NAG Library Function Document

nag_dopgtr (f08gfc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

+− 10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1 Purpose

nag_dopgtr (f08gfc) generates the real orthogonal matrix

Q

, which was determined by nag_dsptrd (f08gec) when reducing a symmetric matrix to tridiagonal form.

2 Specification

#include <nag.h>

#include <nagf08.h>

void	nag_dopgtr (Nag_OrderType order, Nag_UploType uplo, Integer n, const double ap[], const double tau[], double q[], Integer pdq, NagError *fail)

3 Description

nag_dopgtr (f08gfc) is intended to be used after a call to nag_dsptrd (f08gec), which reduces a real symmetric matrix

A

to symmetric tridiagonal form

T

by an orthogonal similarity transformation:

A = Q T Q^{T}

. nag_dsptrd (f08gec) represents the orthogonal matrix

Q

as a product of

n - 1

elementary reflectors.

This function may be used to generate

Q

explicitly as a square matrix.

4 References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

5 Arguments

1: order – 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

order = Nag_RowMajor

. See Section 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.

Constraint:

order = Nag_RowMajor

Nag_ColMajor

2: uplo – Nag_UploTypeInput

On entry: this must be the same argument uplo as supplied to nag_dsptrd (f08gec).

Constraint:

uplo = Nag_Upper

Nag_Lower

3: n – IntegerInput

On entry:

n

, the order of the matrix

Q

Constraint:

n \geq 0

4: ap[ $\dim$ ] – const doubleInput

Note: the dimension, dim, of the array ap must be at least

\max (1, n \times (n + 1) / 2)

On entry: details of the vectors which define the elementary reflectors, as returned by nag_dsptrd (f08gec).

5: tau[ $\dim$ ] – const doubleInput

Note: the dimension, dim, of the array tau must be at least

\max (1, n - 1)

On entry: further details of the elementary reflectors, as returned by nag_dsptrd (f08gec).

6: q[ $\dim$ ] – doubleOutput

Note: the dimension, dim, of the array q must be at least

\max (1, pdq \times n)

The

(i, j)

th element of the matrix

Q

is stored in

$q [(j - 1) \times pdq + i - 1]$ when $order = Nag_ColMajor$ ;
$q [(i - 1) \times pdq + j - 1]$ when $order = Nag_RowMajor$ .

On exit: the

n

n

orthogonal matrix

Q

7: pdq – IntegerInput

On entry: the stride separating row or column elements (depending on the value of order) in the array q.

Constraint:

pdq \geq \max (1, n)

8: fail – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
NE_BAD_PARAM: On entry, argument $⟨value⟩$ had an illegal value.
NE_INT: On entry, $n = ⟨value⟩$ .
Constraint: $n \geq 0$ .
On entry, $pdq = ⟨value⟩$ .
Constraint: $pdq > 0$ .
NE_INT_2: On entry, $pdq = ⟨value⟩$ and $n = ⟨value⟩$ .
Constraint: $pdq \geq \max (1, n)$ .
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.

7 Accuracy

The computed matrix

Q

differs from an exactly orthogonal matrix by a matrix

E

such that

{‖E‖}_{2} = O (ε),

where

ε

is the machine precision.

8 Parallelism and Performance

nag_dopgtr (f08gfc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

nag_dopgtr (f08gfc) 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 Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

The total number of floating-point operations is approximately

\frac{4}{3} n^{3}

The complex analogue of this function is nag_zupgtr (f08gtc).

10 Example

This example computes all the eigenvalues and eigenvectors of the matrix

A

, where

A = (\begin{array}{r} 2.07 & 3.87 & 4.20 & - 1.15 \\ 3.87 & - 0.21 & 1.87 & 0.63 \\ 4.20 & 1.87 & 1.15 & 2.06 \\ - 1.15 & 0.63 & 2.06 & - 1.81 \end{array}),

using packed storage. Here

A

is symmetric and must first be reduced to tridiagonal form by nag_dsptrd (f08gec). The program then calls nag_dopgtr (f08gfc) to form

Q

, and passes this matrix to nag_dsteqr (f08jec) which computes the eigenvalues and eigenvectors of

A

NAG Library Function Documentnag_dopgtr (f08gfc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG Library Function Document

nag_dopgtr (f08gfc)