naginterfaces.library.lapackeig.dspgvd

naginterfaces.library.lapackeig.dspgvd(itype, jobz, uplo, n, ap, bp)

dspgvd computes all the eigenvalues and, optionally, the eigenvectors of a real generalized symmetric-definite eigenproblem, of the form

$$Az=\lambda Bz\text{,}ABz=\lambda z\text{or}BAz=\lambda z\text{,}$$

where $A$ and $B$ are symmetric, stored in packed format, 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 f08tc

https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/f08/f08tcf.html

Parameters

itypeint

Specifies the problem type to be solved.

$itype=1$

$Az=\lambda Bz$.

$itype=2$

$ABz=\lambda z$.

$itype=3$

$BAz=\lambda z$.

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.

nint

$n$, the order of the matrices $A$ and $B$.

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.

bpfloat, array-like, shape $(n\times (n+1)/2)$

The upper or lower triangle of the $n\times n$ symmetric matrix $B$, packed by columns.

Returns

apfloat, ndarray, shape $(n\times (n+1)/2)$

The contents of $ap$ are destroyed.

bpfloat, ndarray, shape $(n\times (n+1)/2)$

The triangular factor $U$ or $L$ from the Cholesky factorization $B=U^{T}U$ or $B=L{L}^T$, in the same storage format as $B$.

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. The eigenvectors are normalized as follows:

if $itype=1$ or $2$, $Z^{T}BZ=I$;

if $itype=3$, $Z^T{B}^{-1}Z=I$.

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

Raises

NagValueError

(errno $-1$)

On entry, error in parameter $itype$.

Constraint: $itype=1$, $2$ or $3$.

(errno $-2$)

On entry, error in parameter $jobz$.

Constraint: $jobz=\text{'N'}$ or $\text{'V'}$.

(errno $-3$)

On entry, error in parameter $uplo$.

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

(errno $-4$)

On entry, error in parameter $n$.

Constraint: $n\ge 0$.

(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 $1\le ⟨value⟩\le n$, then the leading minor of order $⟨value⟩$ of $B$ is not positive definite. The factorization of $B$ could not be completed and no eigenvalues or eigenvectors were computed.

Notes

dspgvd first performs a Cholesky factorization of the matrix $B$ as $B=U^{T}U$, when $uplo=\text{'U'}$ or $B=L{L}^T$, when $uplo=\text{'L'}$. The generalized problem is then reduced to a standard symmetric eigenvalue problem

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

which is solved for the eigenvalues and, optionally, the eigenvectors; the eigenvectors are then backtransformed to give the eigenvectors of the original problem.

For the problem $Az=\lambda Bz$, 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. For the problem $ABz=\lambda z$ we correspondingly have

$$Z^{-1}A{Z}^{-T}=\Lambda \text{and}{Z}^{T}BZ=I\text{,}$$

and for $BAz=\lambda z$ we have

$$Z^{T}AZ=\Lambda \text{and}{Z}^T{B}^{-1}Z=I\text{.}$$

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