NAG Library Function Document

nag_zero_nonlin_eqns_deriv_rcomm (c05rdc)

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

1:    $\mathbf{irevcm}$ – Integer *Input/Output

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

On intermediate exit: specifies what action you must take before re-entering nag_zero_nonlin_eqns_deriv_rcomm (c05rdc) 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 nag_zero_nonlin_eqns_deriv_rcomm (c05rdc), fvec must contain the function values $f_i\left(x\right)$.

$\mathbf{irevcm}=3$

Indicates that before re-entry to nag_zero_nonlin_eqns_deriv_rcomm (c05rdc), $\mathbf{fjac}\left[\left(\mathit{j}-1\right)\times \mathbf{n}+\mathit{i}-1\right]$ 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 nag_zero_nonlin_eqns_deriv_rcomm (c05rdc) 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 nag_zero_nonlin_eqns_deriv_rcomm (c05rdc). If your code inadvertently does return any NaNs or infinities, nag_zero_nonlin_eqns_deriv_rcomm (c05rdc) is likely to produce unexpected results.

2:    $\mathbf{n}$ – IntegerInput

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

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

3:    $\mathbf{x}\left[\mathbf{n}\right]$ – doubleInput/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:    $\mathbf{fvec}\left[\mathbf{n}\right]$ – doubleInput/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:    $\mathbf{fjac}\left[\mathbf{n}\times \mathbf{n}\right]$ – doubleInput/Output

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

On initial entry: need not be set.

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

If $\mathbf{irevcm}=3$, $\mathbf{fjac}\left[\left(\mathit{j}-1\right)\times \mathbf{n}+\mathit{i}-1\right]$ 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, stored by columns.

6:    $\mathbf{xtol}$ – doubleInput

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 nag_machine_precision (X02AJC).

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

7:    $\mathbf{scale\_mode}$ – Nag_ScaleTypeInput

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

If $\mathbf{scale\_mode}=\mathrm{Nag\_ScaleProvided}$, the scaling must have been supplied in diag.

Otherwise, if $\mathbf{scale\_mode}=\mathrm{Nag\_NoScaleProvided}$, the variables will be scaled internally.

Constraint: $\mathbf{scale\_mode}=\mathrm{Nag\_NoScaleProvided}$ or $\mathrm{Nag\_ScaleProvided}$.

8:    $\mathbf{diag}\left[\mathbf{n}\right]$ – doubleInput/Output

On initial entry: if $\mathbf{scale\_mode}=\mathrm{Nag\_ScaleProvided}$, diag must contain multiplicative scale factors for the variables.

If $\mathbf{scale\_mode}=\mathrm{Nag\_NoScaleProvided}$, diag need not be set.

Constraint: if $\mathbf{scale\_mode}=\mathrm{Nag\_ScaleProvided}$,$\mathbf{diag}\left[\mathit{i}-1\right]>0.0$, 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 $\mathbf{scale\_mode}=\mathrm{Nag\_NoScaleProvided}$).

9:    $\mathbf{factor}$ – doubleInput

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:  $\mathbf{r}\left[\mathbf{n}\times \left(\mathbf{n}+1\right)/2\right]$ – doubleInput/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:  $\mathbf{qtf}\left[\mathbf{n}\right]$ – doubleInput/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:  $\mathbf{iwsav}\left[17\right]$ – IntegerCommunication Array

13:  $\mathbf{rwsav}\left[4\times \mathbf{n}+10\right]$ – doubleCommunication Array

The arrays iwsav and rwsav MUST NOT be altered between calls to nag_zero_nonlin_eqns_deriv_rcomm (c05rdc).

14:  $\mathbf{fail}$ – 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_DIAG_ELEMENTS

On entry, $\mathbf{scale\_mode}=\mathrm{Nag\_ScaleProvided}$ and diag contained a non-positive element.

NE_INT

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

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

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_IMPROVEMENT

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 nag_zero_nonlin_eqns_deriv_rcomm (c05rdc) from a different starting point may avoid the region of difficulty.

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 nag_zero_nonlin_eqns_deriv_rcomm (c05rdc) from a different starting point may avoid the region of difficulty.

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

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

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

NE_TOO_SMALL

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

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example

10.1

Program Text

Program Text (c05rdce.c)

10.2

Program Data

10.3

Program Results

Program Results (c05rdce.r)