NAG Library Routine Document

C05AZF

1  Purpose

2  Specification

3  Description

4  References

5  Parameters

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

2:     Y – REAL (KIND=nag_wp)Input/Output

On initial entry: X and Y must define an initial interval $\left[a,b\right]$ containing the zero, such that $f\left(\mathbf{X}\right)\times f\left(\mathbf{Y}\right)\le 0.0$. It is not necessary that $\mathbf{X}<\mathbf{Y}$.

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

On final exit: X and Y define a smaller interval containing the zero, such that $f\left(\mathbf{X}\right)\times f\left(\mathbf{Y}\right)\le 0.0$, and $\left|\mathbf{X}-\mathbf{Y}\right|$ satisfies the accuracy specified by TOLX and IR, unless an error has occurred. If $\mathbf{IFAIL}=\mathbf{4}$, X and Y generally contain very good approximations to a pole; if $\mathbf{IFAIL}=\mathbf{5}$, X and Y generally contain very good approximations to the zero (see Section 6). If a point X is found such that $f\left(\mathbf{X}\right)=0.0$, then on final exit $\mathbf{X}=\mathbf{Y}$ (in this case there is no guarantee that X is a simple zero). In all cases, the value returned in X is the better approximation to the zero.

3:     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.

4:     TOLX – REAL (KIND=nag_wp)Input

On initial entry: the accuracy to which the zero is required. The type of error test is specified by IR.

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

5:     IR – INTEGERInput

On initial entry: indicates the type of error test.

$\mathbf{IR}=0$

The test is: $\left|\mathbf{X}-\mathbf{Y}\right|\le 2.0\times \mathbf{TOLX}\times \max \left(1.0,\left|\mathbf{X}\right|\right)$.

$\mathbf{IR}=1$

The test is: $\left|\mathbf{X}-\mathbf{Y}\right|\le 2.0\times \mathbf{TOLX}$.

$\mathbf{IR}=2$

The test is: $\left|\mathbf{X}-\mathbf{Y}\right|\le 2.0\times \mathbf{TOLX}\times \left|\mathbf{X}\right|$.

Suggested value: $\mathbf{IR}=0$.

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

6:     C($17$) – REAL (KIND=nag_wp) arrayInput/Output

On initial entry: if $\mathbf{IND}=1$, no elements of C need be set.

If $\mathbf{IND}=-1$, $\mathbf{C}\left(1\right)$ must contain $f\left(\mathbf{Y}\right)$, other elements of C need not be set.

On final exit: is undefined.

7:     IND – INTEGERInput/Output

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

$\mathbf{IND}=1$

FX and $\mathbf{C}\left(1\right)$ need not be set.

$\mathbf{IND}=-1$

FX and $\mathbf{C}\left(1\right)$ must contain $f\left(\mathbf{X}\right)$ and $f\left(\mathbf{Y}\right)$ respectively.

On intermediate exit: contains $2$, $3$ or $4$. The calling program must evaluate $f$ at X, storing the result in FX, and re-enter C05AZF 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, $f\left(\mathbf{X}\right)$ and $f\left(\mathbf{Y}\right)$ have the same sign with neither equalling $0.0$.

$\mathbf{IFAIL}=2$

On entry, $\mathbf{IND}\ne -1$, $1$, $2$, $3$ or $4$.

$\mathbf{IFAIL}=3$

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

$\mathbf{IFAIL}=4$

An interval $\left[\mathbf{X},\mathbf{Y}\right]$ has been determined satisfying the error tolerance specified by TOLX and IR and such that $f\left(\mathbf{X}\right)\times f\left(\mathbf{Y}\right)\le 0$. However, from observation of the values of $f$ during the calculation of $\left[\mathbf{X},\mathbf{Y}\right]$, it seems that the interval $\left[\mathbf{X},\mathbf{Y}\right]$ contains a pole rather than a zero. Note that this error exit is not completely reliable: the error exit may be taken in extreme cases when $\left[\mathbf{X},\mathbf{Y}\right]$ contains a zero, or the error exit may not be taken when $\left[\mathbf{X},\mathbf{Y}\right]$ contains a pole. Both these cases occur most frequently when TOLX is large.

$\mathbf{IFAIL}=5$

The tolerance TOLX is too small for the problem being solved. This indicator is only set when the interval containing the zero has been reduced to one of relative length at most $2\epsilon$, where $\epsilon$ is the machine precision, but the exit condition specified by IR is not satisfied. It is unsafe to continue reducing the interval beyond this point, but the final values of X and Y returned are accurate approximations to the zero.

7  Accuracy

8  Further Comments

9  Example

9.1  Program Text

Program Text (c05azfe.f90)

9.2  Program Data

9.3  Program Results

Program Results (c05azfe.r)