naginterfaces.library.lapackeig.zupmtr

naginterfaces.library.lapackeig.zupmtr(side, uplo, trans, ap, tau, c)

zupmtr multiplies an arbitrary complex matrix $C$ by the complex unitary matrix $Q$ which was determined by zhptrd() when reducing a complex Hermitian matrix to tridiagonal form.

For full information please refer to the NAG Library document for f08gu

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/f08/f08guf.html

Parameters

sidestr, length 1

Indicates how $Q$ or $Q^H$ is to be applied to $C$.

$side=\text{'L'}$

$Q$ or $Q^H$ is applied to $C$ from the left.

$side=\text{'R'}$

$Q$ or $Q^H$ is applied to $C$ from the right.

uplostr, length 1

This must be the same argument $uplo$ as supplied to zhptrd().

transstr, length 1

Indicates whether $Q$ or $Q^H$ is to be applied to $C$.

$trans=\text{'N'}$

$Q$ is applied to $C$.

$trans=\text{'C'}$

$Q^H$ is applied to $C$.

apcomplex, array-like, shape $(:)$

Note: the required length for this argument is determined as follows: if $side=\text{'L'}$: $m\times (m+1)/2$; if $side=\text{'R'}$: $n\times (n+1)/2$; otherwise: $0$.

Details of the vectors which define the elementary reflectors, as returned by zhptrd().

taucomplex, array-like, shape $(:)$

Note: the required length for this argument is determined as follows: if $side=\text{'L'}$: $max(1,m-1)$; if $side=\text{'R'}$: $max(1,n-1)$; otherwise: $0$.

Further details of the elementary reflectors, as returned by zhptrd().

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

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

Returns

apcomplex, ndarray, shape $(:)$

Is used as internal workspace prior to being restored and hence is unchanged.

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

$c$ is overwritten by $QC$ or $Q^{H}C$ or $CQ$ or $C{Q}^H$ as specified by $side$ and $trans$.

Raises

NagValueError

(errno $-1$)

On entry, error in parameter $side$.

Constraint: $side=\text{'L'}$ or $\text{'R'}$.

(errno $-2$)

On entry, error in parameter $uplo$.

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

(errno $-3$)

On entry, error in parameter $trans$.

Constraint: $trans=\text{'N'}$ or $\text{'C'}$.

(errno $-4$)

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

Constraint: $m\ge 0$.

(errno $-5$)

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

Constraint: $n\ge 0$.

Notes

zupmtr is intended to be used after a call to zhptrd(), which reduces a complex Hermitian matrix $A$ to real symmetric tridiagonal form $T$ by a unitary similarity transformation: $A=QT{Q}^H$. zhptrd() represents the unitary matrix $Q$ as a product of elementary reflectors.

This function may be used to form one of the matrix products

$$QC,Q^{H}C,CQ\text{or}C{Q}^H\text{,}$$

overwriting the result on $C$ (which may be any complex rectangular matrix).

A common application of this function is to transform a matrix $Z$ of eigenvectors of $T$ to the matrix $QZ$ of eigenvectors of $A$.

References

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