NAG Library Function Document

nag_zgemqrt (f08aqc)

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

1:    $\mathbf{order}$ – 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 $\mathbf{order}=\mathrm{Nag\_RowMajor}$. See Section 3.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.

Constraint: $\mathbf{order}=\mathrm{Nag\_RowMajor}$ or $\mathrm{Nag\_ColMajor}$.

2:    $\mathbf{side}$ – Nag_SideTypeInput

On entry: indicates how $Q$ or $Q^{\mathrm{H}}$ is to be applied to $C$.

$\mathbf{side}=\mathrm{Nag\_LeftSide}$

$Q$ or $Q^{\mathrm{H}}$ is applied to $C$ from the left.

$\mathbf{side}=\mathrm{Nag\_RightSide}$

$Q$ or $Q^{\mathrm{H}}$ is applied to $C$ from the right.

Constraint: $\mathbf{side}=\mathrm{Nag\_LeftSide}$ or $\mathrm{Nag\_RightSide}$.

3:    $\mathbf{trans}$ – Nag_TransTypeInput

On entry: indicates whether $Q$ or $Q^{\mathrm{H}}$ is to be applied to $C$.

$\mathbf{trans}=\mathrm{Nag\_NoTrans}$

$Q$ is applied to $C$.

$\mathbf{trans}=\mathrm{Nag\_ConjTrans}$

$Q^{\mathrm{H}}$ is applied to $C$.

Constraint: $\mathbf{trans}=\mathrm{Nag\_NoTrans}$ or $\mathrm{Nag\_ConjTrans}$.

4:    $\mathbf{m}$ – IntegerInput

On entry: $m$, the number of rows of the matrix $C$.

Constraint: $\mathbf{m}\ge 0$.

5:    $\mathbf{n}$ – IntegerInput

On entry: $n$, the number of columns of the matrix $C$.

Constraint: $\mathbf{n}\ge 0$.

6:    $\mathbf{k}$ – IntegerInput

On entry: $k$, the number of elementary reflectors whose product defines the matrix $Q$. Usually $\mathbf{k}=\min \left(m_A,n_A\right)$ where $m_A$, $n_A$ are the dimensions of the matrix $A$ supplied in a previous call to nag_zgeqrt (f08apc).

Constraints:

• if $\mathbf{side}=\mathrm{Nag\_LeftSide}$, $\mathbf{m}\ge \mathbf{k}\ge 0$;
• if $\mathbf{side}=\mathrm{Nag\_RightSide}$, $\mathbf{n}\ge \mathbf{k}\ge 0$.

7:    $\mathbf{nb}$ – IntegerInput

On entry: the block size used in the $QR$ factorization performed in a previous call to nag_zgeqrt (f08apc); this value must remain unchanged from that call.

Constraints:

• $\mathbf{nb}\ge 1$;
• if $\mathbf{k}>0$, $\mathbf{nb}\le \mathbf{k}$.

8:    $\mathbf{v}\left[\mathit{dim}\right]$ – const ComplexInput

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

• $\max \left(1,\mathbf{pdv}\times \mathbf{k}\right)$ when $\mathbf{order}=\mathrm{Nag\_ColMajor}$;
• $\max \left(1,\mathbf{m}\times \mathbf{pdv}\right)$ when $\mathbf{order}=\mathrm{Nag\_RowMajor}$ and $\mathbf{side}=\mathrm{Nag\_LeftSide}$;
• $\max \left(1,\mathbf{n}\times \mathbf{pdv}\right)$ when $\mathbf{order}=\mathrm{Nag\_RowMajor}$ and $\mathbf{side}=\mathrm{Nag\_RightSide}$.

On entry: details of the vectors which define the elementary reflectors, as returned by nag_zgeqrt (f08apc) in the first $k$ columns of its array argument a.

9:    $\mathbf{pdv}$ – IntegerInput

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

Constraints:

• if $\mathbf{order}=\mathrm{Nag\_ColMajor}$,
  • if $\mathbf{side}=\mathrm{Nag\_LeftSide}$, $\mathbf{pdv}\ge \max \left(1,\mathbf{m}\right)$;
  • if $\mathbf{side}=\mathrm{Nag\_RightSide}$, $\mathbf{pdv}\ge \max \left(1,\mathbf{n}\right)$;
• if $\mathbf{order}=\mathrm{Nag\_RowMajor}$, $\mathbf{pdv}\ge \max \left(1,\mathbf{k}\right)$.

10:  $\mathbf{t}\left[\mathit{dim}\right]$ – const ComplexInput

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

• $\max \left(1,\mathbf{pdt}\times \mathbf{k}\right)$ when $\mathbf{order}=\mathrm{Nag\_ColMajor}$;
• $\max \left(1,\mathbf{nb}\times \mathbf{pdt}\right)$ when $\mathbf{order}=\mathrm{Nag\_RowMajor}$.

The $\left(i,j\right)$th element of the matrix $T$ is stored in

• $\mathbf{t}\left[\left(j-1\right)\times \mathbf{pdt}+i-1\right]$ when $\mathbf{order}=\mathrm{Nag\_ColMajor}$;
• $\mathbf{t}\left[\left(i-1\right)\times \mathbf{pdt}+j-1\right]$ when $\mathbf{order}=\mathrm{Nag\_RowMajor}$.

On entry: further details of the unitary matrix $Q$ as returned by nag_zgeqrt (f08apc). The number of blocks is $b=⌈\frac{k}{\mathbf{nb}}⌉$, where $k=\min \left(m,n\right)$ and each block is of order nb except for the last block, which is of order $k-\left(b-1\right)\times \mathbf{nb}$. For the $b$ blocks the upper triangular block reflector factors ${\mathit{T}}_1,{\mathit{T}}_2,\dots ,{\mathit{T}}_b$ are stored in the $\mathbf{nb}$ by $n$ matrix $T$ as $\mathit{T}=\left[{\mathit{T}}_1\left|{\mathit{T}}_2\right|\dots |{\mathit{T}}_b\right]$.

11:  $\mathbf{pdt}$ – IntegerInput

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

Constraints:

• if $\mathbf{order}=\mathrm{Nag\_ColMajor}$, $\mathbf{pdt}\ge \mathbf{nb}$;
• if $\mathbf{order}=\mathrm{Nag\_RowMajor}$, $\mathbf{pdt}\ge \max \left(1,\mathbf{k}\right)$.

12:  $\mathbf{c}\left[\mathit{dim}\right]$ – ComplexInput/Output

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

• $\max \left(1,\mathbf{pdc}\times \mathbf{n}\right)$ when $\mathbf{order}=\mathrm{Nag\_ColMajor}$;
• $\max \left(1,\mathbf{m}\times \mathbf{pdc}\right)$ when $\mathbf{order}=\mathrm{Nag\_RowMajor}$.

The $\left(i,j\right)$th element of the matrix $C$ is stored in

• $\mathbf{c}\left[\left(j-1\right)\times \mathbf{pdc}+i-1\right]$ when $\mathbf{order}=\mathrm{Nag\_ColMajor}$;
• $\mathbf{c}\left[\left(i-1\right)\times \mathbf{pdc}+j-1\right]$ when $\mathbf{order}=\mathrm{Nag\_RowMajor}$.

On entry: the $m$ by $n$ matrix $C$.

On exit: c is overwritten by $QC$ or $Q^{\mathrm{H}}C$ or $CQ$ or $C{Q}^{\mathrm{H}}$ as specified by side and trans.

13:  $\mathbf{pdc}$ – IntegerInput

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

Constraints:

• if $\mathbf{order}=\mathrm{Nag\_ColMajor}$, $\mathbf{pdc}\ge \max \left(1,\mathbf{m}\right)$;
• if $\mathbf{order}=\mathrm{Nag\_RowMajor}$, $\mathbf{pdc}\ge \max \left(1,\mathbf{n}\right)$.

14:  $\mathbf{fail}$ – NagError *Input/Output

The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6

Error Indicators and Warnings

NE_ALLOC_FAIL

Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.

NE_BAD_PARAM

On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.

NE_ENUM_INT_3

On entry, $\mathbf{side}=〈\mathit{\text{value}}〉$, $\mathbf{m}=〈\mathit{\text{value}}〉$, $\mathbf{n}=〈\mathit{\text{value}}〉$ and $\mathbf{k}=〈\mathit{\text{value}}〉$.
Constraint: if $\mathbf{side}=\mathrm{Nag\_LeftSide}$, $\mathbf{m}\ge \mathbf{k}\ge 0$;
if $\mathbf{side}=\mathrm{Nag\_RightSide}$, $\mathbf{n}\ge \mathbf{k}\ge 0$.

On entry, $\mathbf{side}=〈\mathit{\text{value}}〉$, $\mathbf{m}=〈\mathit{\text{value}}〉$, $\mathbf{n}=〈\mathit{\text{value}}〉$ and $\mathbf{pdv}=〈\mathit{\text{value}}〉$.
Constraint: if $\mathbf{side}=\mathrm{Nag\_LeftSide}$, $\mathbf{pdv}\ge \max \left(1,\mathbf{m}\right)$;
if $\mathbf{side}=\mathrm{Nag\_RightSide}$, $\mathbf{pdv}\ge \max \left(1,\mathbf{n}\right)$.

NE_INT

On entry, $\mathbf{m}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{m}\ge 0$.

On entry, $\mathbf{n}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{n}\ge 0$.

NE_INT_2

On entry, $\mathbf{nb}=〈\mathit{\text{value}}〉$ and $\mathbf{k}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{nb}\ge 1$ and
if $\mathbf{k}>0$, $\mathbf{nb}\le \mathbf{k}$.

On entry, $\mathbf{pdc}=〈\mathit{\text{value}}〉$ and $\mathbf{m}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{pdc}\ge \max \left(1,\mathbf{m}\right)$.

On entry, $\mathbf{pdc}=〈\mathit{\text{value}}〉$ and $\mathbf{n}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{pdc}\ge \max \left(1,\mathbf{n}\right)$.

On entry, $\mathbf{pdt}=〈\mathit{\text{value}}〉$ and $\mathbf{k}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{pdt}\ge \max \left(1,\mathbf{k}\right)$.

On entry, $\mathbf{pdt}=〈\mathit{\text{value}}〉$ and $\mathbf{nb}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{pdt}\ge \mathbf{nb}$.

On entry, $\mathbf{pdv}=〈\mathit{\text{value}}〉$ and $\mathbf{k}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{pdv}\ge \max \left(1,\mathbf{k}\right)$.

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 2.7.6 in How to Use the NAG Library and its Documentation for further information.

NE_NO_LICENCE

Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.

7

Accuracy

8

Parallelism and Performance

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Further Comments

10

Example