NAG Library Routine Document

e04pcf  (bnd_lin_lsq)

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

1:     $\mathbf{itype}$ – IntegerInput

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:     $\mathbf{m}$ – IntegerInput

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

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

3:     $\mathbf{n}$ – IntegerInput

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

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

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

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

On entry: the $m$ by $n$ matrix $A$.

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

5:     $\mathbf{lda}$ – IntegerInput

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) arrayInput/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) arrayInput

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

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) arrayOutput

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:   $\mathbf{nfree}$ – IntegerOutput

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) arrayOutput

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 arrayOutput

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:   $\mathbf{ifail}$ – IntegerInput/Output

On entry: ifail must be set to $0$, $-1\text{ or }1$. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{ or }1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, because for this routine the values of the output arguments may be useful even if $\mathbf{ifail}\ne \mathbf{0}$ on exit, the recommended value is $-1$. When the value $-\mathbf{1}\text{ or }1$ 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 3.9 in How to Use the NAG Library and its Documentation for further information.

$\mathbf{ifail}=-399$

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

See Section 3.8 in How to Use the NAG Library and its Documentation for further information.

$\mathbf{ifail}=-999$

Dynamic memory allocation failed.

See Section 3.7 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 (e04pcfe.f90)

10.2

Program Data

Program Data (e04pcfe.d)

10.3

Program Results

Program Results (e04pcfe.r)