naginterfaces.library.lapackeig.zgebrd

naginterfaces.library.lapackeig.zgebrd(a)

zgebrd reduces a complex $m\times n$ matrix to bidiagonal form.

For full information please refer to the NAG Library document for f08ks

https://www.nag.com/numeric/nl/nagdoc_29.3/flhtml/f08/f08ksf.html

Parameters

acomplex, array-like, shape $(m,n)$

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

Returns

acomplex, ndarray, shape $(m,n)$

If $m\ge n$, the diagonal and first superdiagonal are overwritten by the upper bidiagonal matrix $B$, elements below the diagonal are overwritten by details of the unitary matrix $Q$ and elements above the first superdiagonal are overwritten by details of the unitary matrix $P$.

If $m<n$, the diagonal and first subdiagonal are overwritten by the lower bidiagonal matrix $B$, elements below the first subdiagonal are overwritten by details of the unitary matrix $Q$ and elements above the diagonal are overwritten by details of the unitary matrix $P$.

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

The diagonal elements of the bidiagonal matrix $B$.

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

The off-diagonal elements of the bidiagonal matrix $B$.

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

Further details of the unitary matrix $Q$.

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

Further details of the unitary matrix $P$.

Raises

NagValueError

(errno $-1$)

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

Constraint: $m\ge 0$.

(errno $-2$)

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

Constraint: $n\ge 0$.

Notes

zgebrd reduces a complex $m\times n$ matrix $A$ to real bidiagonal form $B$ by a unitary transformation: $A=QB{P}^H$, where $Q$ and $P^H$ are unitary matrices of order $m$ and $n$ respectively.

If $m\ge n$, the reduction is given by:

$$\begin{array}{c}A=Q\left(\begin{array}{c}{B}_1\\ 0\end{array}\right)P^H=Q_1{B}_1{P}^H\text{,}\end{array}$$

where $B_1$ is a real $n\times n$ upper bidiagonal matrix and $Q_1$ consists of the first $n$ columns of $Q$.

If $m<n$, the reduction is given by

$$A=Q\left(\begin{array}{cc}{B}_1& 0\end{array}\right)P^H=Q{B}_1{P}_1^H\text{,}$$

where $B_1$ is a real $m\times m$ lower bidiagonal matrix and $P_1^H$ consists of the first $m$ rows of $P^H$.

The unitary matrices $Q$ and $P$ are not formed explicitly but are represented as products of elementary reflectors (see the F08 Introduction for details). Functions are provided to work with $Q$ and $P$ in this representation (see Further Comments).

References

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