NAG Library Function Document

nag_dgglse (f08zac)

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:    $\mathbf{m}$ – IntegerInput

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

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

3:    $\mathbf{n}$ – IntegerInput

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

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

4:    $\mathbf{p}$ – IntegerInput

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

Constraint: $0\le \mathbf{p}\le \mathbf{n}\le \mathbf{m}+\mathbf{p}$.

5:    $\mathbf{a}\left[\mathit{dim}\right]$ – 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: a is overwritten.

6:    $\mathbf{pda}$ – 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 $\mathbf{order}=\mathrm{Nag\_RowMajor}$, $\mathbf{pda}\ge \max \left(1,\mathbf{n}\right)$.

7:    $\mathbf{b}\left[\mathit{dim}\right]$ – 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: b is overwritten.

8:    $\mathbf{pdb}$ – 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 $\mathbf{order}=\mathrm{Nag\_RowMajor}$, $\mathbf{pdb}\ge \max \left(1,\mathbf{n}\right)$.

9:    $\mathbf{c}\left[\mathbf{m}\right]$ – doubleInput/Output

On entry: the right-hand side vector $c$ for the least squares part of the LSE problem.

On exit: the residual sum of squares for the solution vector $x$ is given by the sum of squares of elements $\mathbf{c}\left[\mathbf{n}-\mathbf{p}\right],\mathbf{c}\left[\mathbf{n}-\mathbf{p}+1\right],\dots ,\mathbf{c}\left[\mathbf{m}-1\right]$; the remaining elements are overwritten.

10:  $\mathbf{d}\left[\mathbf{p}\right]$ – doubleInput/Output

On entry: the right-hand side vector $d$ for the equality constraints.

On exit: d is overwritten.

11:  $\mathbf{x}\left[\mathbf{n}\right]$ – doubleOutput

On exit: the solution vector $x$ of the LSE problem.

12:  $\mathbf{fail}$ – 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_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, $\mathbf{pda}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{pda}>0$.

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

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_INT_3

On entry, $\mathbf{p}=〈\mathit{\text{value}}〉$, $\mathbf{m}=〈\mathit{\text{value}}〉$ and $\mathbf{n}=〈\mathit{\text{value}}〉$.
Constraint: $0\le \mathbf{p}\le \mathbf{n}\le \mathbf{m}+\mathbf{p}$.

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.

NE_SINGULAR

The $\left(N-P\right)$ by $\left(N-P\right)$ part of the upper trapezoidal factor $T$ associated with $A$ in the generalized $RQ$ factorization of the pair $\left(B,A\right)$ is singular, so that the rank of the matrix ($E$) comprising the rows of $A$ and $B$ is less than $n$; the least squares solutions could not be computed.

The upper triangular factor $R$ associated with $B$ in the generalized $RQ$ factorization of the pair $\left(B,A\right)$ is singular, so that $\mathrm{rank}\left(B\right)<p$; the least squares solution could not be computed.

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example

10.1

Program Text

Program Text (f08zace.c)

10.2

Program Data

Program Data (f08zace.d)

10.3

Program Results

Program Results (f08zace.r)