naginterfaces.library.lapackeig.dsbgvx

naginterfaces.library.lapackeig.dsbgvx(jobz, erange, uplo, ka, kb, ab, bb, vl, vu, il, iu, abstol)

dsbgvx computes selected eigenvalues and, optionally, eigenvectors of a real generalized symmetric-definite banded eigenproblem, of the form

$$Az=\lambda Bz\text{,}$$

where $A$ and $B$ are symmetric and banded, and $B$ is also positive definite. Eigenvalues and eigenvectors can be selected by specifying either all eigenvalues, a range of values or a range of indices for the desired eigenvalues.

For full information please refer to the NAG Library document for f08ub

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/f08/f08ubf.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 triangles of $A$ and $B$ are stored.

If $uplo=\text{'L'}$, the lower triangles of $A$ and $B$ are stored.

kaint

If $uplo=\text{'U'}$, the number of superdiagonals, $k_a$, of the matrix $A$.

If $uplo=\text{'L'}$, the number of subdiagonals, $k_a$, of the matrix $A$.

kbint

If $uplo=\text{'U'}$, the number of superdiagonals, $k_b$, of the matrix $B$.

If $uplo=\text{'L'}$, the number of subdiagonals, $k_b$, of the matrix $B$.

abfloat, array-like, shape $(ka+1,n)$

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

bbfloat, array-like, shape $(kb+1,n)$

The upper or lower triangle of the $n\times n$ symmetric positive definite band matrix $B$.

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 $C$ 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 $errno$ = 1 … $n$, 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

abfloat, ndarray, shape $(ka+1,n)$

The contents of $ab$ are overwritten.

bbfloat, ndarray, shape $(kb+1,n)$

The factor $S$ from the split Cholesky factorization $B=S^{T}S$, as returned by dpbstf().

qfloat, ndarray, shape $(:,:)$

If $jobz=\text{'V'}$, the $n\times n$ matrix, $Q$ used in the reduction of the standard form, i.e., $Cx=\lambda x$, from symmetric banded 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 eigenvalues in ascending order.

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

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

if no exception or warning is raised, the first $m$ columns of $Z$ contain the eigenvectors corresponding to the selected eigenvalues, with the $i$th column of $Z$ holding the eigenvector associated with $w[i-1]$. The eigenvectors are normalized so that $Z^{T}BZ=I$;

if an eigenvector fails to converge ($errno$ = 1 … $n$), 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 $errno$ = 1 … $n$, the first $\text{errno}$ elements of $jfail$ 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 $ka$.

Constraint: $ka\ge 0$.

(errno $-6$)

On entry, error in parameter $kb$.

Constraint: $ka\ge kb\ge 0$.

(errno $-14$)

On entry, error in parameter $vu$.

Constraint: $vl<vu$.

(errno $-15$)

On entry, error in parameter $il$.

(errno $-16$)

On entry, error in parameter $iu$.

(errno $i>n$)

If $\text{errno}=n+⟨value⟩$, for $1\le ⟨value⟩\le n$, dpbstf() returned $\text{errno}=⟨value⟩$: $B$ is not positive definite. The factorization of $B$ could not be completed and no eigenvalues or eigenvectors were computed.

Warns

NagAlgorithmicWarning

(errno $(i>0)\text{and}(i\le n)$)

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

Notes

The generalized symmetric-definite band problem

$$Az=\lambda Bz$$

is first reduced to a standard band symmetric problem

$$Cx=\lambda x\text{,}$$

where $C$ is a symmetric band matrix, using Wilkinson’s modification to Crawford’s algorithm (see Crawford (1973) and Wilkinson (1977)). The symmetric eigenvalue problem is then solved for the required eigenvalues and eigenvectors, and the eigenvectors are then backtransformed to the eigenvectors of the original problem.

The eigenvectors are normalized so that

$$z^{T}Az=\lambda \text{and}{z}^{T}Bz=1\text{.}$$

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

Crawford, C R, 1973, Reduction of a band-symmetric generalized eigenvalue problem, Comm. ACM (16), 41–44

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

Wilkinson, J H, 1977, Some recent advances in numerical linear algebra, The State of the Art in Numerical Analysis, (ed D A H Jacobs), Academic Press