NAG FL Interface
f08zaf (dgglse)

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

1: $\mathbf{m}$ – Integer Input

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

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

2: $\mathbf{n}$ – Integer Input

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

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

3: $\mathbf{p}$ – Integer Input

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

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

4: $\mathbf{a}(\mathbf{lda},*)$ – Real (Kind=nag_wp) array Input/Output

Note: the second dimension of the array a must be at least $\max (1,\mathbf{n})$.

On entry: the $m\times n$ matrix $A$.

On exit: a is overwritten.

5: $\mathbf{lda}$ – Integer Input

On entry: the first dimension of the array a as declared in the (sub)program from which f08zaf is called.

Constraint: $\mathbf{lda}\ge \max (1,\mathbf{m})$.

6: $\mathbf{b}(\mathbf{ldb},*)$ – Real (Kind=nag_wp) array Input/Output

Note: the second dimension of the array b must be at least $\max (1,\mathbf{n})$.

On entry: the $p\times n$ matrix $B$.

On exit: b is overwritten.

7: $\mathbf{ldb}$ – Integer Input

On entry: the first dimension of the array b as declared in the (sub)program from which f08zaf is called.

Constraint: $\mathbf{ldb}\ge \max (1,\mathbf{p})$.

8: $\mathbf{c}\left(\mathbf{m}\right)$ – Real (Kind=nag_wp) array Input/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 $\mathbf{c}\left(\mathbf{n}-\mathbf{p}+1\right),\mathbf{c}\left(\mathbf{n}-\mathbf{p}+2\right),\dots ,\mathbf{c}\left(\mathbf{m}\right)$; the remaining elements are overwritten.

9: $\mathbf{d}\left(\mathbf{p}\right)$ – Real (Kind=nag_wp) array Input/Output

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

On exit: d is overwritten.

10: $\mathbf{x}\left(\mathbf{n}\right)$ – Real (Kind=nag_wp) array Output

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

11: $\mathbf{work}\left(\max (1,\mathbf{lwork})\right)$ – Real (Kind=nag_wp) array Workspace

On exit: if $\mathbf{info}=\mathbf{0}$, $\mathbf{work}\left(1\right)$ contains the minimum value of lwork required for optimal performance.

12: $\mathbf{lwork}$ – Integer Input

On entry: the dimension of the array work as declared in the (sub)program from which f08zaf is called.

If $\mathbf{lwork}=-1$, a workspace query is assumed; the routine only calculates the optimal size of the work array, returns this value as the first entry of the work array, and no error message related to lwork is issued.

Suggested value: for optimal performance, $\mathbf{lwork}\ge \mathbf{p}+\min (\mathbf{m},\mathbf{n})+\max (\mathbf{m},\mathbf{n})\times \mathit{nb}$, where $\mathit{nb}$ is the optimal block size.

Constraint: $\mathbf{lwork}\ge \max (1,\mathbf{m}+\mathbf{n}+\mathbf{p})$ or $\mathbf{lwork}=-1$.

13: $\mathbf{info}$ – Integer Output

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.

$\mathbf{info}=1$

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

$\mathbf{info}=2$

The $(N-P)\times (N-P)$ part of the upper trapezoidal factor $T$ associated with $A$ in the generalized $RQ$ factorization of the pair $(B,A)$ 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.

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results