void	f08bqc (Nag_OrderType order, Nag_SideType side, Nag_TransType trans, Integer m, Integer n, Integer k, Integer l, Integer nb, const Complex v[], Integer pdv, const Complex t[], Integer pdt, Complex c1[], Integer pdc1, Complex c2[], Integer pdc2, NagError *fail)

The function may be called by the names: f08bqc, nag_lapackeig_ztpmqrt or nag_ztpmqrt.

3 Description

f08bqc is intended to be used after a call to f08bpc which performs a

Q R

factorization of a triangular-pentagonal matrix containing an upper triangular matrix

A

over a pentagonal matrix

B

. The unitary matrix

Q

is represented as a product of elementary reflectors.

This function may be used to form the matrix products

Q C, Q^{H} C, C Q ​ or ​ C Q^{H},

where the complex rectangular

m_{c} \times n_{c}

matrix

C

is split into component matrices

C_{1}

and

C_{2}

Q

is being applied from the left (

Q C

Q^{H} C

) then

C = (\begin{matrix} C_{1} \\ C_{2} \end{matrix})

where

C_{1}

k \times n_{c}

C_{2}

m_{v} \times n_{c}

m_{c} = k + m_{v}

is fixed and

m_{v}

is the number of rows of the matrix

V

containing the elementary reflectors (i.e., m as passed to f08bpc); the number of columns of

V

n_{v}

(i.e., n as passed to f08bpc).

Q

is being applied from the right (

C Q

C Q^{H}

) then

C = (\begin{matrix} C_{1} & C_{2} \end{matrix})

where

C_{1}

m_{c} \times k

, and

C_{2}

m_{c} \times m_{v}

and

n_{c} = k + m_{v}

is fixed.

The matrices

C_{1}

and

C_{2}

are overwriten by the result of the matrix product.

A common application of this routine is in updating the solution of a linear least squares problem as illustrated in Section 10 in f08bpc.

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: the number of rows of the matrix

C_{2}

, that is,

if $side = Nag_LeftSide$: then $m_{v}$ , the number of rows of the matrix $V$ ;
if $side = Nag_RightSide$: then $m_{c}$ , the number of rows of the matrix $C$ .

Constraint:

m \geq 0

5: $n$ – Integer Input

On entry: the number of columns of the matrix

C_{2}

, that is,

if $side = Nag_LeftSide$: then $n_{c}$ , the number of columns of the matrix $C$ ;
if $side = Nag_RightSide$: then $n_{v}$ , the number of columns of the matrix $V$ .

Constraint:

n \geq 0

6: $k$ – Integer Input

On entry:

k

, the number of elementary reflectors whose product defines the matrix

Q

Constraint:

k \geq 0

7: $l$ – Integer Input

On entry:

l

, the number of rows of the upper trapezoidal part of the pentagonal composite matrix

V

, passed (as b) in a previous call to f08bpc. This must be the same value used in the previous call to f08bpc (see l in f08bpc).

Constraint:

0 \leq l \leq k

8: $nb$ – Integer Input

On entry:

nb

, the blocking factor used in a previous call to f08bpc to compute the

Q R

factorization of a triangular-pentagonal matrix containing composite matrices

A

and

B

Constraints:

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

9: $v [\dim]$ – 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 $side = Nag_RightSide$ .

The

(i, j)

th element of the matrix

V

is stored in

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

On entry: the

m_{v} \times n_{v}

matrix

V

; this should remain unchanged from the array b returned by a previous call to f08bpc.

10: $pdv$ – 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 $order = Nag_RowMajor$ , $pdv \geq \max (1, k)$ .

11: $t [\dim]$ – 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: this must remain unchanged from a previous call to f08bpc (see t in f08bpc).

12: $pdt$ – 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 $order = Nag_RowMajor$ , $pdt \geq \max (1, k)$ .

13: $c1 [\dim]$ – Complex Input/Output

Note: the dimension, dim, of the array c1 must be at least

$\max (1, pdc1 \times n)$ when $side = Nag_LeftSide$ and $order = Nag_ColMajor$ ;
$\max (1, k \times pdc1)$ when $side = Nag_LeftSide$ and $order = Nag_RowMajor$ ;
$\max (1, pdc1 \times k)$ when $side = Nag_RightSide$ and $order = Nag_ColMajor$ ;
$\max (1, m \times pdc1)$ when $side = Nag_RightSide$ and $order = Nag_RowMajor$ .

On entry:

C_{1}

, the first part of the composite matrix

C

if $side = Nag_LeftSide$: then c1 contains the first $k$ rows of $C$ ;
if $side = Nag_RightSide$: then c1 contains the first $k$ columns of $C$ .

On exit: c1 is overwritten by the corresponding block of

Q C

Q^{H} C

C Q

C Q^{H}

14: $pdc1$ – Integer Input

On entry: the stride separating row or column elements (depending on the value of order) in the array c1.

Constraints:

if $order = Nag_ColMajor$ ,
- if $side = Nag_LeftSide$ , $pdc1 \geq \max (1, k)$ ;
- if $side = Nag_RightSide$ , $pdc1 \geq \max (1, m)$ ;
if $order = Nag_RowMajor$ ,
- if $side = Nag_LeftSide$ , $pdc1 \geq \max (1, n)$ ;
- if $side = Nag_RightSide$ , $pdc1 \geq \max (1, k)$ .

15: $c2 [\dim]$ – Complex Input/Output

Note: the dimension, dim, of the array c2 must be at least

$\max (1, pdc2 \times n)$ when $order = Nag_ColMajor$ ;
$\max (1, m \times pdc2)$ when $order = Nag_RowMajor$ .

On entry:

C_{2}

, the second part of the composite matrix

C

if $side = Nag_LeftSide$: then c2 contains the remaining $m_{v}$ rows of $C$ ;
if $side = Nag_RightSide$: then c2 contains the remaining $m_{v}$ columns of $C$ ;

On exit: c2 is overwritten by the corresponding block of

Q C

Q^{H} C

C Q

C Q^{H}

16: $pdc2$ – Integer Input

On entry: the stride separating row or column elements (depending on the value of order) in the array c2.

Constraints:

if $order = Nag_ColMajor$ , $pdc2 \geq \max (1, m)$ ;
if $order = Nag_RowMajor$ , $pdc2 \geq \max (1, n)$ .

17: $fail$ – 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, $side = ⟨ value ⟩$ , $k = ⟨ value ⟩$ , $m = ⟨ value ⟩$ and $pdc1 = ⟨ value ⟩$ .
Constraint: if $side = Nag_LeftSide$ , $pdc1 \geq \max (1, k)$ ;
if $side = Nag_RightSide$ , $pdc1 \geq \max (1, m)$ .

On entry, $side = ⟨ value ⟩$ , $m = ⟨ value ⟩$ , $n = ⟨ value ⟩$ and $pdv = ⟨ value ⟩$ .
Constraint: if $side = Nag_LeftSide$ , $pdv \geq \max (1, m)$ ;
if $side = Nag_RightSide$ , $pdv \geq \max (1, n)$ .

On entry, $side = ⟨ value ⟩$ , $pdc1 = ⟨ value ⟩$ , $n = ⟨ value ⟩$ and $k = ⟨ value ⟩$ .
Constraint: if $side = Nag_LeftSide$ , $pdc1 \geq \max (1, n)$ ;
if $side = Nag_RightSide$ , $pdc1 \geq \max (1, k)$ .
NE_INT: On entry, $k = ⟨ value ⟩$ .
Constraint: $k \geq 0$ .

On entry, $m = ⟨ value ⟩$ .
Constraint: $m \geq 0$ .

On entry, $n = ⟨ value ⟩$ .
Constraint: $n \geq 0$ .
NE_INT_2: On entry, $l = ⟨ value ⟩$ and $k = ⟨ value ⟩$ .
Constraint: $0 \leq l \leq k$ .

On entry, $m = ⟨ value ⟩$ and $pdc2 = ⟨ value ⟩$ .
Constraint: $pdc2 \geq \max (1, m)$ .

On entry, $nb = ⟨ value ⟩$ and $k = ⟨ value ⟩$ .
Constraint: $nb \geq 1$ and
if $k > 0$ , $nb \leq k$ .

On entry, $pdc2 = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $pdc2 \geq \max (1, n)$ .

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

Background information to multithreading can be found in the Multithreading documentation.

f08bqc 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.