NAG Library Function Document

nag_zunglq (f08awc)

+− 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_zunglq (f08awc) generates all or part of the complex unitary matrix

Q

from an

L Q

factorization computed by nag_zgelqf (f08avc).

2 Specification

#include <nag.h>

#include <nagf08.h>

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

3 Description

nag_zunglq (f08awc) is intended to be used after a call to nag_zgelqf (f08avc), which performs an

L Q

factorization of a complex matrix

A

. The unitary 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

n

matrix

A

with

p \leq n

. The whole of

Q

may be computed by:

nag_zunglq(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_zunglq(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 nag_zgelqf (f08avc) followed by nag_zunglq (f08awc) can be used to orthogonalise 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 unitary matrix arising from this factorization can be computed by:

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

or its leading

k

rows by:

nag_zunglq(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_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

Q

Constraint:

m \geq 0

3: n – IntegerInput

On entry:

n

, the number of columns of the matrix

Q

Constraint:

n \geq m

4: k – IntegerInput

On entry:

k

, the number of elementary reflectors whose product defines the matrix

Q

Constraint:

m \geq k \geq 0

5: 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 $order = Nag_RowMajor$ .

On entry: details of the vectors which define the elementary reflectors, as returned by nag_zgelqf (f08avc).

On exit: the

m

n

matrix

Q

order = Nag_ColMajor

, the

(i, j)

th element of the matrix is stored in

a [(j - 1) \times pda + i - 1]

order = Nag_RowMajor

, the

(i, j)

th element of the matrix is stored in

a [(i - 1) \times pda + j - 1]

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

7: tau[ $\dim$ ] – const ComplexInput

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 nag_zgelqf (f08avc).

8: 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, $pda = 〈value〉$ .
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.

7 Accuracy

The computed matrix

Q

differs from an exactly unitary matrix by a matrix

E

such that

{‖E‖}_{2} = O (ε),

where

ε

is the machine precision.

8 Further Comments

The total number of real floating point operations is approximately

16 m n k - 8 (m + n) k^{2} + \frac{16}{3} k^{3}

; when

m = k

, the number is approximately

\frac{8}{3} m^{2} (3 n - m)

The real analogue of this function is nag_dorglq (f08ajc).

9 Example

This example forms the leading

4

rows of the unitary matrix

Q

from the

L Q

factorization of the matrix

A

, 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}) .

The rows of

Q

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

A

NAG Library Function Documentnag_zunglq (f08awc)

+− 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_zunglq (f08awc)