NAG Library Function Document

nag_opt_bnd_lin_lsq (e04pcc)

1  Purpose

2  Specification

3  Description

4  References

5  Arguments

1:     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:     m – IntegerInput

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

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

3:     n – IntegerInput

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

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

4:     a[$\mathbf{pda}\times \mathbf{n}$] – 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:     pda – IntegerInput

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

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

6:     b[m] – 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:     bl[n] – const doubleInput

8:     bu[n] – 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:     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 $\mathbf{tol}<\sqrt{\mathit{machine precision}}$, then $\sqrt{\mathit{machine precision}}$ is used.

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

10:   x[n] – doubleOutput

On exit: the solution vector $x$.

11:   rnorm – double *Output

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

12:   nfree – Integer *Output

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

13:   w[n] – 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:   indx[n] – 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:   fail – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

NE_ALLOC_FAIL

Dynamic memory allocation failed.

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.

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)