NAG Library Routine Document

f08acf (dgemqrt)

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

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

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

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

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

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

$Q$ or $Q^{\mathrm{T}}$ 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 $Q^{\mathrm{T}}$ is to be applied to $C$.

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

$Q$ is applied to $C$.

$\mathbf{trans}=\text{'T'}$

$Q^{\mathrm{T}}$ is applied to $C$.

Constraint: $\mathbf{trans}=\text{'N'}$ or $\text{'T'}$.

3:     $\mathbf{m}$ – IntegerInput

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

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

4:     $\mathbf{n}$ – IntegerInput

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

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

5:     $\mathbf{k}$ – 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 f08abf (dgeqrt).

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:     $\mathbf{nb}$ – IntegerInput

On entry: the block size used in the $QR$ factorization performed in a previous call to f08abf (dgeqrt); this value must remain unchanged from that call.

Constraints:

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

7:     $\mathbf{v}\left(\mathbf{ldv},*\right)$ – Real (Kind=nag_wp) arrayInput

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 f08abf (dgeqrt) in the first $k$ columns of its array argument a.

8:     $\mathbf{ldv}$ – IntegerInput

On entry: the first dimension of the array v as declared in the (sub)program from which f08acf (dgemqrt) 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:     $\mathbf{t}\left(\mathbf{ldt},*\right)$ – Real (Kind=nag_wp) arrayInput

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

On entry: further details of the orthogonal matrix $Q$ as returned by f08abf (dgeqrt). 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 $\mathit{T}=\left[{\mathit{T}}_1\left|{\mathit{T}}_2\right|\dots |{\mathit{T}}_b\right]$.

10:   $\mathbf{ldt}$ – IntegerInput

On entry: the first dimension of the array t as declared in the (sub)program from which f08acf (dgemqrt) is called.

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

11:   $\mathbf{c}\left(\mathbf{ldc},*\right)$ – Real (Kind=nag_wp) arrayInput/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 $Q^{\mathrm{T}}C$ or $CQ$ or $C{Q}^{\mathrm{T}}$ as specified by side and trans.

12:   $\mathbf{ldc}$ – IntegerInput

On entry: the first dimension of the array c as declared in the (sub)program from which f08acf (dgemqrt) is called.

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

13:   $\mathbf{work}\left(*\right)$ – Real (Kind=nag_wp) arrayWorkspace

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:   $\mathbf{info}$ – IntegerOutput

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