Integer, Intent (In)	::	nfmax
Integer, Intent (Inout)	::	iuser(*), ifail
Real (Kind=nag_wp), External	::	f
Real (Kind=nag_wp), Intent (In)	::	eps, eta
Real (Kind=nag_wp), Intent (Inout)	::	x, ruser(*)

C Header Interface

#include <nag.h>

void	c05awf_ (double x, const double eps, const double eta, double (NAG_CALL f)(const double x, Integer iuser[], double ruser[]), const Integer nfmax, Integer iuser[], double ruser[], Integer *ifail)

The routine may be called by the names c05awf or nagf_roots_contfn_cntin.

3 Description

c05awf attempts to obtain an approximation to a simple zero

α

of the function

f (x)

given an initial approximation

x

α

. The zero is found by a call to c05axf whose specification should be consulted for details of the method used.

The approximation

x

to the zero

α

is determined so that at least one of the following criteria is satisfied:

(i) $| x - α | \sim eps$ ,
(ii) $| f (x) | < eta$ .

4 References

None.

5 Arguments

1: $x$ – Real (Kind=nag_wp) Input/Output

On entry: an initial approximation to the zero.

On exit: the final approximation to the zero, unless

ifail = 1

2

, in which case it contains no useful information.

2: $eps$ – Real (Kind=nag_wp) Input

On entry: an absolute tolerance to control the accuracy to which the zero is determined. In general, the smaller the value of eps the more accurate x will be as an approximation to

α

. Indeed, for very small positive values of eps, it is likely that the final approximation will satisfy

| x - α | < eps

. You are advised to call the routine with more than one value for eps to check the accuracy obtained.

Constraint:

eps > 0.0

3: $eta$ – Real (Kind=nag_wp) Input

On entry: a value such that if

| f (x) | < eta

x

is accepted as the zero. eta may be specified as

0.0

(see Section 7).

4: $f$ – real (Kind=nag_wp) Function, supplied by the user. External Procedure

f must evaluate the function

f

whose zero is to be determined.

The specification of f is:

Fortran Interface

Function f (

x, iuser, ruser)

Real (Kind=nag_wp)	::	f
Integer, Intent (Inout)	::	iuser(*)
Real (Kind=nag_wp), Intent (In)	::	x
Real (Kind=nag_wp), Intent (Inout)	::	ruser(*)

C Header Interface

double

f (const double *x, Integer iuser[], double ruser[])

1: $x$ – Real (Kind=nag_wp) Input: On entry: the point at which the function must be evaluated.
2: $iuser (*)$ – Integer array User Workspace
3: $ruser (*)$ – Real (Kind=nag_wp) array User Workspace: f is called with the arguments iuser and ruser as supplied to c05awf. You should use the arrays iuser and ruser to supply information to f.

f must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which c05awf is called. Arguments denoted as Input must not be changed by this procedure.

Note: f should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by c05awf. If your code inadvertently does return any NaNs or infinities, c05awf is likely to produce unexpected results.

5: $nfmax$ – Integer Input

On entry: the maximum permitted number of calls to f from c05awf. If f is inexpensive to evaluate, nfmax should be given a large value (say

> 1000

Constraint:

nfmax > 0

6: $iuser (*)$ – Integer array User Workspace

7: $ruser (*)$ – Real (Kind=nag_wp) array User Workspace

iuser and ruser are not used by c05awf, but are passed directly to f and may be used to pass information to this routine.

8: $ifail$ – Integer Input/Output

On entry: ifail must be set to

0

- 1

1

to set behaviour on detection of an error; these values have no effect when no error is detected.

A value of

0

causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of

- 1

means that an error message is printed while a value of

1

means that it is not.

If halting is not appropriate, the value

- 1

1

is recommended. If message printing is undesirable, then the value

1

is recommended. Otherwise, the value

0

is recommended. When the value $- 1$ or $1$ is used it is essential to test the value of ifail on exit.

On exit:

ifail = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

ifail = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by x04aaf).

Errors or warnings detected by the routine:

$ifail = 1$: On entry, $eps = ⟨ value ⟩$ .
Constraint: $eps > 0.0$ .

On entry, $nfmax = ⟨ value ⟩$ .
Constraint: $nfmax > 0$ .

$ifail = 2$: Internal scale factor invalid for this problem. Consider using c05axf instead and setting scal.

$ifail = 3$: Either f has no zero near x or too much accuracy has been requested. Check the coding of f or increase eps.

$ifail = 4$: More than nfmax calls have been made to f.

nfmax may be too small for the problem (because x is too far away from the zero), or f has no zero near x, or too much accuracy has been requested in calculating the zero. Increase nfmax, check the coding of f or increase eps.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

The levels of accuracy depend on the values of eps and eta. If full machine accuracy is required, they may be set very small, resulting in an exit with

ifail = 3

4

, although this may involve many more iterations than a lesser accuracy. You are recommended to set

eta = 0.0

and to use eps to control the accuracy, unless you have considerable knowledge of the size of

f (x)

for values of

x

near the zero.

8 Parallelism and Performance

c05awf is not threaded in any implementation.

9 Further Comments

The time taken by c05awf depends primarily on the time spent evaluating the function

f

(see Section 5) and on how close the initial value of x is to the zero.

If a more flexible way of specifying the function

f

is required or if you wish to have closer control of the calculation, then the reverse communication routine c05axf is recommended instead of c05awf.