NAG Library Function Document
nag_dormbr (f08kgc)
1
Purpose
nag_dormbr (f08kgc) multiplies an arbitrary real
by
matrix
by one of the real orthogonal matrices
or
which were determined by
nag_dgebrd (f08kec) when reducing a real matrix to bidiagonal form.
2
Specification
#include <nag.h> |
#include <nagf08.h> |
void |
nag_dormbr (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) |
|
3
Description
nag_dormbr (f08kgc) is intended to be used after a call to
nag_dgebrd (f08kec), which reduces a real rectangular matrix
to bidiagonal form
by an orthogonal transformation:
.
nag_dgebrd (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_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 3.3.1.3 in How to Use the NAG Library and its Documentation 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 doubleInput
-
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
nag_dgebrd (f08kec).
- 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 doubleInput
-
Note: the dimension,
dim, of the array
tau
must be at least
.
On entry: further details of the elementary reflectors, as returned by
nag_dgebrd (f08kec) in its argument
tauq if
, or in its argument
taup if
.
- 11:
– doubleInput/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 3.7 in How to Use the NAG Library and its Documentation).
6
Error Indicators and Warnings
- NE_ALLOC_FAIL
-
Dynamic memory allocation failed.
See
Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
- NE_BAD_PARAM
-
On entry, argument had an illegal value.
- NE_ENUM_INT_2
-
On entry, , , .
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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
- NE_NO_LICENCE
-
Your licence key may have expired or may not have been installed correctly.
See
Section 2.7.5 in How to Use the NAG Library and its Documentation 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
nag_dormbr (f08kgc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_dormbr (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
nag_zunmbr (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
nag_dormbr (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
nag_dormbr (f08kgc) to form
.
10.1
Program Text
Program Text (f08kgce.c)
10.2
Program Data
Program Data (f08kgce.d)
10.3
Program Results
Program Results (f08kgce.r)