naginterfaces.library.lapackeig.zgbbrd

naginterfaces.library.lapackeig.zgbbrd(vect, m, kl, ku, ab, c)

zgbbrd reduces a complex $m\times n$ band matrix to real upper bidiagonal form.

For full information please refer to the NAG Library document for f08ls

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/f08/f08lsf.html

Parameters

vectstr, length 1

Indicates whether the matrices $Q$ and/or $P^H$ are generated.

$vect=\text{'N'}$

Neither $Q$ nor $P^H$ is generated.

$vect=\text{'Q'}$

$Q$ is generated.

$vect=\text{'P'}$

$P^H$ is generated.

$vect=\text{'B'}$

Both $Q$ and $P^H$ are generated.

mint

$m$, the number of rows of the matrix $A$.

klint

The number of subdiagonals, $k_l$, within the band of $A$.

kuint

The number of superdiagonals, $k_u$, within the band of $A$.

abcomplex, array-like, shape $(kl+ku+1,n)$

The original $m\times n$ band matrix $A$.

ccomplex, array-like, shape $(:,\text{ncc})$

Note: the required extent for this argument in dimension 1 is determined as follows: if $\text{ncc}>0$: $m$; if $\text{ncc}=0$: $1$; otherwise: $0$.

An $m\times n_C$ matrix $C$.

Returns

abcomplex, ndarray, shape $(kl+ku+1,n)$

$ab$ is overwritten by values generated during the reduction.

dfloat, ndarray, shape $(min(m,n))$

The diagonal elements of the bidiagonal matrix $B$.

efloat, ndarray, shape $(min(m,n)-1)$

The superdiagonal elements of the bidiagonal matrix $B$.

qcomplex, ndarray, shape $(:,:)$

If $vect=\text{'Q'}$ or $\text{'B'}$, contains the $m\times m$ unitary matrix $Q$.

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

ptcomplex, ndarray, shape $(:,:)$

The $n\times n$ unitary matrix $P^H$, if $vect=\text{'P'}$ or $\text{'B'}$. If $vect=\text{'N'}$ or $\text{'Q'}$, $pt$ is not referenced.

ccomplex, ndarray, shape $(:,\text{ncc})$

$c$ is overwritten by $Q^{H}C$. If $\text{ncc}=0$, $c$ is not referenced.

Raises

NagValueError

(errno $-1$)

On entry, error in parameter $vect$.

Constraint: $vect=\text{'N'}$, $\text{'Q'}$, $\text{'P'}$ or $\text{'B'}$.

(errno $-2$)

On entry, error in parameter $m$.

Constraint: $m\ge 0$.

(errno $-3$)

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

Constraint: $n\ge 0$.

(errno $-4$)

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

Constraint: $\text{ncc}\ge 0$.

(errno $-5$)

On entry, error in parameter $kl$.

Constraint: $kl\ge 0$.

(errno $-6$)

On entry, error in parameter $ku$.

Constraint: $ku\ge 0$.

Notes

zgbbrd reduces a complex $m\times n$ band matrix to real upper bidiagonal form $B$ by a unitary transformation: $A=QB{P}^H$. The unitary matrices $Q$ and $P^H$, of order $m$ and $n$ respectively, are determined as a product of Givens rotation matrices, and may be formed explicitly by the function if required. A matrix $C$ may also be updated to give $\tilde{C}=Q^{H}C$.

The function uses a vectorizable form of the reduction.