NAG CL Interface
f08fgc (dormtr)
1
Purpose
f08fgc multiplies an arbitrary real matrix
by the real orthogonal matrix
which was determined by
f08fec when reducing a real symmetric matrix to tridiagonal form.
2
Specification
void |
f08fgc (Nag_OrderType order,
Nag_SideType side,
Nag_UploType uplo,
Nag_TransType trans,
Integer m,
Integer n,
const double a[],
Integer pda,
const double tau[],
double c[],
Integer pdc,
NagError *fail) |
|
The function may be called by the names: f08fgc, nag_lapackeig_dormtr or nag_dormtr.
3
Description
f08fgc is intended to be used after a call to
f08fec, which reduces a real symmetric matrix
to symmetric tridiagonal form
by an orthogonal similarity transformation:
.
f08fec represents the orthogonal matrix
as a product 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).
A common application of this function is to transform a matrix of eigenvectors of to the matrix of eigenvectors of .
4
References
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
5
Arguments
-
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_SideType
Input
-
On entry: indicates how
or
is to be applied to
.
- or is applied to from the left.
- or is applied to from the right.
Constraint:
or .
-
3:
– Nag_UploType
Input
-
On entry: this
must be the same argument
uplo as supplied to
f08fec.
Constraint:
or .
-
4:
– Nag_TransType
Input
-
On entry: indicates whether
or
is to be applied to
.
- is applied to .
- is applied to .
Constraint:
or .
-
5:
– Integer
Input
-
On entry: , the number of rows of the matrix ; is also the order of if .
Constraint:
.
-
6:
– Integer
Input
-
On entry: , the number of columns of the matrix ; is also the order of if .
Constraint:
.
-
7:
– const double
Input
-
Note: the dimension,
dim, of the array
a
must be at least
- when
;
- when
.
On entry: details of the vectors which define the elementary reflectors, as returned by
f08fec.
-
8:
– Integer
Input
On entry: the stride separating row or column elements (depending on the value of
order) of the matrix
in the array
a.
Constraints:
- if , ;
- if , .
-
9:
– const double
Input
-
Note: the dimension,
dim, of the array
tau
must be at least
- when ;
- when .
On entry: further details of the elementary reflectors, as returned by
f08fec.
-
10:
– 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 by matrix .
On exit:
c is overwritten by
or
or
or
as specified by
side and
trans.
-
11:
– Integer
Input
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
c.
Constraints:
- if ,
;
- if , .
-
12:
– 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_3
-
On entry, , , and .
Constraint: if , ;
if , .
- NE_INT
-
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
f08fgc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08fgc 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 .
The complex analogue of this function is
f08fuc.
10
Example
This example computes the two smallest eigenvalues, and the associated eigenvectors, of the matrix
, where
Here
is symmetric and must first be reduced to tridiagonal form
by
f08fec. The program then calls
f08jjc to compute the requested eigenvalues and
f08jkc to compute the associated eigenvectors of
. Finally
f08fgc is called to transform the eigenvectors to those of
.
10.1
Program Text
10.2
Program Data
10.3
Program Results