void	f08cgc (Nag_OrderType order, Nag_SideType side, Nag_TransType trans, Integer m, Integer n, Integer k, const double a[], Integer pda, const double tau[], double c[], Integer pdc, NagError *fail)

The function may be called by the names: f08cgc, nag_lapackeig_dormql or nag_dormql.

3 Description

f08cgc is intended to be used following a call to f08cec, which performs a

Q L

factorization of a real matrix

A

and represents the orthogonal matrix

Q

as a product of elementary reflectors.

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

Q C, Q^{T} C, C Q, C Q^{T},

overwriting the result on

C

, which may be any real rectangular

m \times n

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 f08cec.

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 https://www.netlib.org/lapack/lug

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^{T}

is to be applied to

C

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

Constraint:

side = Nag_LeftSide

Nag_RightSide

3: $trans$ – Nag_TransType Input

On entry: indicates whether

Q

Q^{T}

is to be applied to

C

$trans = Nag_NoTrans$: $Q$ is applied to $C$ .
$trans = Nag_Trans$: $Q^{T}$ is applied to $C$ .

Constraint:

trans = Nag_NoTrans

Nag_Trans

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

Constraints:

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

7: $a [\dim]$ – const double Input

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 f08cec.

On exit: is modified by f08cgc but restored on exit.

8: $pda$ – Integer Input

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 double Input

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 f08cec.

10: $c [\dim]$ – double 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 \times n

matrix

C

On exit: c is overwritten by

Q C

Q^{T} C

C Q

C Q^{T}

as specified by side and trans.

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

12: $fail$ – 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, $side = ⟨ value ⟩$ , $m = ⟨ value ⟩$ , $pda = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
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.
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.

f08cgc 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.