NAG CL Interface
f08kgc (dormbr)
1
Purpose
f08kgc multiplies an arbitrary real
by
matrix
by one of the real orthogonal matrices
or
which were determined by
f08kec when reducing a real matrix to bidiagonal form.
2
Specification
void |
f08kgc (Nag_OrderType order,
Nag_VectType vect,
Nag_SideType side,
Nag_TransType trans,
Integer m,
Integer n,
Integer k,
const double a[],
Integer pda,
const double tau[],
double c[],
Integer pdc,
NagError *fail) |
|
The function may be called by the names: f08kgc, nag_lapackeig_dormbr or nag_dormbr.
3
Description
f08kgc is intended to be used after a call to
f08kec, which reduces a real rectangular matrix
to bidiagonal form
by an orthogonal transformation:
.
f08kec 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 real rectangular matrix).
4
References
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
5
Arguments
Note: in the descriptions below, denotes the order of or : if , and if , .
-
1:
– Nag_OrderType
Input
-
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_VectType
Input
-
On entry: indicates whether
or
or
or
is to be applied to
.
- or is applied to .
- or is applied to .
Constraint:
or .
-
3:
– Nag_SideType
Input
-
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_TransType
Input
-
On entry: indicates whether
or
or
or
is to be applied to
.
- or is applied to .
- or is applied to .
Constraint:
or .
-
5:
– Integer
Input
-
On entry: , the number of rows of the matrix .
Constraint:
.
-
6:
– Integer
Input
-
On entry: , the number of columns of the matrix .
Constraint:
.
-
7:
– Integer
Input
-
On entry: if
, the number of columns in the original matrix
.
If , the number of rows in the original matrix .
Constraint:
.
-
8:
– const double
Input
-
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
f08kec.
-
9:
– Integer
Input
-
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 double
Input
-
Note: the dimension,
dim, of the array
tau
must be at least
.
On entry: further details of the elementary reflectors, as returned by
f08kec in its argument
tauq if
, or in its argument
taup if
.
-
11:
– double
Input/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:
– Integer
Input
-
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).
6
Error 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.
7
Accuracy
The computed result differs from the exact result by a matrix
such that
where
is the
machine precision.
8
Parallelism and Performance
f08kgc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08kgc 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.
The total number of floating-point operations is approximately
- if and , ;
- if and , ;
- if and , ;
- if and , ,
where
is the value of the argument
k.
The complex analogue of this function is
f08kuc.
10
Example
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
f08kgc 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
f08kgc to form
.
10.1
Program Text
10.2
Program Data
10.3
Program Results