f08ktc generates one of the complex unitary matrices
or
which were determined by
f08ksc when reducing a complex matrix to bidiagonal form.
f08ktc is intended to be used after a call to
f08ksc, which reduces a complex rectangular matrix
to real bidiagonal form
by a unitary transformation:
.
f08ksc represents the matrices
and
as products of elementary reflectors.
The various possibilities are specified by the arguments
vect,
m,
n and
k. The appropriate values to cover the most likely cases are as follows (assuming that
was an
matrix):
-
1.To form the full matrix :
nag_lapackeig_zungbr(order,Nag_FormQ,m,m,n,...)
(note that the array a must have at least columns).
-
2.If , to form the leading columns of :
nag_lapackeig_zungbr(order,Nag_FormQ,m,n,n,...)
-
3.To form the full matrix :
nag_lapackeig_zungbr(order,Nag_FormP,n,n,m,...)
(note that the array a must have at least rows).
-
4.If , to form the leading rows of :
nag_lapackeig_zungbr(order,Nag_FormP,m,n,m,...)
- NE_ALLOC_FAIL
-
Dynamic memory allocation failed.
See
Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
- NE_BAD_PARAM
-
On entry, argument had an illegal value.
- NE_ENUM_INT_3
-
On entry, , , and .
Constraint: and
if and , ;
if and , ;
if and , ;
if and , .
- NE_INT
-
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
- NE_INT_2
-
On entry, and .
Constraint: .
- NE_INTERNAL_ERROR
-
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact
NAG for assistance.
See
Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
- NE_NO_LICENCE
-
Your licence key may have expired or may not have been installed correctly.
See
Section 8 in the Introduction to the NAG Library CL Interface for further information.
The computed matrix
differs from an exactly unitary matrix by a matrix
such that
where
is the
machine precision. A similar statement holds for the computed matrix
.
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 function. Please also consult the
Users' Note for your implementation for any additional implementation-specific information.
The total number of real floating-point operations for the cases listed in
Section 3 are approximately as follows:
-
1.To form the whole of :
- if ,
- if ;
-
2.To form the leading columns of when :
-
3.To form the whole of :
- if ,
- if ;
-
4.To form the leading rows of when :
The real analogue of this function is
f08kfc.
For this function two examples are presented, both of which involve computing the singular value decomposition of a matrix
, where
in the first example and
in the second.
must first be reduced to tridiagonal form by
f08ksc. The program then calls
f08ktc twice to form
and
, and passes these matrices to
f08msc, which computes the singular value decomposition of
.