naginterfaces.library.lapackeig.dspgst

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

dspgst reduces a real symmetric-definite generalized eigenproblem $Az=\lambda Bz$, $ABz=\lambda z$ or $BAz=\lambda z$ to the standard form $Cy=\lambda y$, where $A$ is a real symmetric matrix and $B$ has been factorized by lapacklin.dpptrf, using packed storage.

For full information please refer to the NAG Library document for f08te

https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/f08/f08tef.html

Parameters

itypeint

Indicates how the standard form is computed.

$itype=1$

if $uplo=\text{'U'}$, $C=U^{-T}A{U}^{-1}$;

if $uplo=\text{'L'}$, $C=L^{-1}A{L}^{-T}$.

$itype=2$ or $3$

if $uplo=\text{'U'}$, $C=UA{U}^T$;

if $uplo=\text{'L'}$, $C=L^{T}AL$.

uplostr, length 1

Indicates whether the upper or lower triangular part of $A$ is stored and how $B$ has been factorized.

$uplo=\text{'U'}$

The upper triangular part of $A$ is stored and $B=U^{T}U$.

$uplo=\text{'L'}$

The lower triangular part of $A$ is stored and $B=L{L}^T$.

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 Cholesky factor of $B$ as specified by $uplo$ and returned by lapacklin.dpptrf.

Returns

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

The upper or lower triangle of $ap$ is overwritten by the corresponding upper or lower triangle of $C$ as specified by $itype$ and $uplo$, using the same packed storage format as described above.

Raises

NagValueError

(errno $-1$)

On entry, error in parameter $itype$.

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

(errno $-2$)

On entry, error in parameter $uplo$.

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

(errno $-3$)

On entry, error in parameter $n$.

Constraint: $n\ge 0$.

Notes

To reduce the real symmetric-definite generalized eigenproblem $Az=\lambda Bz$, $ABz=\lambda z$ or $BAz=\lambda z$ to the standard form $Cy=\lambda y$ using packed storage, dspgst must be preceded by a call to lapacklin.dpptrf which computes the Cholesky factorization of $B$; $B$ must be positive definite.

The different problem types are specified by the argument $itype$, as indicated in the table below. The table shows how $C$ is computed by the function, and also how the eigenvectors $z$ of the original problem can be recovered from the eigenvectors of the standard form.

$itype$	Problem	$uplo$	$B$	$C$	$z$
$1$	$Az=\lambda Bz$	‘U’ ‘L’	$U^{T}U$ $L{L}^T$	$U^{-T}A{U}^{-1}$ $L^{-1}A{L}^{-T}$	$U^{-1}y$ $L^{-T}y$
$2$	$ABz=\lambda z$	‘U’ ‘L’	$U^{T}U$ $L{L}^T$	$UA{U}^T$ $L^{T}AL$	$U^{-1}y$ $L^{-T}y$
$3$	$BAz=\lambda z$	‘U’ ‘L’	$U^{T}U$ $L{L}^T$	$UA{U}^T$ $L^{T}AL$	$U^{T}y$ $Ly$

References

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