NAG Library Function Document

nag_zero_nonlin_eqns_deriv_expert (c05rcc)

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

1:    $\mathbf{fcn}$ – function, supplied by the userExternal Function

Depending upon the value of iflag, fcn must either return the values of the functions $f_i$ at a point $x$ or return the Jacobian at $x$.

The specification of fcn is:

void  fcn (Integer n, const double x[], double fvec[], double fjac[], Nag_Comm *comm, Integer *iflag)

1:    $\mathbf{n}$ – IntegerInput

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

2:    $\mathbf{x}\left[\mathbf{n}\right]$ – const doubleInput

On entry: the components of the point $x$ at which the functions or the Jacobian must be evaluated.

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

On entry: if $\mathbf{iflag}=0$ or $2$, fvec contains the function values $f_i\left(x\right)$ and must not be changed.

On exit: if $\mathbf{iflag}=1$ on entry, fvec must contain the function values $f_i\left(x\right)$ (unless iflag is set to a negative value by fcn).

4:    $\mathbf{fjac}\left[\mathbf{n}\times \mathbf{n}\right]$ – 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 entry: if $\mathbf{iflag}=0$, $\mathbf{fjac}\left[\left(\mathit{j}-1\right)\times \mathbf{n}+\mathit{i}-1\right]$ contains the value of $\frac{\partial f_{\mathit{i}}}{\partial x_{\mathit{j}}}$ at the point $x$, for $\mathit{i}=1,2,\dots ,n$ and $\mathit{j}=1,2,\dots ,n$. When $\mathbf{iflag}=0$ or $1$, fjac must not be changed.

On exit: if $\mathbf{iflag}=2$ on entry, $\mathbf{fjac}\left[\left(\mathit{j}-1\right)\times \mathbf{n}+\mathit{i}-1\right]$ 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 $\mathit{j}=1,2,\dots ,n$, (unless iflag is set to a negative value by fcn).

5:    $\mathbf{comm}$ – Nag_Comm *

Pointer to structure of type Nag_Comm; the following members are relevant to fcn.

user – double *

iuser – Integer *

p – Pointer 

The type Pointer will be void *. Before calling nag_zero_nonlin_eqns_deriv_expert (c05rcc) you may allocate memory and initialize these pointers with various quantities for use by fcn when called from nag_zero_nonlin_eqns_deriv_expert (c05rcc) (see Section 3.3.1.1 in How to Use the NAG Library and its Documentation).

6:    $\mathbf{iflag}$ – Integer *Input/Output

On entry: $\mathbf{iflag}=0$, $1$ or $2$.

$\mathbf{iflag}=0$

x, fvec and fjac are available for printing (see nprint).

$\mathbf{iflag}=1$

fvec is to be updated.

$\mathbf{iflag}=2$

fjac is to be updated.

On exit: in general, iflag should not be reset by fcn. If, however, you wish to terminate execution (perhaps because some illegal point x has been reached), iflag should be set to a negative integer value.

Note: fcn should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by nag_zero_nonlin_eqns_deriv_expert (c05rcc). If your code inadvertently does return any NaNs or infinities, nag_zero_nonlin_eqns_deriv_expert (c05rcc) 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 entry: an initial guess at the solution vector.

On exit: the final estimate of the solution vector.

4:    $\mathbf{fvec}\left[\mathbf{n}\right]$ – doubleOutput

On exit: the function values at the final point returned in x.

5:    $\mathbf{fjac}\left[\mathbf{n}\times \mathbf{n}\right]$ – doubleOutput

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

6:    $\mathbf{xtol}$ – doubleInput

On 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:    $\mathbf{maxfev}$ – IntegerInput

On entry: the maximum number of calls to fcn with $\mathbf{iflag}\ne 0$. nag_zero_nonlin_eqns_deriv_expert (c05rcc) will exit with $\mathbf{fail}\mathbf{.}\mathbf{code}=$ NE_TOO_MANY_FEVALS, if, at the end of an iteration, the number of calls to fcn exceeds maxfev.

Suggested value: $\mathbf{maxfev}=100\times \left(\mathbf{n}+1\right)$.

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

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

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

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

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

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

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

On 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 exit: the scale factors actually used (computed internally if $\mathbf{scale\_mode}=\mathrm{Nag\_NoScaleProvided}$).

10:  $\mathbf{factor}$ – doubleInput

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

11:  $\mathbf{nprint}$ – IntegerInput

On entry: indicates whether (and how often) special calls to fcn, with iflag set to $0$, are to be made for printing purposes.

$\mathbf{nprint}\le 0$

No calls are made.

$\mathbf{nprint}>0$

fcn is called at the beginning of the first iteration, every nprint iterations thereafter and immediately before the return from nag_zero_nonlin_eqns_deriv_expert (c05rcc).

12:  $\mathbf{nfev}$ – Integer *Output

On exit: the number of calls made to fcn to evaluate the functions.

13:  $\mathbf{njev}$ – Integer *Output

On exit: the number of calls made to fcn to evaluate the Jacobian.

14:  $\mathbf{r}\left[\mathbf{n}\times \left(\mathbf{n}+1\right)/2\right]$ – doubleOutput

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

15:  $\mathbf{qtf}\left[\mathbf{n}\right]$ – doubleOutput

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

16:  $\mathbf{comm}$ – Nag_Comm *

The NAG communication argument (see Section 3.3.1.1 in How to Use the NAG Library and its Documentation).

17:  $\mathbf{fail}$ – 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, $\mathbf{maxfev}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{maxfev}>0$.

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_expert (c05rcc) 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_expert (c05rcc) 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_MANY_FEVALS

There have been at least maxfev calls to fcn: $\mathbf{maxfev}=〈\mathit{\text{value}}〉$. Consider restarting the calculation from the final point held in x.

NE_TOO_SMALL

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

NE_USER_STOP

iflag was set negative in fcn. $\mathbf{iflag}=〈\mathit{\text{value}}〉$.

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example

10.1

Program Text

Program Text (c05rcce.c)

10.2

Program Data

10.3

Program Results

Program Results (c05rcce.r)