NAG Library Function Document

nag_zgerqf (f08cvc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

nag_zgerqf (f08cvc) computes an RQ factorization of a complex

m

n

matrix

A

2 Specification

#include <nag.h>

#include <nagf08.h>

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

3 Description

nag_zgerqf (f08cvc) 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

unitary 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

unitary 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 8).

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

or 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[ $\dim$ ] – ComplexInput/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 unitary 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[ $\dim$ ] – ComplexOutput

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 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 unitary matrix

Q

nag_zgerqf (f08cvc) may be followed by a call to nag_zungrq (f08cwc):

nag_zungrq(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_zgerqf (f08cvc). 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_zungrq(order, m, n, m, a, pda, tau, c, pdc, &fail)

To apply

Q

to an arbitrary real rectangular matrix

C

, nag_zgerqf (f08cvc) may be followed by a call to nag_zunmrq (f08cxc). For example:

nag_zunmrq(Nag_LeftSide, Nag_ConjTrans, n, p, minmn, a, pda, tau, c, pdc, &fail)

forms

C = Q^{H} C

, where

C

n

p

The real analogue of this function is nag_dgerqf (f08chc).

9 Example

This example finds the minimum norm solution to the underdetermined equations

A x = b

where

A = (\begin{array}{r} 0.28 - 0.36 i & 0.50 - 0.86 i & - 0.77 - 0.48 i & 1.58 + 0.66 i \\ - 0.50 - 1.10 i & - 1.21 + 0.76 i & - 0.32 - 0.24 i & - 0.27 - 1.15 i \\ 0.36 - 0.51 i & - 0.07 + 1.33 i & - 0.75 + 0.47 i & - 0.08 + 1.01 i \end{array})

and

b = (\begin{array}{r} - 1.35 + 0.19 i \\ 9.41 - 3.56 i \\ - 7.57 + 6.93 i \end{array}) .

The solution is obtained by first obtaining an

R Q

factorization of the matrix

A

NAG Library Function Documentnag_zgerqf (f08cvc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Function Document

nag_zgerqf (f08cvc)