f08kuc multiplies an arbitrary complex matrix by one of the complex unitary matrices or which were determined by f08ksc when reducing a complex matrix to bidiagonal form.
The function may be called by the names: f08kuc, nag_lapackeig_zunmbr or nag_zunmbr.
3Description
f08kuc 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.
This function may be used to form one of the matrix products
overwriting the result on (which may be any complex rectangular matrix).
4References
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
5Arguments
Note: in the descriptions below, denotes the order of or : if , and if , .
1: – Nag_OrderTypeInput
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by . See Section 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.
Constraint:
or .
2: – Nag_VectTypeInput
On entry: indicates whether or or or is to be applied to .
or is applied to .
or is applied to .
Constraint:
or .
3: – Nag_SideTypeInput
On entry: indicates how or or or is to be applied to .
or or or is applied to from the left.
or or or is applied to from the right.
Constraint:
or .
4: – Nag_TransTypeInput
On entry: indicates whether or or or is to be applied to .
or is applied to .
or is applied to .
Constraint:
or .
5: – IntegerInput
On entry: , the number of rows of the matrix .
Constraint:
.
6: – IntegerInput
On entry: , the number of columns of the matrix .
Constraint:
.
7: – IntegerInput
On entry: if , the number of columns in the original matrix .
If , the number of rows in the original matrix .
Constraint:
.
8: – const ComplexInput
Note: the dimension, dim, of the array a
must be at least
when
and
;
when
and
;
when
and
;
when
and
.
On entry: details of the vectors which define the elementary reflectors, as returned by f08ksc.
9: – IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array a.
Constraints:
if ,
if , ;
if , ;
if ,
if ,
;
if ,
.
10: – const ComplexInput
Note: the dimension, dim, of the array tau
must be at least
.
On entry: further details of the elementary reflectors, as returned by f08ksc in its argument tauq if , or in its argument taup if .
11: – ComplexInput/Output
Note: the dimension, dim, of the array c
must be at least
when
;
when
.
The th element of the matrix is stored in
when ;
when .
On entry: the matrix .
On exit: c is overwritten by or or or or or or or as specified by vect, side and trans.
12: – IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array c.
Constraints:
if ,
;
if , .
13: – NagError *Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).
6Error Indicators and Warnings
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_2
On entry, , and .
Constraint: if ,
; if ,
.
On entry, , and .
Constraint: if , ;
if , .
NE_INT
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, . Constraint: .
On entry, . Constraint: .
NE_INT_2
On entry, and .
Constraint: .
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.
7Accuracy
The computed result differs from the exact result by a matrix such that
where is the machine precision.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
f08kuc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08kuc makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
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.
9Further Comments
The total number of real floating-point operations is approximately
For this function two examples are presented. Both illustrate how the reduction to bidiagonal form of a matrix may be preceded by a or factorization of .
In the first example, , and
The function first performs a factorization of as and then reduces the factor to bidiagonal form : . Finally it forms and calls f08kuc to form .
In the second example, , and
The function first performs an factorization of as and then reduces the factor to bidiagonal form : . Finally it forms and calls f08kuc to form .