NAG FL Interface
e04pcf (bnd_​lin_​lsq)

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

1: $\mathbf{itype}$ – Integer Input

On entry: provides the choice of returning a regularized solution if the matrix is not of full rank.

$\mathbf{itype}=0$

Specifies that a regularized solution is to be computed.

$\mathbf{itype}=1$

Specifies that no regularization is to take place.

Suggested value: unless there is a definite need for a minimal length solution we recommend that $\mathbf{itype}=1$ is used.

Constraint: $\mathbf{itype}=0$ or $1$.

2: $\mathbf{m}$ – Integer Input

On entry: $m$, the number of linear equations.

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

3: $\mathbf{n}$ – Integer Input

On entry: $n$, the number of variables.

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

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

Note: the second dimension of the array a must be at least $\mathbf{n}$.

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

On exit: if $\mathbf{itype}=1$, a contains the product matrix $QA$, where $Q$ is an $m\times m$ orthogonal matrix generated by e04pcf; otherwise, a is unchanged.

5: $\mathbf{lda}$ – Integer Input

On entry: the first dimension of the array a as declared in the (sub)program from which e04pcf is called.

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

6: $\mathbf{b}\left(\mathbf{m}\right)$ – Real (Kind=nag_wp) array Input/Output

On entry: the right-hand side vector $b$.

On exit: if $\mathbf{itype}=1$, the product of $Q$ times the original vector $b$, where $Q$ is as described in argument a; otherwise, b is unchanged.

7: $\mathbf{bl}\left(\mathbf{n}\right)$ – Real (Kind=nag_wp) array Input

8: $\mathbf{bu}\left(\mathbf{n}\right)$ – Real (Kind=nag_wp) array Input

On entry: $\mathbf{bl}\left(\mathit{i}\right)$ and $\mathbf{bu}\left(\mathit{i}\right)$ must specify the lower and upper bounds, $l_i$ and $u_i$ respectively, to be imposed on the solution vector $x_i$.

Constraint: $\mathbf{bl}\left(\mathit{i}\right)\le \mathbf{bu}\left(\mathit{i}\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{n}$.

9: $\mathbf{tol}$ – Real (Kind=nag_wp) Input

On entry: tol specifies a parameter used to determine the relative linear dependence of a column vector for a variable moved from its initial value. It determines the computational rank of the matrix. Increasing its value from $\sqrt{\mathit{machine precision}}$ will increase the likelihood of additional elements of $x$ being set to zero. It may be worth experimenting with increasing values of tol to determine whether the nature of the solution, $x$, changes significantly. In practice a value of $\sqrt{\mathit{machine precision}}$ is recommended (see x02ajf).

If on entry $\mathbf{tol}<\sqrt{\mathit{machine precision}}$, $\sqrt{\mathit{machine precision}}$ is used.

Suggested value: $\mathbf{tol}=0.0$

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

On exit: the solution vector $x$.

11: $\mathbf{rnorm}$ – Real (Kind=nag_wp) Output

On exit: the Euclidean norm of the residual vector $b-Ax$.

12: $\mathbf{nfree}$ – Integer Output

On exit: indicates the number of components of the solution vector that are not at one of the constraints.

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

On exit: contains the dual solution vector. The magnitude of $\mathbf{w}\left(i\right)$ gives a measure of the improvement in the objective value if the corresponding bound were to be relaxed so that $x_i$ could take different values.

A value of $\mathbf{w}\left(i\right)$ equal to the special value $-999.0$ is indicative of the matrix $A$ not having full rank. It is only likely to occur when $\mathbf{itype}=1$. However a matrix may have less than full rank without $\mathbf{w}\left(i\right)$ being set to $-999.0$. If $\mathbf{itype}=1$, then the values contained in w (other than those set to $-999.0$) may be unreliable; the corresponding values in indx may likewise be unreliable. If you have any doubts set $\mathbf{itype}=0$. Otherwise, the values of $\mathbf{w}\left(i\right)$ have the following meaning:

$\mathbf{w}\left(i\right)=0$

if $x_i$ is unconstrained.

$\mathbf{w}\left(i\right)<0$

if $x_i$ is constrained by its lower bound.

$\mathbf{w}\left(i\right)>0$

if $x_i$ is constrained by its upper bound.

$\mathbf{w}\left(i\right)$

may be any value if $l_i=u_i$.

14: $\mathbf{indx}\left(\mathbf{n}\right)$ – Integer array Output

On exit: the contents of this array describe the components of the solution vector as follows:

$\mathbf{indx}\left(\mathit{i}\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{nfree}$

These elements of the solution have not hit a constraint; i.e., $\mathbf{w}\left(i\right)=0$.

$\mathbf{indx}\left(\mathit{i}\right)$, for $\mathit{i}=\mathbf{nfree}+1,\dots ,k$

These elements of the solution have been constrained by either the lower or upper bound.

$\mathbf{indx}\left(\mathit{i}\right)$, for $\mathit{i}=k+1,\dots ,\mathbf{n}$

These elements of the solution are fixed by the bounds; i.e., $\mathbf{bl}\left(i\right)=\mathbf{bu}\left(i\right)$.

Here $k$ is determined from nfree and the number of fixed components. (Often the latter will be $0$, so $k$ will be $\mathbf{n}-\mathbf{nfree}$.)

15: $\mathbf{ifail}$ – Integer Input/Output

On entry: ifail must be set to $0$, $\mathrm{-1}$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.

A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $\mathrm{-1}$ means that an error message is printed while a value of $1$ means that it is not.

If halting is not appropriate, the value $\mathrm{-1}$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $\mathrm{-1}$ is recommended since useful values can be provided in some output arguments even when $\mathbf{ifail}\ne \mathbf{0}$ on exit. When the value $-\mathbf{1}$ or $\mathbf{1}$ is used it is essential to test the value of ifail on exit.

On exit: $\mathbf{ifail}=\mathbf{0}$ unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

$\mathbf{ifail}=1$

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

On entry, $\mathbf{m}=⟨\mathit{\text{value}}⟩$ and $\mathbf{lda}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{lda}\ge \mathbf{m}$.

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

On entry, when $i=⟨\mathit{\text{value}}⟩$, $\mathbf{bl}\left(i\right)=⟨\mathit{\text{value}}⟩$ and $\mathbf{bu}\left(i\right)=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{bl}\left(i\right)\le \mathbf{bu}\left(i\right)$.

$\mathbf{ifail}=2$

The routine failed to converge in $3\times n$ iterations. This is not expected. Please contact NAG.

$\mathbf{ifail}=-99$

An unexpected error has been triggered by this routine. Please contact NAG.

See Section 7 in the Introduction to the NAG Library FL Interface for further information.

$\mathbf{ifail}=-399$

Your licence key may have expired or may not have been installed correctly.

See Section 8 in the Introduction to the NAG Library FL Interface for further information.

$\mathbf{ifail}=-999$

Dynamic memory allocation failed.

See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results