NAG Library Function Document

nag_dgemqrt (f08acc)

1  Purpose

2  Specification

3  Description

4  References

5  Arguments

1:     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 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.

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

2:     side – Nag_SideTypeInput

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

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

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

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

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

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

3:     trans – Nag_TransTypeInput

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

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

$Q$ is applied to $C$.

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

$Q^{\mathrm{T}}$ is applied to $C$.

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

4:     m – IntegerInput

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

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

5:     n – IntegerInput

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

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

6:     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_dgeqrt (f08abc).

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:     nb – IntegerInput

On entry: the block size used in the $QR$ factorization performed in a previous call to nag_dgeqrt (f08abc); this value must remain unchanged from that call.

Constraints:

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

8:     v[$\mathit{dim}$] – const doubleInput

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_dgeqrt (f08abc) in the first $k$ columns of its array argument a.

9:     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:   t[$\mathit{dim}$] – const doubleInput

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 orthogonal matrix $Q$ as returned by nag_dgeqrt (f08abc). 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:   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:   c[$\mathit{dim}$] – doubleInput/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 $Q^{\mathrm{T}}C$ or $CQ$ or $C{Q}^{\mathrm{T}}$ as specified by side and trans.

13:   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:   fail – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

NE_ALLOC_FAIL

Dynamic memory allocation failed.

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.

7  Accuracy

8  Parallelism and Performance

9  Further Comments

10  Example