naginterfaces.library.lapackeig.dsbgvd

naginterfaces.library.lapackeig.dsbgvd(jobz, uplo, ka, kb, ab, bb)

dsbgvd computes all the eigenvalues and, optionally, the 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. If eigenvectors are desired, it uses a divide-and-conquer algorithm.

For full information please refer to the NAG Library document for f08uc

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/f08/f08ucf.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.

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 band matrix $B$.

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().

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

The eigenvalues in ascending order.

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

If $jobz=\text{'V'}$, $z$ contains the matrix $Z$ of eigenvectors, 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 $jobz=\text{'N'}$, $z$ 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 $uplo$.

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

(errno $-3$)

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

Constraint: $n\ge 0$.

(errno $-4$)

On entry, error in parameter $ka$.

Constraint: $ka\ge 0$.

(errno $-5$)

On entry, error in parameter $kb$.

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

(errno $i\le n$)

The algorithm failed to converge; $⟨value⟩$ off-diagonal elements of an intermediate tridiagonal form did not converge to zero.

(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.

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 eigenvalues and the eigenvectors, if required, which are then backtransformed to the eigenvectors of the original problem.

The eigenvectors are normalized so that the matrix of eigenvectors, $Z$, satisfies

$$Z^{T}AZ=\Lambda \text{and}{Z}^{T}BZ=I\text{,}$$

where $\Lambda$ is the diagonal matrix whose diagonal elements are the eigenvalues.

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

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