NAG Library Routine Document

F08ACF (DGEMQRT)

1  Purpose

2  Specification

3  Description

4  References

5  Parameters

1:     $\mathrm{SIDE}$ – 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:     $\mathrm{TRANS}$ – 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:     $\mathrm{M}$ – INTEGERInput

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

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

4:     $\mathrm{N}$ – INTEGERInput

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

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

5:     $\mathrm{K}$ – 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:     $\mathrm{NB}$ – 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:     $\mathrm{V}\left(\mathbf{LDV},*\right)$ – 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 parameter A.

8:     $\mathrm{LDV}$ – 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:     $\mathrm{T}\left(\mathbf{LDT},*\right)$ – 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:   $\mathrm{LDT}$ – 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:   $\mathrm{C}\left(\mathbf{LDC},*\right)$ – 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:   $\mathrm{LDC}$ – 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:   $\mathrm{WORK}\left(*\right)$ – 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:   $\mathrm{INFO}$ – 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