NAG Library Function Document

nag_dgerqf (f08chc)

+− 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_dgerqf (f08chc) computes an RQ factorization of a real

m

n

matrix

A

2 Specification

#include <nag.h>

#include <nagf08.h>

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

3 Description

nag_dgerqf (f08chc) forms the

R Q

factorization of an arbitrary rectangular real

m

n

matrix. If

m \leq n

, the factorization is given by

A = (\begin{matrix} 0 & R \end{matrix}) Q,

where

R

is an

m

m

lower triangular matrix and

Q

is an

n

n

orthogonal matrix. If

m > n

the factorization is given by

A = R Q,

where

R

is an

m

n

upper trapezoidal matrix and

Q

is again an

n

n

orthogonal matrix. In the case where

m < n

the factorization can be expressed as

A = (\begin{matrix} 0 & R \end{matrix}) (\begin{matrix} Q_{1} \\ Q_{2} \end{matrix}) = R Q_{2},

where

Q_{1}

consists of the first

(n - m)

rows of

Q

and

Q_{2}

the remaining

m

rows.

The matrix

Q

is not formed explicitly, but is represented as a product of

\min (m, n)

elementary reflectors (see the f08 Chapter Introduction for details). Functions are provided to work with

Q

in this representation (see Section 9).

4 References

Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia http://www.netlib.org/lapack/lug

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: 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 0

4: a[ $\dim$ ] – doubleInput/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 $order = Nag_RowMajor$ .

Where

A (i, j)

appears in this document, it refers to the array element

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

On entry: the

m

n

matrix

A

On exit: if

m \leq n

, the upper triangle of the subarray

A (1 : m, n - m + 1 : n)

contains the

m

m

upper triangular matrix

R

m \geq n

, the elements on and above the

(m - n)

th subdiagonal contain the

m

n

upper trapezoidal matrix

R

; the remaining elements, with the array tau, represent the orthogonal matrix

Q

as a product of

\min (m, n)

elementary reflectors (see Section 3.3.6 in the f08 Chapter Introduction).

5: pda – IntegerInput

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 $order = Nag_RowMajor$ , $pda \geq \max (1, n)$ .

6: tau[ $\dim$ ] – doubleOutput

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

\max (1, \min (m, n))

On exit: the scalar factors of the elementary reflectors.

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

7 Accuracy

The computed factorization is the exact factorization of a nearby matrix

A + E

, where

{‖E‖}_{2} = O ε {‖A‖}_{2}

and

ε

is the machine precision.

8 Parallelism and Performance

nag_dgerqf (f08chc) is not threaded by NAG in any implementation.

nag_dgerqf (f08chc) 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{2}{3} m^{2} (3 n - m)

m \leq n

, or

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

m > n

To form the orthogonal matrix

Q

nag_dgerqf (f08chc) may be followed by a call to nag_dorgrq (f08cjc):

nag_dorgrq(order, n, n, minmn, a, pda, tau, &fail)

where

minmn = \min (m, n)

, but note that the first dimension of the array a must be at least n, which may be larger than was required by nag_dgerqf (f08chc). When

m \leq n

, it is often only the first

m

rows of

Q

that are required and they may be formed by the call:

nag_dorgrq(order, m, n, m, a, pda, tau, c, pdc, &fail)

To apply

Q

to an arbitrary real rectangular matrix

C

, nag_dgerqf (f08chc) may be followed by a call to nag_dormrq (f08ckc). For example:

nag_dormrq(Nag_LeftSide, Nag_Trans, n, p, minmn, a, pda, tau, c, pdc, &fail)

forms

C = Q^{T} C

, where

C

n

p

The complex analogue of this function is nag_zgerqf (f08cvc).

10 Example

This example finds the minimum norm solution to the underdetermined equations

A x = b

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}) and b = (\begin{array}{r} - 2.87 \\ 1.63 \\ - 3.52 \\ 0.45 \end{array}) .

The solution is obtained by first obtaining an

R Q

factorization of the matrix

A

NAG Library Function Documentnag_dgerqf (f08chc)

+− 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_dgerqf (f08chc)