NAG CL Interfacec05azc (contfn_​brent_​rcomm)

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1Purpose

c05azc locates a simple zero of a continuous function in a given interval by using Brent's method, which is a combination of nonlinear interpolation, linear extrapolation and bisection. It uses reverse communication for evaluating the function.

2Specification

 #include
 void c05azc (double *x, double *y, double fx, double tolx, Nag_ErrorControl ir, double c[], Integer *ind, NagError *fail)
The function may be called by the names: c05azc, nag_roots_contfn_brent_rcomm or nag_zero_cont_func_brent_rcomm.

3Description

You must supply x and y to define an initial interval $\left[a,b\right]$ containing a simple zero of the function $f\left(x\right)$ (the choice of x and y must be such that $f\left({\mathbf{x}}\right)×f\left({\mathbf{y}}\right)\le 0.0$). The function combines the methods of bisection, nonlinear interpolation and linear extrapolation (see Dahlquist and Björck (1974)), to find a sequence of sub-intervals of the initial interval such that the final interval $\left[{\mathbf{x}},{\mathbf{y}}\right]$ contains the zero and $|{\mathbf{x}}-{\mathbf{y}}|$ is less than some tolerance specified by tolx and ir (see Section 5). In fact, since the intermediate intervals $\left[{\mathbf{x}},{\mathbf{y}}\right]$ are determined only so that $f\left({\mathbf{x}}\right)×f\left({\mathbf{y}}\right)\le 0.0$, it is possible that the final interval may contain a discontinuity or a pole of $f$ (violating the requirement that $f$ be continuous). c05azc checks if the sign change is likely to correspond to a pole of $f$ and gives an error return in this case.
A feature of the algorithm used by this function is that unlike some other methods it guarantees convergence within about ${\left({\mathrm{log}}_{2}\left[\left(b-a\right)/\delta \right]\right)}^{2}$ function evaluations, where $\delta$ is related to the argument tolx. See Brent (1973) for more details.
c05azc returns to the calling program for each evaluation of $f\left(x\right)$. On each return you should set ${\mathbf{fx}}=f\left({\mathbf{x}}\right)$ and call c05azc again.
The function is a modified version of procedure ‘zeroin’ given by Brent (1973).
Brent R P (1973) Algorithms for Minimization Without Derivatives Prentice–Hall
Bus J C P and Dekker T J (1975) Two efficient algorithms with guaranteed convergence for finding a zero of a function ACM Trans. Math. Software 1 330–345
Dahlquist G and Björck Å (1974) Numerical Methods Prentice–Hall

5Arguments

Note: this function uses reverse communication. Its use involves an initial entry, intermediate exits and re-entries, and a final exit, as indicated by the argument ind. Between intermediate exits and re-entries, all arguments other than fx must remain unchanged.
1: $\mathbf{x}$double * Input/Output
2: $\mathbf{y}$double * Input/Output
On initial entry: x and y must define an initial interval $\left[a,b\right]$ containing the zero, such that $f\left({\mathbf{x}}\right)×f\left({\mathbf{y}}\right)\le 0.0$. 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 function.
On final exit: x and y define a smaller interval containing the zero, such that $f\left({\mathbf{x}}\right)×f\left({\mathbf{y}}\right)\le 0.0$, and $|{\mathbf{x}}-{\mathbf{y}}|$ satisfies the accuracy specified by tolx and ir, unless an error has occurred. If ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_PROBABLE_POLE, x and y generally contain very good approximations to a pole; if ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NW_TOO_MUCH_ACC_REQUESTED, x and y generally contain very good approximations to the zero (see Section 6). If a point x is found such that $f\left({\mathbf{x}}\right)=0.0$, 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: $\mathbf{fx}$double 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: $\mathbf{tolx}$double 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: $\mathbf{ir}$Nag_ErrorControl Input
On initial entry: indicates the type of error test.
${\mathbf{ir}}=\mathrm{Nag_Mixed}$
The test is: $|{\mathbf{x}}-{\mathbf{y}}|\le 2.0×{\mathbf{tolx}}×\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1.0,|{\mathbf{x}}|\right)$.
${\mathbf{ir}}=\mathrm{Nag_Absolute}$
The test is: $|{\mathbf{x}}-{\mathbf{y}}|\le 2.0×{\mathbf{tolx}}$.
${\mathbf{ir}}=\mathrm{Nag_Relative}$
The test is: $|{\mathbf{x}}-{\mathbf{y}}|\le 2.0×{\mathbf{tolx}}×|{\mathbf{x}}|$.
Suggested value: ${\mathbf{ir}}=\mathrm{Nag_Mixed}$.
Constraint: ${\mathbf{ir}}=\mathrm{Nag_Mixed}$, $\mathrm{Nag_Absolute}$ or $\mathrm{Nag_Relative}$.
6: $\mathbf{c}\left[17\right]$double Input/Output
On initial entry: if ${\mathbf{ind}}=1$, no elements of c need be set.
If ${\mathbf{ind}}=-1$, ${\mathbf{c}}\left[0\right]$ must contain $f\left({\mathbf{y}}\right)$, other elements of c need not be set.
On final exit: is undefined.
7: $\mathbf{ind}$Integer * Input/Output
On initial entry: must be set to $1$ or $-1$.
${\mathbf{ind}}=1$
fx and ${\mathbf{c}}\left[0\right]$ need not be set.
${\mathbf{ind}}=-1$
fx and ${\mathbf{c}}\left[0\right]$ 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 c05azc with all other arguments unchanged.
On final exit: contains $0$.
Constraint: on entry ${\mathbf{ind}}=-1$, $1$, $2$, $3$ or $4$.
Note: any values you return to c05azc as part of the reverse communication procedure should not include floating-point NaN (Not a Number) or infinity values, since these are not handled by c05azc. If your code inadvertently does return any NaNs or infinities, c05azc is likely to produce unexpected results.
8: $\mathbf{fail}$NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument $⟨\mathit{\text{value}}⟩$ had an illegal value.
NE_INT
On entry, ${\mathbf{ind}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{ind}}=-1$, $1$, $2$, $3$ or $4$.
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 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_NOT_SIGN_CHANGE
On entry, $f\left({\mathbf{x}}\right)$ and $f\left({\mathbf{y}}\right)$ have the same sign with neither equalling $0.0$: $f\left({\mathbf{x}}\right)=⟨\mathit{\text{value}}⟩$ and $f\left({\mathbf{y}}\right)=⟨\mathit{\text{value}}⟩$.
NE_PROBABLE_POLE
The final interval may contain a pole rather than a zero. Note that this error exit is not completely reliable: it may be taken in extreme cases when $\left[{\mathbf{x}},{\mathbf{y}}\right]$ contains a zero, or it may not be taken when $\left[{\mathbf{x}},{\mathbf{y}}\right]$ contains a pole. Both these cases occur most frequently when tolx is large.
NE_REAL
On entry, ${\mathbf{tolx}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{tolx}}>0.0$.
NW_TOO_MUCH_ACC_REQUESTED
The tolerance tolx has been set too small for the problem being solved. However, the values x and y returned may well be good approximations to the zero. ${\mathbf{tolx}}=⟨\mathit{\text{value}}⟩$.

7Accuracy

The accuracy of the final value x as an approximation of the zero is determined by tolx and ir (see Section 5). A relative accuracy criterion (${\mathbf{ir}}=2$) should not be used when the initial values x and y are of different orders of magnitude. In this case a change of origin of the independent variable may be appropriate. For example, if the initial interval $\left[{\mathbf{x}},{\mathbf{y}}\right]$ is transformed linearly to the interval $\left[1,2\right]$, then the zero can be determined to a precise number of figures using an absolute (${\mathbf{ir}}=1$) or relative (${\mathbf{ir}}=2$) error test and the effect of the transformation back to the original interval can also be determined. Except for the accuracy check, such a transformation has no effect on the calculation of the zero.

8Parallelism and Performance

c05azc is not threaded in any implementation.

For most problems, the time taken on each call to c05azc will be negligible compared with the time spent evaluating $f\left(x\right)$ between calls to c05azc.
If the calculation terminates because $f\left({\mathbf{x}}\right)=0.0$, then on return y is set to x. (In fact, ${\mathbf{y}}={\mathbf{x}}$ on return only in this case and, possibly, when ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NW_TOO_MUCH_ACC_REQUESTED.) There is no guarantee that the value returned in x corresponds to a simple root and you should check whether it does. One way to check this is to compute the derivative of $f$ at the point x, preferably analytically, or, if this is not possible, numerically, perhaps by using a central difference estimate. If ${f}^{\prime }\left({\mathbf{x}}\right)=0.0$, then x must correspond to a multiple zero of $f$ rather than a simple zero.

10Example

This example calculates a zero of ${e}^{-x}-x$ with an initial interval $\left[0,1\right]$, ${\mathbf{tolx}}=\text{1.0e−5}$ and a mixed error test.

10.1Program Text

Program Text (c05azce.c)

None.

10.3Program Results

Program Results (c05azce.r)