NAG Library Routine Document

c05rdf  (sys_deriv_rcomm)

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

1:     $\mathbf{irevcm}$ – IntegerInput/Output

On initial entry: must have the value $0$.

On intermediate exit: specifies what action you must take before re-entering c05rdf with irevcm unchanged. The value of irevcm should be interpreted as follows:

$\mathbf{irevcm}=1$

Indicates the start of a new iteration. No action is required by you, but x and fvec are available for printing.

$\mathbf{irevcm}=2$

Indicates that before re-entry to c05rdf, fvec must contain the function values $f_i\left(x\right)$.

$\mathbf{irevcm}=3$

Indicates that before re-entry to c05rdf, $\mathbf{fjac}\left(\mathit{i},\mathit{j}\right)$ must contain the value of $\frac{\partial f_{\mathit{i}}}{\partial x_{\mathit{j}}}$ at the point $x$, for $\mathit{i}=1,2,\dots ,n$ and $\mathit{j}=1,2,\dots ,n$.

On final exit: $\mathbf{irevcm}=0$ and the algorithm has terminated.

Constraint: $\mathbf{irevcm}=0$, $1$, $2$ or $3$.

Note: any values you return to c05rdf as part of the reverse communication procedure should not include floating-point NaN (Not a Number) or infinity values, since these are not handled by c05rdf. If your code inadvertently does return any NaNs or infinities, c05rdf is likely to produce unexpected results.

2:     $\mathbf{n}$ – IntegerInput

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

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

3:     $\mathbf{x}\left(\mathbf{n}\right)$ – Real (Kind=nag_wp) arrayInput/Output

On initial entry: an initial guess at the solution vector.

On intermediate exit: contains the current point.

On final exit: the final estimate of the solution vector.

4:     $\mathbf{fvec}\left(\mathbf{n}\right)$ – Real (Kind=nag_wp) arrayInput/Output

On initial entry: need not be set.

On intermediate re-entry: if $\mathbf{irevcm}\ne 2$, fvec must not be changed.

If $\mathbf{irevcm}=2$, fvec must be set to the values of the functions computed at the current point x.

On final exit: the function values at the final point, x.

5:     $\mathbf{fjac}\left(\mathbf{n},\mathbf{n}\right)$ – Real (Kind=nag_wp) arrayInput/Output

On initial entry: need not be set.

On intermediate re-entry: if $\mathbf{irevcm}\ne 3$, fjac must not be changed.

If $\mathbf{irevcm}=3$, $\mathbf{fjac}\left(\mathit{i},\mathit{j}\right)$ must contain the value of $\frac{\partial f_{\mathit{i}}}{\partial x_{\mathit{j}}}$ at the point $x$, for $\mathit{i}=1,2,\dots ,n$ and $\mathit{j}=1,2,\dots ,n$.

On final exit: the orthogonal matrix $Q$ produced by the $QR$ factorization of the final approximate Jacobian.

6:     $\mathbf{xtol}$ – Real (Kind=nag_wp)Input

On initial entry: the accuracy in x to which the solution is required.

Suggested value: $\sqrt{\epsilon }$, where $\epsilon$ is the machine precision returned by x02ajf.

Constraint: $\mathbf{xtol}\ge 0.0$.

7:     $\mathbf{mode}$ – IntegerInput

On initial entry: indicates whether or not you have provided scaling factors in diag.

If $\mathbf{mode}=2$, the scaling must have been supplied in diag.

Otherwise, if $\mathbf{mode}=1$, the variables will be scaled internally.

Constraint: $\mathbf{mode}=1$ or $2$.

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

On initial entry: if $\mathbf{mode}=2$, diag must contain multiplicative scale factors for the variables.

If $\mathbf{mode}=1$, diag need not be set.

Constraint: if $\mathbf{mode}=2$,$\mathbf{diag}\left(\mathit{i}\right)>0.0$, for $\mathit{i}=1,2,\dots ,n$.

On intermediate exit: diag must not be changed.

On final exit: the scale factors actually used (computed internally if $\mathbf{mode}=1$).

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

On initial entry: a quantity to be used in determining the initial step bound. In most cases, factor should lie between $0.1$ and $100.0$. (The step bound is $\mathbf{factor}\times {‖\mathbf{diag}\times \mathbf{x}‖}_2$ if this is nonzero; otherwise the bound is factor.)

Suggested value: $\mathbf{factor}=100.0$.

Constraint: $\mathbf{factor}>0.0$.

10:   $\mathbf{r}\left(\mathbf{n}\times \left(\mathbf{n}+1\right)/2\right)$ – Real (Kind=nag_wp) arrayInput/Output

On initial entry: need not be set.

On intermediate exit: must not be changed.

On final exit: the upper triangular matrix $R$ produced by the $QR$ factorization of the final approximate Jacobian, stored row-wise.

11:   $\mathbf{qtf}\left(\mathbf{n}\right)$ – Real (Kind=nag_wp) arrayInput/Output

On initial entry: need not be set.

On intermediate exit: must not be changed.

On final exit: the vector $Q^{\mathrm{T}}f$.

12:   $\mathbf{iwsav}\left(17\right)$ – Integer arrayCommunication Array

13:   $\mathbf{rwsav}\left(4\times \mathbf{n}+10\right)$ – Real (Kind=nag_wp) arrayCommunication Array

The arrays iwsav and rwsav must not be altered between calls to c05rdf.

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

On initial 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 final 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

$\mathbf{ifail}=2$

On entry, $\mathbf{irevcm}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{irevcm}=0$, $1$, $2$ or $3$.

$\mathbf{ifail}=3$

No further improvement in the solution is possible. xtol is too small: $\mathbf{xtol}=〈\mathit{\text{value}}〉$.

$\mathbf{ifail}=4$

The iteration is not making good progress, as measured by the improvement from the last $〈\mathit{\text{value}}〉$ Jacobian evaluations. This failure exit may indicate that the system does not have a zero, or that the solution is very close to the origin (see Section 7). Otherwise, rerunning c05rdf from a different starting point may avoid the region of difficulty.

$\mathbf{ifail}=5$

The iteration is not making good progress, as measured by the improvement from the last $〈\mathit{\text{value}}〉$ iterations. This failure exit may indicate that the system does not have a zero, or that the solution is very close to the origin (see Section 7). Otherwise, rerunning c05rdf from a different starting point may avoid the region of difficulty.

$\mathbf{ifail}=11$

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

$\mathbf{ifail}=12$

On entry, $\mathbf{xtol}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{xtol}\ge 0.0$.

$\mathbf{ifail}=13$

On entry, $\mathbf{mode}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{mode}=1$ or $2$.

$\mathbf{ifail}=14$

On entry, $\mathbf{factor}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{factor}>0.0$.

$\mathbf{ifail}=15$

On entry, $\mathbf{mode}=2$ and diag contained a non-positive element.

$\mathbf{ifail}=-99$

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 (c05rdfe.f90)

10.2

Program Data

10.3

Program Results

Program Results (c05rdfe.r)