NAG C Library Function Document

nag_dggsvp3 (f08vgc)

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:    $\mathbf{jobu}$ – Nag_ComputeUTypeInput

On entry: if $\mathbf{jobu}=\mathrm{Nag\_AllU}$, the orthogonal matrix $U$ is computed.

If $\mathbf{jobu}=\mathrm{Nag\_NotU}$, $U$ is not computed.

Constraint: $\mathbf{jobu}=\mathrm{Nag\_AllU}$ or $\mathrm{Nag\_NotU}$.

3:    $\mathbf{jobv}$ – Nag_ComputeVTypeInput

On entry: if $\mathbf{jobv}=\mathrm{Nag\_ComputeV}$, the orthogonal matrix $V$ is computed.

If $\mathbf{jobv}=\mathrm{Nag\_NotV}$, $V$ is not computed.

Constraint: $\mathbf{jobv}=\mathrm{Nag\_ComputeV}$ or $\mathrm{Nag\_NotV}$.

4:    $\mathbf{jobq}$ – Nag_ComputeQTypeInput

On entry: if $\mathbf{jobq}=\mathrm{Nag\_ComputeQ}$, the orthogonal matrix $Q$ is computed.

If $\mathbf{jobq}=\mathrm{Nag\_NotQ}$, $Q$ is not computed.

Constraint: $\mathbf{jobq}=\mathrm{Nag\_ComputeQ}$ or $\mathrm{Nag\_NotQ}$.

5:    $\mathbf{m}$ – IntegerInput

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

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

6:    $\mathbf{p}$ – IntegerInput

On entry: $p$, the number of rows of the matrix $B$.

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

7:    $\mathbf{n}$ – IntegerInput

On entry: $n$, the number of columns of the matrices $A$ and $B$.

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

8:    $\mathbf{a}\left[\mathit{dim}\right]$ – doubleInput/Output

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

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

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

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

On entry: the $m$ by $n$ matrix $A$.

On exit: contains the triangular (or trapezoidal) matrix described in Section 3.

9:    $\mathbf{pda}$ – IntegerInput

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

Constraints:

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

10:  $\mathbf{b}\left[\mathit{dim}\right]$ – doubleInput/Output

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

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

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

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

On entry: the $p$ by $n$ matrix $B$.

On exit: contains the triangular matrix described in Section 3.

11:  $\mathbf{pdb}$ – IntegerInput

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

Constraints:

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

12:  $\mathbf{tola}$ – doubleInput

13:  $\mathbf{tolb}$ – doubleInput

On entry: tola and tolb are the thresholds to determine the effective numerical rank of matrix $B$ and a subblock of $A$. Generally, they are set to

$$\begin{array}{c}\mathbf{tola}=\max \left(\mathbf{m},\mathbf{n}\right)‖A‖\epsilon \text{,}\\ \mathbf{tolb}=\max \left(\mathbf{p},\mathbf{n}\right)‖B‖\epsilon \text{,}\end{array}$$

where $\epsilon$ is the machine precision.

The size of tola and tolb may affect the size of backward errors of the decomposition.

14:  $\mathbf{k}$ – Integer *Output

15:  $\mathbf{l}$ – Integer *Output

On exit: k and l specify the dimension of the subblocks $k$ and $l$ as described in Section 3; $\left(k+l\right)$ is the effective numerical rank of ${\left(\begin{array}{cc}{\mathbf{a}}^{\mathrm{T}}& {\mathbf{b}}^{\mathrm{T}}\end{array}\right)}^{\mathrm{T}}$.

16:  $\mathbf{u}\left[\mathit{dim}\right]$ – doubleOutput

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

• $\max \left(1,\mathbf{pdu}\times \mathbf{m}\right)$ when $\mathbf{jobu}=\mathrm{Nag\_AllU}$;
• $1$ otherwise.

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

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

On exit: if $\mathbf{jobu}=\mathrm{Nag\_AllU}$, u contains the orthogonal matrix $U$.

If $\mathbf{jobu}=\mathrm{Nag\_NotU}$, u is not referenced.

17:  $\mathbf{pdu}$ – IntegerInput

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

Constraints:

• if $\mathbf{jobu}=\mathrm{Nag\_AllU}$, $\mathbf{pdu}\ge \max \left(1,\mathbf{m}\right)$;
• otherwise $\mathbf{pdu}\ge 1$.

18:  $\mathbf{v}\left[\mathit{dim}\right]$ – doubleOutput

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

• $\max \left(1,\mathbf{pdv}\times \mathbf{p}\right)$ when $\mathbf{jobv}=\mathrm{Nag\_ComputeV}$;
• $1$ otherwise.

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

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

On exit: if $\mathbf{jobv}=\mathrm{Nag\_ComputeV}$, v contains the orthogonal matrix $V$.

If $\mathbf{jobv}=\mathrm{Nag\_NotV}$, v is not referenced.

19:  $\mathbf{pdv}$ – IntegerInput

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

Constraints:

• if $\mathbf{jobv}=\mathrm{Nag\_ComputeV}$, $\mathbf{pdv}\ge \max \left(1,\mathbf{p}\right)$;
• otherwise $\mathbf{pdv}\ge 1$.

20:  $\mathbf{q}\left[\mathit{dim}\right]$ – doubleOutput

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

• $\max \left(1,\mathbf{pdq}\times \mathbf{n}\right)$ when $\mathbf{jobq}=\mathrm{Nag\_ComputeQ}$;
• $1$ otherwise.

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

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

On exit: if $\mathbf{jobq}=\mathrm{Nag\_ComputeQ}$, q contains the orthogonal matrix $Q$.

If $\mathbf{jobq}=\mathrm{Nag\_NotQ}$, q is not referenced.

21:  $\mathbf{pdq}$ – IntegerInput

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

Constraints:

• if $\mathbf{jobq}=\mathrm{Nag\_ComputeQ}$, $\mathbf{pdq}\ge \max \left(1,\mathbf{n}\right)$;
• otherwise $\mathbf{pdq}\ge 1$.

22:  $\mathbf{fail}$ – 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_2

On entry, $\mathbf{jobq}=〈\mathit{\text{value}}〉$, $\mathbf{pdq}=〈\mathit{\text{value}}〉$ and $\mathbf{n}=〈\mathit{\text{value}}〉$.
Constraint: if $\mathbf{jobq}=\mathrm{Nag\_ComputeQ}$, $\mathbf{pdq}\ge \max \left(1,\mathbf{n}\right)$;
otherwise $\mathbf{pdq}\ge 1$.

On entry, $\mathbf{jobu}=〈\mathit{\text{value}}〉$, $\mathbf{pdu}=〈\mathit{\text{value}}〉$ and $\mathbf{m}=〈\mathit{\text{value}}〉$.
Constraint: if $\mathbf{jobu}=\mathrm{Nag\_AllU}$, $\mathbf{pdu}\ge \max \left(1,\mathbf{m}\right)$;
otherwise $\mathbf{pdu}\ge 1$.

On entry, $\mathbf{jobv}=〈\mathit{\text{value}}〉$, $\mathbf{pdv}=〈\mathit{\text{value}}〉$ and $\mathbf{p}=〈\mathit{\text{value}}〉$.
Constraint: if $\mathbf{jobv}=\mathrm{Nag\_ComputeV}$, $\mathbf{pdv}\ge \max \left(1,\mathbf{p}\right)$;
otherwise $\mathbf{pdv}\ge 1$.

NE_INT

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

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

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

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

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

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

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

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

NE_INT_2

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

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

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

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

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

9

Further Comments

10

Example

10.1

Program Text

Program Text (f08vgce.c)

10.2

Program Data

Program Data (f08vgce.d)

10.3

Program Results

Program Results (f08vgce.r)