NAG Library Function Document

nag_zunmqr (f08auc)

void	nag_zunmqr (Nag_OrderType order, Nag_SideType side, Nag_TransType trans, Integer m, Integer n, Integer k, const Complex a[], Integer pda, const Complex tau[], Complex c[], Integer pdc, NagError *fail)

3 Description

nag_zunmqr (f08auc) is intended to be used after a call to nag_zgeqrf (f08asc), nag_zgeqpf (f08bsc) or nag_zgeqp3 (f08btc), which perform a

Q R

factorization of a complex matrix

A

. The unitary matrix

Q

is represented as a product of elementary reflectors.

This function may be used to form one of the matrix products

Q C, Q^{H} C, C Q ​ or ​ C Q^{H},

overwriting the result on

c

(which may be any complex rectangular matrix).

A common application of this function is in solving linear least squares problems, as described in the f08 Chapter Introduction and illustrated in Section 10 in nag_zgeqrf (f08asc).

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

Nag_ColMajor

2: side – Nag_SideTypeInput

On entry: indicates how

Q

Q^{H}

is to be applied to

C

$side = Nag_LeftSide$: $Q$ or $Q^{H}$ is applied to $C$ from the left.
$side = Nag_RightSide$: $Q$ or $Q^{H}$ is applied to $C$ from the right.

Constraint:

side = Nag_LeftSide

Nag_RightSide

3: trans – Nag_TransTypeInput

On entry: indicates whether

Q

Q^{H}

is to be applied to

C

$trans = Nag_NoTrans$: $Q$ is applied to $C$ .
$trans = Nag_ConjTrans$: $Q^{H}$ is applied to $C$ .

Constraint:

trans = Nag_NoTrans

Nag_ConjTrans

4: m – IntegerInput

On entry:

m

, the number of rows of the matrix

C

Constraint:

m \geq 0

5: n – IntegerInput

On entry:

n

, the number of columns of the matrix

C

Constraint:

n \geq 0

6: k – IntegerInput

On entry:

k

, the number of elementary reflectors whose product defines the matrix

Q

Constraints:

if $side = Nag_LeftSide$ , $m \geq k \geq 0$ ;
if $side = Nag_RightSide$ , $n \geq k \geq 0$ .

7: a[ $\dim$ ] – const ComplexInput

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

$\max (1, pda \times k)$ when $order = Nag_ColMajor$ ;
$\max (1, m \times pda)$ when $order = Nag_RowMajor$ and $side = Nag_LeftSide$ ;
$\max (1, n \times pda)$ when $order = Nag_RowMajor$ and $side = Nag_RightSide$ .

On entry: details of the vectors which define the elementary reflectors, as returned by nag_zgeqrf (f08asc), nag_zgeqpf (f08bsc) or nag_zgeqp3 (f08btc).

8: 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,
- if $side = Nag_LeftSide$ , $pda \geq \max (1, m)$ ;
- if $side = Nag_RightSide$ , $pda \geq \max (1, n)$ ;
if $order = Nag_RowMajor$ , $pda \geq \max (1, k)$ .

9: 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_zgeqrf (f08asc), nag_zgeqpf (f08bsc) or nag_zgeqp3 (f08btc).

10: c[ $\dim$ ] – ComplexInput/Output

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

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

The

(i, j)

th element of the matrix

C

is stored in

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

On entry: the

m

n

matrix

C

On exit: c is overwritten by

Q C

Q^{H} C

C Q

C Q^{H}

as specified by side and trans.

11: pdc – IntegerInput

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

Constraints:

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

12: 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_ENUM_INT_3: On entry, $side = ⟨value⟩$ , $m = ⟨value⟩$ , $n = ⟨value⟩$ and $k = ⟨value⟩$ .
Constraint: if $side = Nag_LeftSide$ , $m \geq k \geq 0$ ;
if $side = Nag_RightSide$ , $n \geq k \geq 0$ .
On entry, $side = ⟨value⟩$ , $m = ⟨value⟩$ , $n = ⟨value⟩$ and $pda = ⟨value⟩$ .
Constraint: if $side = Nag_LeftSide$ , $pda \geq \max (1, m)$ ;
if $side = Nag_RightSide$ , $pda \geq \max (1, n)$ .
NE_INT: On entry, $m = ⟨value⟩$ .
Constraint: $m \geq 0$ .
On entry, $n = ⟨value⟩$ .
Constraint: $n \geq 0$ .
On entry, $pda = ⟨value⟩$ .
Constraint: $pda > 0$ .
On entry, $pdc = ⟨value⟩$ .
Constraint: $pdc > 0$ .
NE_INT_2: On entry, $pda = ⟨value⟩$ and $k = ⟨value⟩$ .
Constraint: $pda \geq \max (1, k)$ .
On entry, $pdc = ⟨value⟩$ and $m = ⟨value⟩$ .
Constraint: $pdc \geq \max (1, m)$ .
On entry, $pdc = ⟨value⟩$ and $n = ⟨value⟩$ .
Constraint: $pdc \geq \max (1, n)$ .
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 result differs from the exact result by a matrix

E

such that

{‖E‖}_{2} = O (ε) {‖C‖}_{2},

where

ε

is the machine precision.

8 Parallelism and Performance

nag_zunmqr (f08auc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

nag_zunmqr (f08auc) 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 real floating-point operations is approximately

8 n k (2 m - k)

side = Nag_LeftSide

and

8 m k (2 n - k)

side = Nag_RightSide

The real analogue of this function is nag_dormqr (f08agc).

10 Example

See Section 10 in nag_zgeqrf (f08asc).

nag_zunmqr (f08auc) (PDF version)

f08 Chapter Contents

f08 Chapter Introduction

NAG Library Manual

NAG Library Function Documentnag_zunmqr (f08auc)

+− 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

NAG Library Function Document

nag_zunmqr (f08auc)