naginterfaces.library.lapackeig.dsbgst

naginterfaces.library.lapackeig.dsbgst(vect, uplo, ka, kb, ab, bb)

dsbgst reduces a real symmetric-definite generalized eigenproblem $Az=\lambda Bz$ to the standard form $Cy=\lambda y$, where $A$ and $B$ are band matrices, $A$ is a real symmetric matrix, and $B$ has been factorized by dpbstf().

For full information please refer to the NAG Library document for f08ue

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/f08/f08uef.html

Parameters

vectstr, length 1

Indicates whether $X$ is to be returned.

$vect=\text{'N'}$

$X$ is not returned.

$vect=\text{'V'}$

$X$ is returned.

uplostr, length 1

Indicates whether the upper or lower triangular part of $A$ is stored.

$uplo=\text{'U'}$

The upper triangular part of $A$ is stored.

$uplo=\text{'L'}$

The lower triangular part of $A$ is 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 banded split Cholesky factor of $B$ as specified by $uplo$, $n$ and $kb$ and returned by dpbstf().

Returns

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

The upper or lower triangle of $ab$ is overwritten by the corresponding upper or lower triangle of $C$ as specified by $uplo$.

xfloat, ndarray, shape $(:,:)$

The $n\times n$ matrix $X=S^{-1}Q$, if $vect=\text{'V'}$.

If $vect=\text{'N'}$, $x$ is not referenced.

Raises

NagValueError

(errno $-1$)

On entry, error in parameter $vect$.

Constraint: $vect=\text{'N'}$ 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: $ka\ge kb\ge 0$.

Notes

To reduce the real symmetric-definite generalized eigenproblem $Az=\lambda Bz$ to the standard form $Cy=\lambda y$, where $A$, $B$ and $C$ are banded, dsbgst must be preceded by a call to dpbstf() which computes the split Cholesky factorization of the positive definite matrix $B$: $B=S^{T}S$. The split Cholesky factorization, compared with the ordinary Cholesky factorization, allows the work to be approximately halved.

This function overwrites $A$ with $C=X^{T}AX$, where $X=S^{-1}Q$ and $Q$ is a orthogonal matrix chosen (implicitly) to preserve the bandwidth of $A$. The function also has an option to allow the accumulation of $X$, and then, if $z$ is an eigenvector of $C$, $Xz$ is an eigenvector of the original system.

References

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

Kaufman, L, 1984, Banded eigenvalue solvers on vector machines, ACM Trans. Math. Software (10), 73–86