void	f08aqc (Nag_OrderType order, Nag_SideType side, Nag_TransType trans, Integer m, Integer n, Integer k, Integer nb, const Complex v[], Integer pdv, const Complex t[], Integer pdt, Complex c[], Integer pdc, NagError *fail)

The function may be called by the names: f08aqc, nag_lapackeig_zgemqrt or nag_zgemqrt.

3 Description

f08aqc is intended to be used after a call to f08apc, which performs 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 f08apc.

4 References

Golub G H and Van Loan C F (2012) Matrix Computations (4th 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: $side$ – Nag_SideType Input

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_TransType Input

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: $m$ – Integer Input

On entry:

m

, the number of rows of the matrix

C

Constraint:

m \geq 0

5: $n$ – Integer Input

On entry:

n

, the number of columns of the matrix

C

Constraint:

n \geq 0

6: $k$ – Integer Input

On entry:

k

, the number of elementary reflectors whose product defines the matrix

Q

. Usually

k = \min (m_{A}, n_{A})

where

m_{A}

n_{A}

are the dimensions of the matrix

A

supplied in a previous call to f08apc.

Constraints:

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

7: $nb$ – Integer Input

On entry: the block size used in the

Q R

factorization performed in a previous call to f08apc; this value must remain unchanged from that call.

Constraints:

$nb \geq 1$ ;
if $k > 0$ , $nb \leq k$ .

8: $v [\dim]$ – const Complex Input

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

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

On entry: details of the vectors which define the elementary reflectors, as returned by f08apc in the first

k

columns of its array argument a.

9: $pdv$ – Integer Input

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

Constraints:

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

10: $t [\dim]$ – const Complex Input

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

$\max (1, pdt \times k)$ when $order = Nag_ColMajor$ ;
$\max (1, nb \times pdt)$ when $order = Nag_RowMajor$ .

The

(i, j)

th element of the matrix

T

is stored in

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

On entry: further details of the unitary matrix

Q

as returned by f08apc. The number of blocks is

b = ⌈\frac{k}{nb}⌉

, where

k = \min (m, n)

and each block is of order nb except for the last block, which is of order

k - (b - 1) \times nb

. For the

b

blocks the upper triangular block reflector factors

T_{1}, T_{2}, \dots, T_{b}

are stored in the

nb

n

matrix

T

T = [T_{1} | T_{2} | \dots | T_{b}]

11: $pdt$ – Integer Input

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

Constraints:

if $order = Nag_ColMajor$ , $pdt \geq nb$ ;
if $order = Nag_RowMajor$ , $pdt \geq \max (1, k)$ .

12: $c [\dim]$ – Complex Input/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.

13: $pdc$ – Integer Input

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

14: $fail$ – 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_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 $pdv = 〈value〉$ .
Constraint: if $side = Nag_LeftSide$ , $pdv \geq \max (1, m)$ ;
if $side = Nag_RightSide$ , $pdv \geq \max (1, n)$ .
NE_INT: On entry, $m = 〈value〉$ .
Constraint: $m \geq 0$ .

On entry, $n = 〈value〉$ .
Constraint: $n \geq 0$ .
NE_INT_2: On entry, $nb = 〈value〉$ and $k = 〈value〉$ .
Constraint: $nb \geq 1$ and
if $k > 0$ , $nb \leq k$ .

On entry, $pdc = 〈value〉$ and $m = 〈value〉$ .
Constraint: $pdc \geq \max (1, m)$ .

On entry, $pdc = 〈value〉$ and $n = 〈value〉$ .
Constraint: $pdc \geq \max (1, n)$ .

On entry, $pdt = 〈value〉$ and $k = 〈value〉$ .
Constraint: $pdt \geq \max (1, k)$ .

On entry, $pdt = 〈value〉$ and $nb = 〈value〉$ .
Constraint: $pdt \geq nb$ .

On entry, $pdv = 〈value〉$ and $k = 〈value〉$ .
Constraint: $pdv \geq \max (1, k)$ .
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 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

f08aqc 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 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 f08acc.

10 Example

See Section 10 in f08apc.

NAG Library Manual, Mark 27

Interfaces: FL CL AD

NAG CL Interface Introduction

F08 (Lapackeig) Chapter Contents

F08 (Lapackeig) Chapter Introduction

f08aq: FL CL AD

NAG CL Interfacef08aqc (zgemqrt)

▸▿ 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 CL Interface
f08aqc (zgemqrt)