c05 Chapter Contents
c05 Chapter Introduction
NAG Library Manual

# NAG Library Function Documentnag_zero_cont_func_cntin (c05awc)

## 1  Purpose

nag_zero_cont_func_cntin (c05awc) attempts to locate a zero of a continuous function using a continuation method based on a secant iteration.

## 2  Specification

 #include #include
void  nag_zero_cont_func_cntin (double *x, double eps, double eta,
 double (*f)(double x, Nag_Comm *comm),
Integer nfmax, Nag_Comm *comm, NagError *fail)

## 3  Description

nag_zero_cont_func_cntin (c05awc) 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 nag_zero_cont_func_cntin_rcomm (c05axc) 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:
 (i) $\left|x-\alpha \right|\sim {\mathbf{eps}}$, (ii) $\left|f\left(x\right)\right|<{\mathbf{eta}}$.

None.

## 5  Arguments

1:    $\mathbf{x}$double *Input/Output
On entry: an initial approximation to the zero.
On exit: if ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_NOERROR, NE_SECANT_ITER_FAILED or NE_TOO_MANY_CALLS it contains the approximation to the zero, otherwise it contains no useful information.
2:    $\mathbf{eps}$doubleInput
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 $\left|{\mathbf{x}}-\alpha \right|<{\mathbf{eps}}$. You are advised to call the function with more than one value for eps to check the accuracy obtained.
Constraint: ${\mathbf{eps}}>0.0$.
3:    $\mathbf{eta}$doubleInput
On entry: a value such that if $\left|f\left(x\right)\right|<{\mathbf{eta}}$, $x$ is accepted as the zero. eta may be specified as $0.0$ (see Section 7).
4:    $\mathbf{f}$function, supplied by the userExternal Function
f must evaluate the function $f$ whose zero is to be determined.
The specification of f is:
 double f (double x, Nag_Comm *comm)
1:    $\mathbf{x}$doubleInput
On entry: the point at which the function must be evaluated.
2:    $\mathbf{comm}$Nag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to f.
userdouble *
iuserInteger *
pPointer
The type Pointer will be void *. Before calling nag_zero_cont_func_cntin (c05awc) you may allocate memory and initialize these pointers with various quantities for use by f when called from nag_zero_cont_func_cntin (c05awc) (see Section 2.3.1.1 in How to Use the NAG Library and its Documentation).
5:    $\mathbf{nfmax}$IntegerInput
On entry: the maximum permitted number of calls to f from nag_zero_cont_func_cntin (c05awc). If f is inexpensive to evaluate, nfmax should be given a large value (say $\text{}>1000$).
Constraint: ${\mathbf{nfmax}}>0$.
6:    $\mathbf{comm}$Nag_Comm *
The NAG communication argument (see Section 2.3.1.1 in How to Use the NAG Library and its Documentation).
7:    $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 2.7 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.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT
On entry, ${\mathbf{nfmax}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{nfmax}}>0$.
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.
A serious error occurred in an internal call to an auxiliary function.
Internal scale factor invalid for this problem. Consider using nag_zero_cont_func_cntin_rcomm (c05axc) instead and setting scal.
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, ${\mathbf{eps}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{eps}}>0.0$.
NE_SECANT_ITER_FAILED
Either f has no zero near x or too much accuracy has been requested. Check the coding of f or increase eps.
NE_TOO_MANY_CALLS
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.

## 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 ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_SECANT_ITER_FAILED or NE_TOO_MANY_CALLS, 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.

## 8  Parallelism and Performance

nag_zero_cont_func_cntin (c05awc) is not threaded in any implementation.

The time taken by nag_zero_cont_func_cntin (c05awc) 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 function nag_zero_cont_func_cntin_rcomm (c05axc) is recommended instead of nag_zero_cont_func_cntin (c05awc).

## 10  Example

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.

### 10.1  Program Text

Program Text (c05awce.c)

None.

### 10.3  Program Results

Program Results (c05awce.r)