for this argument in dimension 2 is determined as follows: if $compz\text{in}(\text{'V'},\text{'I'})$: $n$; otherwise: $1$.

If $compz=\text{'V'}$, $z$ must contain the unitary matrix $Q$ used in the reduction to tridiagonal form.

Returns

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

If no exception or warning is raised, the eigenvalues in ascending order.

efloat, ndarray, shape $(max(1,n-1))$

$e$ is overwritten.

zcomplex, ndarray, shape $(:,:)$

If $compz=\text{'V'}$, $z$ contains the orthonormal eigenvectors of the original Hermitian matrix $A$, and if $compz=\text{'I'}$, $z$ contains the orthonormal eigenvectors of the symmetric tridiagonal matrix $T$.

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

Raises

NagValueError

(errno $-1$)

On entry, error in parameter $compz$.

Constraint: $compz=\text{'N'}$, $\text{'V'}$ or $\text{'I'}$.

(errno $-2$)

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

Constraint: $n\ge 0$.

(errno $i>0$)

The algorithm failed to compute an eigenvalue while working on the submatrix lying in rows and columns $⟨value⟩/(n+1)$ through $mod(⟨value⟩,n+1)$.

Notes

zstedc computes all the eigenvalues and, optionally, the eigenvectors of a real symmetric tridiagonal matrix $T$. That is, the function computes the spectral factorization of $T$ given by

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

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

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

The function may also be used to compute all the eigenvalues and eigenvectors of a complex full, or banded, Hermitian matrix $A$ which has been reduced to real tridiagonal form $T$ as

$$A=QT{Q}^H\text{,}$$

where $Q$ is unitary. The spectral factorization of $A$ is then given by

$$A=\left(QZ\right)\Lambda {\left(QZ\right)}^H\text{.}$$

In this case $Q$ must be formed explicitly and passed to zstedc in the array $z$, and the function called with $compz=\text{'V'}$. Functions which may be called to form $T$ and $Q$ are

full matrix	zhetrd() and zungtr()
full matrix, packed storage	zhptrd() and zupgtr()
band matrix	zhbtrd(), with $\text{vect}=\text{'V'}$

When only eigenvalues are required then this function calls dsterf() to compute the eigenvalues of the tridiagonal matrix $T$, but when eigenvectors of $T$ are also required and the matrix is not too small, then a divide and conquer method is used, which can be much faster than zsteqr(), although more storage is required.

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