f08nuf multiplies an arbitrary complex matrix
by the complex unitary matrix
which was determined by
f08nsf when reducing a complex general matrix to Hessenberg form.
f08nuf is intended to be used following a call to
f08nsf, which reduces a complex general matrix
to upper Hessenberg form
by a unitary similarity transformation:
.
f08nsf represents the matrix
as a product of
elementary reflectors. Here
and
are values determined by
f08nvf when balancing the matrix; if the matrix has not been balanced,
and
.
This routine may be used to form one of the matrix products
overwriting the result on
(which may be any complex rectangular matrix).
The computed result differs from the exact result by a matrix
such that
where
is the
machine precision.
Background information to multithreading can be found in the
Multithreading documentation.
Please consult the
X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the
Users' Note for your implementation for any additional implementation-specific information.
The real analogue of this routine is
f08ngf.
This example computes all the eigenvalues of the matrix
, where
and those eigenvectors which correspond to eigenvalues
such that
. Here
is general and must first be reduced to upper Hessenberg form
by
f08nsf. The program then calls
f08psf to compute the eigenvalues, and
f08pxf to compute the required eigenvectors of
by inverse iteration. Finally
f08nuf is called to transform the eigenvectors of
back to eigenvectors of the original matrix
.