naginterfaces.library.lapackeig.dsbtrd

naginterfaces.library.lapackeig.dsbtrd(vect, uplo, kd, ab, q=None)

dsbtrd reduces a real symmetric band matrix to tridiagonal form.

For full information please refer to the NAG Library document for f08he

https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/f08/f08hef.html

Parameters

vectstr, length 1

Indicates whether $Q$ is to be returned.

$vect=\text{'V'}$

$Q$ is returned.

$vect=\text{'U'}$

$Q$ is updated (and the array $q$ must contain a matrix on entry).

$vect=\text{'N'}$

$Q$ is not required.

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.

kdint

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

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

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

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

qNone or float, array-like, shape $(:,:)$, optional

Note: the required extent for this argument in dimension 1 is determined as follows: if $vect\text{in}(\text{'V'},\text{'U'})$: $max(1,n)$; if $vect=\text{'N'}$: $1$; otherwise: $0$.

Note: the required extent for this argument in dimension 2 is determined as follows: if $vect\text{in}(\text{'V'},\text{'U'})$: $n$; if $vect=\text{'N'}$: $1$; otherwise: $0$.

If $vect=\text{'U'}$, $q$ must contain the matrix formed in a previous stage of the reduction (for example, the reduction of a banded symmetric-definite generalized eigenproblem); otherwise $q$ need not be set.

Returns

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

$ab$ is overwritten by values generated during the reduction to tridiagonal form.

The first superdiagonal or subdiagonal and the diagonal of the tridiagonal matrix $T$ are returned in $ab$ using the same storage format as described above.

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

The diagonal elements of the tridiagonal matrix $T$.

efloat, ndarray, shape $(max(1,n-1))$

The off-diagonal elements of the tridiagonal matrix $T$.

qNone or float, ndarray, shape $(:,:)$

If $vect=\text{'V'}$ or $\text{'U'}$, the $n\times n$ matrix $Q$.

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

Raises

NagValueError

(errno $-1$)

On entry, error in parameter $vect$.

Constraint: $vect=\text{'V'}$, $\text{'U'}$ or $\text{'N'}$.

(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 $kd$.

Constraint: $kd\ge 0$.

Notes

dsbtrd reduces a symmetric band matrix $A$ to symmetric tridiagonal form $T$ by an orthogonal similarity transformation:

$$T=Q^{T}AQ\text{.}$$

The orthogonal matrix $Q$ is determined as a product of Givens rotation matrices, and may be formed explicitly by the function if required.

The function uses a vectorizable form of the reduction, due to Kaufman (1984).

References

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

Parlett, B N, 1998, The Symmetric Eigenvalue Problem, SIAM, Philadelphia