NAG Library Function Document
nag_dopmtr (f08ggc)
1
Purpose
nag_dopmtr (f08ggc) multiplies an arbitrary real matrix
by the real orthogonal matrix
which was determined by
nag_dsptrd (f08gec) when reducing a real symmetric matrix to tridiagonal form.
2
Specification
#include <nag.h> |
#include <nagf08.h> |
void |
nag_dopmtr (Nag_OrderType order,
Nag_SideType side,
Nag_UploType uplo,
Nag_TransType trans,
Integer m,
Integer n,
double ap[],
const double tau[],
double c[],
Integer pdc,
NagError *fail) |
|
3
Description
nag_dopmtr (f08ggc) is intended to be used after a call to
nag_dsptrd (f08gec), which reduces a real symmetric matrix
to symmetric tridiagonal form
by an orthogonal similarity transformation:
.
nag_dsptrd (f08gec) 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_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_SideTypeInput
-
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_UploTypeInput
-
On entry: this
must be the same argument
uplo as supplied to
nag_dsptrd (f08gec).
Constraint:
or .
- 4:
– Nag_TransTypeInput
-
On entry: indicates whether
or
is to be applied to
.
- is applied to .
- is applied to .
Constraint:
or .
- 5:
– IntegerInput
-
On entry: , the number of rows of the matrix ; is also the order of if .
Constraint:
.
- 6:
– IntegerInput
-
On entry: , the number of columns of the matrix ; is also the order of if .
Constraint:
.
- 7:
– doubleInput/Output
-
Note: the dimension,
dim, of the array
ap
must be at least
- when ;
- when .
On entry: details of the vectors which define the elementary reflectors, as returned by
nag_dsptrd (f08gec).
On exit: is used as internal workspace prior to being restored and hence is unchanged.
- 8:
– const doubleInput
-
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
nag_dsptrd (f08gec).
- 9:
– 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 by matrix .
On exit:
c is overwritten by
or
or
or
as specified by
side and
trans.
- 10:
– IntegerInput
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
c.
Constraints:
- if ,
;
- if , .
- 11:
– 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_INT
-
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_dopmtr (f08ggc) 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
nag_zupmtr (f08guc).
10
Example
This example computes the two smallest eigenvalues, and the associated eigenvectors, of the matrix
, where
using packed storage. Here
is symmetric and must first be reduced to tridiagonal form
by
nag_dsptrd (f08gec). The program then calls
nag_dstebz (f08jjc) to compute the requested eigenvalues and
nag_dstein (f08jkc) to compute the associated eigenvectors of
. Finally
nag_dopmtr (f08ggc) is called to transform the eigenvectors to those of
.
10.1
Program Text
Program Text (f08ggce.c)
10.2
Program Data
Program Data (f08ggce.d)
10.3
Program Results
Program Results (f08ggce.r)