f08aj: FL CL CPP AD PY MB

NAG CL Interface
f08ajc (dorglq)

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NAG Library Manual, Mark 30.1

Interfaces: FL CL CPP AD PY MB

NAG CL Interface Introduction

F08 (Lapackeig) Chapter Contents

F08 (Lapackeig) Chapter Introduction

f08aj: FL CL CPP AD PY MB

▸▿ 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

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1 Purpose

f08ajc generates all or part of the real orthogonal matrix

Q

from an

L Q

factorization computed by f08ahc.

2 Specification

#include <nag.h>

void	f08ajc (Nag_OrderType order, Integer m, Integer n, Integer k, double a[], Integer pda, const double tau[], NagError *fail)

The function may be called by the names: f08ajc, nag_lapackeig_dorglq or nag_dorglq.

3 Description

f08ajc is intended to be used after a call to f08ahc, which performs an

L Q

factorization of a real matrix

A

. The orthogonal matrix

Q

is represented as a product of elementary reflectors.

This function may be used to generate

Q

explicitly as a square matrix, or to form only its leading rows.

Usually

Q

is determined from the

L Q

factorization of a

p \times n

matrix

A

with

p \leq n

. The whole of

Q

may be computed by :

nag_lapackeig_dorglq(order,n,n,p,a,pda,tau,&fail)

(note that the array a must have at least

n

rows) or its leading

p

rows by :

nag_lapackeig_dorglq(order,p,n,p,a,pda,tau,&fail)

The rows of

Q

returned by the last call form an orthonormal basis for the space spanned by the rows of

A

; thus f08ahc followed by f08ajc can be used to orthogonalize the rows of

A

The information returned by the

L Q

factorization functions also yields the

L Q

factorization of the leading

k

rows of

A

, where

k < p

. The orthogonal matrix arising from this factorization can be computed by :

nag_lapackeig_dorglq(order,n,n,k,a,pda,tau,&fail)

or its leading

k

rows by :

nag_lapackeig_dorglq(order,k,n,k,a,pda,tau,&fail)

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_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

order = Nag_RowMajor

. 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:

order = Nag_RowMajor

Nag_ColMajor

2: $m$ – Integer Input

On entry:

m

, the number of rows of the matrix

Q

Constraint:

m \geq 0

3: $n$ – Integer Input

On entry:

n

, the number of columns of the matrix

Q

Constraint:

n \geq m

4: $k$ – Integer Input

On entry:

k

, the number of elementary reflectors whose product defines the matrix

Q

Constraint:

m \geq k \geq 0

5: $a [\dim]$ – double Input/Output

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

$\max (1, pda \times n)$ when $order = Nag_ColMajor$ ;
$\max (1, m \times pda)$ when $order = Nag_RowMajor$ .

On entry: details of the vectors which define the elementary reflectors, as returned by f08ahc.

On exit: the

m \times n

matrix

Q

6: $pda$ – Integer Input

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

Constraints:

if $order = Nag_ColMajor$ , $pda \geq \max (1, m)$ ;
if $order = Nag_RowMajor$ , $pda \geq \max (1, n)$ .

7: $tau [\dim]$ – const double Input

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

\max (1, k)

On entry: further details of the elementary reflectors, as returned by f08ahc.

8: $fail$ – 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 $⟨ value ⟩$ had an illegal value.
NE_INT: On entry, $m = ⟨ value ⟩$ .
Constraint: $m \geq 0$ .

On entry, $pda = ⟨ value ⟩$ .
Constraint: $pda > 0$ .
NE_INT_2: On entry, $m = ⟨ value ⟩$ and $k = ⟨ value ⟩$ .
Constraint: $m \geq k \geq 0$ .

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

On entry, $pda = ⟨ value ⟩$ and $m = ⟨ value ⟩$ .
Constraint: $pda \geq \max (1, m)$ .

On entry, $pda = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $pda \geq \max (1, n)$ .
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.
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 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

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

f08ajc 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.

9 Further Comments

The total number of floating-point operations is approximately

4 m n k - 2 (m + n) k^{2} + \frac{4}{3} k^{3}

; when

m = k

, the number is approximately

\frac{2}{3} m^{2} (3 n - m)

The complex analogue of this function is f08awc.

10 Example

This example forms the leading

4

rows of the orthogonal matrix

Q

from the

L Q

factorization of the matrix

A

, where

A = (\begin{array}{r} - 5.42 & 3.28 & - 3.68 & 0.27 & 2.06 & 0.46 \\ - 1.65 & - 3.40 & - 3.20 & - 1.03 & - 4.06 & - 0.01 \\ - 0.37 & 2.35 & 1.90 & 4.31 & - 1.76 & 1.13 \\ - 3.15 & - 0.11 & 1.99 & - 2.70 & 0.26 & 4.50 \end{array}) .

The rows of

Q

form an orthonormal basis for the space spanned by the rows of

A

f08aj: FL CL CPP AD PY MB

NAG CL Interfacef08ajc (dorglq)

▸▿ 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 CL Interface
f08ajc (dorglq)