NAG Library Routine Document

C05AXF

1  Purpose

2  Specification

3  Description

4  References

5  Parameters

1:     X – REAL (KIND=nag_wp)Input/Output

On initial entry: an initial approximation to the zero.

On intermediate exit: the point at which $f$ must be evaluated before re-entry to the routine.

On final exit: the final approximation to the zero.

2:     FX – REAL (KIND=nag_wp)Input

On initial entry: if $\mathbf{IND}=1$, FX need not be set.

If $\mathbf{IND}=-1$, FX must contain $f\left(\mathbf{X}\right)$ for the initial value of X.

On intermediate re-entry: must contain $f\left(\mathbf{X}\right)$ for the current value of X.

3:     TOL – REAL (KIND=nag_wp)Input

On initial entry: a value that controls the accuracy to which the zero is determined. TOL is used in determining the convergence of the secant iteration used at each stage of the continuation process. It is used directly when solving the last problem (${\theta }_m=0$ in Section 3), and $\sqrt{\mathbf{TOL}}$ is used for the problem defined by ${\theta }_r$, $r<m$. Convergence to the accuracy specified by TOL is not guaranteed, and so you are recommended to find the zero using at least two values for TOL to check the accuracy obtained.

Constraint: $\mathbf{TOL}>0.0$.

4:     IR – INTEGERInput

On initial entry: indicates the type of error test required, as follows. Solving the problem defined by ${\theta }_r$, $1\le r\le m$, involves computing a sequence of secant iterates $x_r^0,x_r^1,\dots$. This sequence will be considered to have converged only if:

for $\mathbf{IR}=0$,

$$\left|x_r^{\left(i+1\right)}-x_r^{\left(i\right)}\right|\le \mathit{eps}\times \max \left(1.0,\left|x_r^{\left(i\right)}\right|\right)\text{,}$$

for $\mathbf{IR}=1$,

$$\left|x_r^{\left(i+1\right)}-x_r^{\left(i\right)}\right|\le \mathit{eps}\text{,}$$

for $\mathbf{IR}=2$,

$$\left|x_r^{\left(i+1\right)}-x_r^{\left(i\right)}\right|\le \mathit{eps}\times \left|x_r^{\left(i\right)}\right|\text{,}$$

for some $i>1$; here $\mathit{eps}$ is either TOL or $\sqrt{\mathbf{TOL}}$ as discussed above. Note that there are other subsidiary conditions (not given here) which must also be satisfied before the secant iteration is considered to have converged.

Constraint: $\mathbf{IR}=0$, $1$ or $2$.

5:     SCAL – REAL (KIND=nag_wp)Input

On initial entry: a factor for use in determining a significant approximation to the derivative of $f\left(x\right)$ at $x=x_0$, the initial value. A number of difference approximations to $f^{\prime }\left(x_0\right)$ are calculated using

$$f^{\prime }\left(x_0\right)\sim \left(f\left(x_0+h\right)-f\left(x_0\right)\right)/h$$

where $\left|h\right|<\left|\mathbf{SCAL}\right|$ and $h$ has the same sign as SCAL. A significance (cancellation) check is made on each difference approximation and the approximation is rejected if insignificant.

Suggested value: $\sqrt{\epsilon }$, where $\epsilon$ is the machine precision returned by X02AJF.

Constraint: $\mathbf{SCAL}$ must be sufficiently large that $\mathbf{X}+\mathbf{SCAL}\ne \mathbf{X}$ on the computer.

6:     C($26$) – REAL (KIND=nag_wp) arrayCommunication Array

($\mathbf{C}\left(5\right)$ contains the current ${\theta }_r$, this value may be useful in the event of an error exit.)

7:     IND – INTEGERInput/Output

On initial entry: must be set to $1$ or $-1$.

$\mathbf{IND}=1$

FX need not be set.

$\mathbf{IND}=-1$

FX must contain $f\left(\mathbf{X}\right)$.

On intermediate exit: contains $2$, $3$ or $4$. The calling program must evaluate $f$ at X, storing the result in FX, and re-enter C05AXF with all other parameters unchanged.

On final exit: contains $0$.

Constraint: on entry $\mathbf{IND}=-1$, $1$, $2$, $3$ or $4$.

8:     IFAIL – INTEGERInput/Output

On initial entry: IFAIL must be set to $0$, $-1\text{ or }1$. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{ or }1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, because for this routine the values of the output parameters may be useful even if $\mathbf{IFAIL}\ne \mathbf{0}$ on exit, the recommended value is $-1$. When the value $-\mathbf{1}\text{ or }1$ is used it is essential to test the value of IFAIL on exit.

On final exit: $\mathbf{IFAIL}=\mathbf{0}$ unless the routine detects an error or a warning has been flagged (see Section 6).

6  Error Indicators and Warnings

$\mathbf{IFAIL}=1$

On entry,	$\mathbf{TOL}\le 0.0$,
or	$\mathbf{IR}\ne 0$, $1$ or $2$.

$\mathbf{IFAIL}=2$

The parameter IND is incorrectly set on initial or intermediate entry.

$\mathbf{IFAIL}=3$

SCAL is too small, or significant derivatives of $f$ cannot be computed (this can happen when $f$ is almost constant and nonzero, for any value of SCAL).

$\mathbf{IFAIL}=4$

The current problem in the continuation sequence cannot be solved, see $\mathbf{C}\left(5\right)$ for the value of ${\theta }_r$. The most likely explanation is that the current problem has no solution, either because the original problem had no solution or because the continuation path passes through a set of insoluble problems. This latter reason for failure should occur rarely, and not at all if the initial approximation to the zero is sufficiently close. Other possible explanations are that TOL is too small and hence the accuracy requirement is too stringent, or that TOL is too large and the initial approximation too poor, leading to successively worse intermediate solutions.

$\mathbf{IFAIL}=5$

Continuation away from the initial point is not possible. This error exit will usually occur if the problem has not been properly posed or the error requirement is extremely stringent.

$\mathbf{IFAIL}=6$

The final problem (with ${\theta }_m=0$) cannot be solved. It is likely that too much accuracy has been requested, or that the zero is at $\alpha =0$ and $\mathbf{IR}=2$.

7  Accuracy

8  Further Comments

9  Example

9.1  Program Text

Program Text (c05axfe.f90)

9.2  Program Data

9.3  Program Results

Program Results (c05axfe.r)