NAG FL Interface
f08aqf (zgemqrt)

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

1: $\mathbf{side}$ – Character(1) Input

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

$\mathbf{side}=\text{'L'}$

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

$\mathbf{side}=\text{'R'}$

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

Constraint: $\mathbf{side}=\text{'L'}$ or $\text{'R'}$.

2: $\mathbf{trans}$ – Character(1) Input

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

$\mathbf{trans}=\text{'N'}$

$Q$ is applied to $C$.

$\mathbf{trans}=\text{'C'}$

$Q^{\mathrm{H}}$ is applied to $C$.

Constraint: $\mathbf{trans}=\text{'N'}$ or $\text{'C'}$.

3: $\mathbf{m}$ – Integer Input

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

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

4: $\mathbf{n}$ – Integer Input

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

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

5: $\mathbf{k}$ – Integer Input

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 f08apf.

Constraints:

• if $\mathbf{side}=\text{'L'}$, $\mathbf{m}\ge \mathbf{k}\ge 0$;
• if $\mathbf{side}=\text{'R'}$, $\mathbf{n}\ge \mathbf{k}\ge 0$.

6: $\mathbf{nb}$ – Integer Input

On entry: the block size used in the $QR$ factorization performed in a previous call to f08apf; this value must remain unchanged from that call.

Constraints:

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

7: $\mathbf{v}\left(\mathbf{ldv},*\right)$ – Complex (Kind=nag_wp) array Input

Note: the second dimension of the array v must be at least $\max \left(1,\mathbf{k}\right)$.

On entry: details of the vectors which define the elementary reflectors, as returned by f08apf in the first $k$ columns of its array argument a.

8: $\mathbf{ldv}$ – Integer Input

On entry: the first dimension of the array v as declared in the (sub)program from which f08aqf is called.

Constraints:

• if $\mathbf{side}=\text{'L'}$, $\mathbf{ldv}\ge \max \left(1,\mathbf{m}\right)$;
• if $\mathbf{side}=\text{'R'}$, $\mathbf{ldv}\ge \max \left(1,\mathbf{n}\right)$.

9: $\mathbf{t}\left(\mathbf{ldt},*\right)$ – Complex (Kind=nag_wp) array Input

Note: the second dimension of the array t must be at least $\max \left(1,\mathbf{k}\right)$.

On entry: further details of the unitary matrix $Q$ as returned by f08apf. 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 $\mathit{T}=\left[{\mathit{T}}_1\left|{\mathit{T}}_2\right|\cdots |{\mathit{T}}_b\right]$.

10: $\mathbf{ldt}$ – Integer Input

On entry: the first dimension of the array t as declared in the (sub)program from which f08aqf is called.

Constraint: $\mathbf{ldt}\ge \mathbf{nb}$.

11: $\mathbf{c}\left(\mathbf{ldc},*\right)$ – Complex (Kind=nag_wp) array Input/Output

Note: the second dimension of the array c must be at least $\max \left(1,\mathbf{n}\right)$.

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.

12: $\mathbf{ldc}$ – Integer Input

On entry: the first dimension of the array c as declared in the (sub)program from which f08aqf is called.

Constraint: $\mathbf{ldc}\ge \max \left(1,\mathbf{m}\right)$.

13: $\mathbf{work}\left(*\right)$ – Complex (Kind=nag_wp) array Workspace

Note: the dimension of the array work must be at least $\mathbf{n}\times \mathbf{nb}$ if $\mathbf{side}=\text{'L'}$ and at least $\mathbf{m}\times \mathbf{nb}$ if $\mathbf{side}=\text{'R'}$.

14: $\mathbf{info}$ – Integer Output

On exit: $\mathbf{info}=0$ unless the routine detects an error (see Section 6).

6 Error Indicators and Warnings

$\mathbf{info}<0$

If $\mathbf{info}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example