NAG Library Function Document

nag_opt_bnd_lin_lsq (e04pcc)

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

1:    $\mathbf{itype}$ – Nag_RegularizedTypeInput

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

$\mathbf{itype}=\mathrm{Nag\_Regularized}$

Specifies that a regularized solution is to be computed.

$\mathbf{itype}=\mathrm{Nag\_NotRegularized}$

Specifies that no regularization is to take place.

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

Constraint: $\mathbf{itype}=\mathrm{Nag\_Regularized}$ or $\mathrm{Nag\_NotRegularized}$.

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:    $\mathbf{a}\left[\mathbf{pda}\times \mathbf{n}\right]$ – doubleInput/Output

Note: the $\left(i,j\right)$th element of the matrix $A$ is stored in $\mathbf{a}\left[\left(j-1\right)\times \mathbf{pda}+i-1\right]$.

On entry: the $m$ by $n$ matrix $A$.

On exit: if $\mathbf{itype}=\mathrm{Nag\_NotRegularized}$, a contains the product matrix $QA$, where $Q$ is an $m$ by $m$ orthogonal matrix generated by nag_opt_bnd_lin_lsq (e04pcc); otherwise a is unchanged.

5:    $\mathbf{pda}$ – IntegerInput

On entry: the stride separating matrix row elements in the array a.

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

6:    $\mathbf{b}\left[\mathbf{m}\right]$ – doubleInput/Output

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

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

7:    $\mathbf{bl}\left[\mathbf{n}\right]$ – const doubleInput

8:    $\mathbf{bu}\left[\mathbf{n}\right]$ – const doubleInput

On entry: $\mathbf{bl}\left[\mathit{i}-1\right]$ and $\mathbf{bu}\left[\mathit{i}-1\right]$ must specify the lower and upper bounds, $l_i$ and $u_i$ respectively, to be imposed on the solution vector $x_i$.

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

9:    $\mathbf{tol}$ – doubleInput

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 nag_machine_precision (X02AJC)).

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

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

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

On exit: the solution vector $x$.

11:  $\mathbf{rnorm}$ – double *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:  $\mathbf{w}\left[\mathbf{n}\right]$ – doubleOutput

On exit: contains the dual solution vector. The magnitude of $\mathbf{w}\left[i-1\right]$ 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 $\mathbf{w}\left[i-1\right]$ equal to the special value $-999.0$ is indicative of the matrix $A$ not having full rank. It is only likely to occur when $\mathbf{itype}=\mathrm{Nag\_NotRegularized}$. However a matrix may have less than full rank without $\mathbf{w}\left[i-1\right]$ being set to $-999.0$. If $\mathbf{itype}=\mathrm{Nag\_NotRegularized}$ 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 $\mathbf{itype}=\mathrm{Nag\_Regularized}$. Otherwise the values of $\mathbf{w}\left[i-1\right]$ have the following meaning:

$\mathbf{w}\left[i-1\right]=0$

if $x_i$ is unconstrained.

$\mathbf{w}\left[i-1\right]<0$

if $x_i$ is constrained by its lower bound.

$\mathbf{w}\left[i-1\right]>0$

if $x_i$ is constrained by its upper bound.

$\mathbf{w}\left[i-1\right]$

may be any value if $l_i=u_i$.

14:  $\mathbf{indx}\left[\mathbf{n}\right]$ – IntegerOutput

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

$\mathbf{indx}\left[\mathit{i}-1\right]$, for $\mathit{i}=1,2,\dots ,\mathbf{nfree}$

These elements of the solution have not hit a constraint; i.e., $\mathbf{w}\left[i-1\right]=0$.

$\mathbf{indx}\left[\mathit{i}-1\right]$, for $\mathit{i}=\mathbf{nfree}+1,\dots ,k$

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

$\mathbf{indx}\left[\mathit{i}-1\right]$, for $\mathit{i}=k+1,\dots ,\mathbf{n}$

These elements of the solution are fixed by the bounds; i.e., $\mathbf{bl}\left[i-1\right]=\mathbf{bu}\left[i-1\right]$.

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:  $\mathbf{fail}$ – 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_CONVERGENCE

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

NE_INT

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

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

NE_INT_2

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

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_REAL_2

On entry, when $i=〈\mathit{\text{value}}〉$, $\mathbf{bl}\left[i-1\right]=〈\mathit{\text{value}}〉$ and $\mathbf{bu}\left[i-1\right]=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{bl}\left[i-1\right]\le \mathbf{bu}\left[i-1\right]$.

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example

10.1

Program Text

Program Text (e04pcce.c)

10.2

Program Data

Program Data (e04pcce.d)

10.3

Program Results

Program Results (e04pcce.r)