Integer, Intent (In)	::	m, n, nrowp, lda
Integer, Intent (Inout)	::	ifail
Real (Kind=nag_wp), Intent (In)	::	zeta(*)
Real (Kind=nag_wp), Intent (Inout)	::	a(lda,*)
Real (Kind=nag_wp), Intent (Out)	::	work(max(m-1,nrowp-m,1))
Character (1), Intent (In)	::	wheret

C Header Interface

#include nagmk26.h

void	f01qkf_ ( const char wheret, const Integer m, const Integer n, const Integer nrowp, double a[], const Integer lda, const double zeta[], double work[], Integer ifail, const Charlen length_wheret)

3

Description

P

is assumed to be given by

P = P_{m} P_{m - 1} \dots P_{1}

where

\begin{matrix} P_{k} = I - u_{k} u_{k}^{T}, \\ u_{k} = (\begin{matrix} w_{k} \\ ζ_{k} \\ 0 \\ z_{k} \end{matrix}), \end{matrix}

ζ_{k}

is a scalar,

w_{k}

is a (

k - 1

) element vector and

z_{k}

is an (

n - m

) element vector.

w_{k}

must be supplied in the

k

th row of a in elements

a (k, 1), \dots, a (k, k - 1)

z_{k}

must be supplied in the

k

th row of a in elements

a (k, m + 1), \dots, a (k, n)

and

ζ_{k}

must be supplied either in

a (k, k)

or in

zeta (k)

, depending upon the argument wheret.

4

References

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

Wilkinson J H (1965) The Algebraic Eigenvalue Problem Oxford University Press, Oxford

5

Arguments

1: $wheret$ – Character(1)Input

On entry: indicates where the elements of

ζ

are to be found.

$wheret ='I'$ (In a): The elements of $ζ$ are in a.
$wheret ='S'$ (Separate): The elements of $ζ$ are separate from a, in zeta.

Constraint:

wheret ='I'

'S'

2: $m$ – IntegerInput

On entry:

m

, the number of rows of the matrix

A

Constraint:

m \geq 0

3: $n$ – IntegerInput

On entry:

n

, the number of columns of the matrix

A

Constraint:

n \geq m

4: $nrowp$ – IntegerInput

On entry:

ℓ

, the required number of rows of

P

nrowp = 0

, an immediate return is effected.

Constraint:

0 \leq nrowp \leq n

5: $a (lda *)$ – Real (Kind=nag_wp) arrayInput/Output

Note: the second dimension of the array a must be at least

\max (1, n)

On entry: the leading

m

m

strictly lower triangular part of the array a, and the

m

(n - m)

rectangular part of a with top left-hand corner at element

a (1, m + 1)

must contain details of the matrix

P

. In addition, if

wheret ='I'

, the diagonal elements of a must contain the elements of

ζ

On exit: the first nrowp rows of the array a are overwritten by the first nrowp rows of the

n

n

orthogonal matrix

P^{T}

6: $lda$ – IntegerInput

On entry: the first dimension of the array a as declared in the (sub)program from which f01qkf is called.

Constraint:

lda \geq \max (1, m, nrowp)

7: $zeta (*)$ – Real (Kind=nag_wp) arrayInput

Note: the dimension of the array zeta must be at least

\max (1, m)

wheret ='S'

, and at least

1

otherwise.

On entry: with

wheret ='S'

, the array zeta must contain the elements of

ζ

. If

zeta (k) = 0.0

then

P_{k}

is assumed to be

I

, otherwise

zeta (k)

is assumed to contain

ζ_{k}

When

wheret ='I'

, the array zeta is not referenced.

8: $work (\max (m - 1, nrowp - m, 1))$ – Real (Kind=nag_wp) arrayWorkspace

Note: the dimension of the array work must be at least

\max (m - 1, nrowp - m, 1)

9: $ifail$ – IntegerInput/Output

On entry: ifail must be set to

0

- 1 ​ or ​ 1

. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value

- 1 ​ or ​ 1

is recommended. If the output of error messages is undesirable, then the value

1

is recommended. Otherwise, if you are not familiar with this argument, the recommended value is

0

. When the value $- 1 or 1$ is used it is essential to test the value of ifail on exit.

On exit:

ifail = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

6

Error Indicators and Warnings

If on entry

ifail = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by x04aaf).

Errors or warnings detected by the routine:

$ifail = - 1$

On entry,	$wheret \neq'I'$ or $'S'$ ,
or	$m < 0$ ,
or	$n < m$ ,
or	$nrowp < 0$ or $nrowp > n$ ,
or	$lda < \max (m, nrowp)$ .

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

7

Accuracy

The computed matrix

P

satisfies the relation

P = Q + E,

where

Q

is an exactly orthogonal matrix and

‖E‖ \leq c ε,

ε

is the machine precision (see x02ajf),

c

is a modest function of

n

, and

‖.‖

denotes the spectral (two) norm. See also Section 7 in f01qjf.

8

Parallelism and Performance

f01qkf 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 routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9

Further Comments

The approximate number of floating-point operations is given by

\begin{array}{l} \frac{2}{3} m \{(3 n - m) (2 ℓ - m) - m (ℓ - m)\}, & if ​ ℓ \geq m, and ​ \\ \frac{2}{3} ℓ^{2} (3 n - ℓ), & if ​ ℓ < m . \end{array}

10

Example

This example obtains the

5

5

orthogonal matrix

P

following the

R Q

factorization of the

3

5

matrix

A

given by

A = (\begin{array}{r} 2.0 & 2.0 & 1.6 & 2.0 & 1.2 \\ 2.5 & 2.5 & - 0.4 & - 0.5 & - 0.3 \\ 2.5 & 2.5 & 2.8 & 0.5 & - 2.9 \end{array}) .

NAG Library Routine Document

f01qkf (real_gen_rq_formq)

▸▿ 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

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