naginterfaces.library.lapackeig.zhbevx

naginterfaces.library.lapackeig.zhbevx(jobz, erange, uplo, kd, ab, vl, vu, il, iu, abstol)

zhbevx computes selected eigenvalues and, optionally, eigenvectors of a complex $n\times n$ Hermitian band matrix $A$ of bandwidth $(2k_d+1)$. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues.

For full information please refer to the NAG Library document for f08hp

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/f08/f08hpf.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.

erangestr, length 1

If $erange=\text{'A'}$, all eigenvalues will be found.

If $erange=\text{'V'}$, all eigenvalues in the half-open interval $(vl,vu]$ will be found.

If $erange=\text{'I'}$, the $il$th to $iu$th eigenvalues will be found.

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 $A$.

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

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

vlfloat

If $erange=\text{'V'}$, the lower and upper bounds of the interval to be searched for eigenvalues.

If $erange=\text{'A'}$ or $\text{'I'}$, $vl$ and $vu$ are not referenced.

vufloat

If $erange=\text{'V'}$, the lower and upper bounds of the interval to be searched for eigenvalues.

If $erange=\text{'A'}$ or $\text{'I'}$, $vl$ and $vu$ are not referenced.

ilint

If $erange=\text{'I'}$, $il$ and $iu$ specify the indices (in ascending order) of the smallest and largest eigenvalues to be returned, respectively.

If $erange=\text{'A'}$ or $\text{'V'}$, $il$ and $iu$ are not referenced.

iuint

If $erange=\text{'I'}$, $il$ and $iu$ specify the indices (in ascending order) of the smallest and largest eigenvalues to be returned, respectively.

If $erange=\text{'A'}$ or $\text{'V'}$, $il$ and $iu$ are not referenced.

abstolfloat

The absolute error tolerance for the eigenvalues. An approximate eigenvalue is accepted as converged when it is determined to lie in an interval $[a,b]$ of width less than or equal to

$$abstol+ϵmax(\left|a\right|,\left|b\right|)\text{,}$$

where $ϵ$ is the machine precision. If $abstol$ is less than or equal to zero, then $ϵ{\parallel T\parallel }_1$ will be used in its place, where $T$ is the tridiagonal matrix obtained by reducing $A$ to tridiagonal form. Eigenvalues will be computed most accurately when $abstol$ is set to twice the underflow threshold $2\times \text{machine.real\_safe}\left(\right)$, not zero. If this function returns with $errno$ > 0, indicating that some eigenvectors did not converge, try setting $abstol$ to $2\times \text{machine.real\_safe}\left(\right)$. See Demmel and Kahan (1990).

Returns

abcomplex, 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.

qcomplex, ndarray, shape $(:,:)$

If $jobz=\text{'V'}$, the $n\times n$ unitary matrix used in the reduction to tridiagonal form.

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

mint

The total number of eigenvalues found. $0\le m\le n$.

If $erange=\text{'A'}$, $m=n$.

If $erange=\text{'I'}$, $m=iu-il+1$.

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

The first $m$ elements contain the selected eigenvalues in ascending order.

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

If $jobz=\text{'V'}$, then

if no exception or warning is raised, the first $m$ columns of $Z$ contain the orthonormal eigenvectors of the matrix $A$ corresponding to the selected eigenvalues, with the $i$th column of $Z$ holding the eigenvector associated with $w[i-1]$;

if an eigenvector fails to converge ($errno$ > 0), then that column of $Z$ contains the latest approximation to the eigenvector, and the index of the eigenvector is returned in $jfail$.

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

Note: you must ensure that at least $max(1,m)$ columns are supplied in the array $z$; if $erange=\text{'V'}$, the exact value of $m$ is not known in advance and an upper bound of at least $\text{n}$ must be used.

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

If $jobz=\text{'V'}$, then

if no exception or warning is raised, the first $m$ elements of $jfail$ are zero;

if $errno$ > 0, $jfail$ contains the indices of the eigenvectors that failed to converge.

If $jobz=\text{'N'}$, $jfail$ 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 $erange$.

Constraint: $erange=\text{'A'}$, $\text{'V'}$ or $\text{'I'}$.

(errno $-3$)

On entry, error in parameter $uplo$.

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

(errno $-4$)

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

Constraint: $n\ge 0$.

(errno $-5$)

On entry, error in parameter $kd$.

Constraint: $kd\ge 0$.

(errno $-11$)

On entry, error in parameter $vu$.

Constraint: $vl<vu$.

(errno $-12$)

On entry, error in parameter $il$.

(errno $-13$)

On entry, error in parameter $iu$.

Warns

NagAlgorithmicWarning

(errno $i>0$)

The algorithm failed to converge; $⟨value⟩$ eigenvectors did not converge. Their indices are stored in array $jfail$.

Notes

The Hermitian band matrix $A$ is first reduced to real tridiagonal form, using unitary similarity transformations. The required eigenvalues and eigenvectors are then computed from the tridiagonal matrix; the method used depends upon whether all, or selected, eigenvalues and eigenvectors are 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

Demmel, J W and Kahan, W, 1990, Accurate singular values of bidiagonal matrices, SIAM J. Sci. Statist. Comput. (11), 873–912

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