# NAG FL Interfacec05awf (contfn_​cntin)

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## 1Purpose

c05awf attempts to locate a zero of a continuous function using a continuation method based on a secant iteration.

## 2Specification

Fortran Interface
 Subroutine c05awf ( x, eps, eta, f,
 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(*)
#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.

## 3Description

c05awf attempts to obtain an approximation to a simple zero $\alpha$ of the function $f\left(x\right)$ given an initial approximation $x$ to $\alpha$. 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 $\alpha$ is determined so that at least one of the following criteria is satisfied:
1. (i)$|x-\alpha |\sim {\mathbf{eps}}$,
2. (ii)$|f\left(x\right)|<{\mathbf{eta}}$.
None.

## 5Arguments

1: $\mathbf{x}$Real (Kind=nag_wp) Input/Output
On entry: an initial approximation to the zero.
On exit: the final approximation to the zero, unless ${\mathbf{ifail}}={\mathbf{1}}$ or ${\mathbf{2}}$, in which case it contains no useful information.
2: $\mathbf{eps}$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 $\alpha$. Indeed, for very small positive values of eps, it is likely that the final approximation will satisfy $|{\mathbf{x}}-\alpha |<{\mathbf{eps}}$. You are advised to call the routine with more than one value for eps to check the accuracy obtained.
Constraint: ${\mathbf{eps}}>0.0$.
3: $\mathbf{eta}$Real (Kind=nag_wp) Input
On entry: a value such that if $|f\left(x\right)|<{\mathbf{eta}}$, $x$ is accepted as the zero. eta may be specified as $0.0$ (see Section 7).
4: $\mathbf{f}$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,
 Real (Kind=nag_wp) :: f Integer, Intent (Inout) :: iuser(*) Real (Kind=nag_wp), Intent (In) :: x Real (Kind=nag_wp), Intent (Inout) :: ruser(*)
 double f (const double *x, Integer iuser[], double ruser[])
1: $\mathbf{x}$Real (Kind=nag_wp) Input
On entry: the point at which the function must be evaluated.
2: $\mathbf{iuser}\left(*\right)$Integer array User Workspace
3: $\mathbf{ruser}\left(*\right)$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: $\mathbf{nfmax}$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 $\text{}>1000$).
Constraint: ${\mathbf{nfmax}}>0$.
6: $\mathbf{iuser}\left(*\right)$Integer array User Workspace
7: $\mathbf{ruser}\left(*\right)$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: $\mathbf{ifail}$Integer Input/Output
On entry: ifail must be set to $0$, $-1$ or $1$ 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 $-1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $-\mathbf{1}$ or $\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
On entry, ${\mathbf{eps}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{eps}}>0.0$.
On entry, ${\mathbf{nfmax}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{nfmax}}>0$.
${\mathbf{ifail}}=2$
Internal scale factor invalid for this problem. Consider using c05axf instead and setting scal.
${\mathbf{ifail}}=3$
Either f has no zero near x or too much accuracy has been requested. Check the coding of f or increase eps.
${\mathbf{ifail}}=4$
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.
${\mathbf{ifail}}=-99$
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
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.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

## 7Accuracy

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 ${\mathbf{ifail}}={\mathbf{3}}$ or ${\mathbf{4}}$, although this may involve many more iterations than a lesser accuracy. You are recommended to set ${\mathbf{eta}}=0.0$ and to use eps to control the accuracy, unless you have considerable knowledge of the size of $f\left(x\right)$ for values of $x$ near the zero.

## 8Parallelism and Performance

c05awf is not threaded in any implementation.

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.

## 10Example

This example calculates the zero of $f\left(x\right)={e}^{-x}-x$ from a starting value ${\mathbf{x}}=1.0$. Two calculations are made with ${\mathbf{eps}}=\text{1.0E−3}$ and $\text{1.0E−4}$ for comparison purposes, with ${\mathbf{eta}}=0.0$ in both cases.

### 10.1Program Text

Program Text (c05awfe.f90)

None.

### 10.3Program Results

Program Results (c05awfe.r)