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 <nag.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)

The routine may be called by the names f01qkf or nagf_matop_real_gen_rq_formq.

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: $m$ – Integer Input

On entry:

m

, the number of rows of the matrix

A

Constraint:

m \geq 0

3: $n$ – Integer Input

On entry:

n

, the number of columns of the matrix

A

Constraint:

n \geq m

4: $nrowp$ – Integer Input

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) array Input/Output

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

\max (1, n)

On entry: the leading

m \times m

strictly lower triangular part of the array a, and the

m \times (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 \times n

orthogonal matrix

P^{T}

6: $lda$ – Integer Input

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) array Input

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) array Workspace

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

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

9: $ifail$ – Integer Input/Output

On entry: ifail must be set to

0

−1

1

to set behaviour on detection of an error; these values have no effect when no error is detected.

A value of

0

causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of

−1

means that an error message is printed while a value of

1

means that it is not.

If halting is not appropriate, the value

−1

1

is recommended. If message printing is undesirable, then the value

1

is recommended. Otherwise, the value

0

is recommended. 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, $lda = ⟨ value ⟩$ , $m = ⟨ value ⟩$ and $nrowp = ⟨ value ⟩$ .
Constraint: $lda \geq \max (m, nrowp)$ .

On entry, $m = ⟨ value ⟩$ .
Constraint: $m \geq 0$ .

On entry, $n = ⟨ value ⟩$ and $m = ⟨ value ⟩$ .
Constraint: $n \geq m$ .

On entry, $nrowp = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $nrowp \geq 0$ and $nrowp \leq n$ .

On entry, $wheret = ⟨ value ⟩$ .
Constraint: $wheret ='I'$ or $'S'$ .

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface 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

Background information to multithreading can be found in the Multithreading documentation.

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 \times 5

orthogonal matrix

P

following the

R Q

factorization of the

3 \times 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}) .

f01qk: FL CL CPP AD PY MB

NAG FL Interfacef01qkf (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

NAG FL Interface
f01qkf (real_gen_rq_formq)