NAG CL Interface
c05axc (contfn_​cntin_​rcomm)

1 Purpose

c05axc attempts to locate a zero of a continuous function using a continuation method based on a secant iteration. It uses reverse communication for evaluating the function.

2 Specification

#include <nag.h>
void  c05axc (double *x, double fx, double tol, Nag_ErrorControl ir, double scal, double c[], Integer *ind, NagError *fail)
The function may be called by the names: c05axc, nag_roots_contfn_cntin_rcomm or nag_zero_cont_func_cntin_rcomm.

3 Description

c05axc uses a modified version of an algorithm given in Swift and Lindfield (1978) to compute a zero α of a continuous function fx . The algorithm used is based on a continuation method in which a sequence of problems
fx-θrfx0,  r=0,1,,m  
are solved, where 1 = θ0 > θ1 > > θm = 0 (the value of m is determined as the algorithm proceeds) and where x0 is your initial estimate for the zero of fx . For each θr the current problem is solved by a robust secant iteration using the solution from earlier problems to compute an initial estimate.
You must supply an error tolerance tol. tol is used directly to control the accuracy of solution of the final problem ( θm=0 ) in the continuation method, and tol is used to control the accuracy in the intermediate problems ( θ1 , θ2 , , θm-1 ).

4 References

Swift A and Lindfield G R (1978) Comparison of a continuation method for the numerical solution of a single nonlinear equation Comput. J. 21 359–362

5 Arguments

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: x double * 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 function.
On final exit: the final approximation to the zero.
2: fx double Input
On initial entry: if ind=1, fx need not be set.
If ind=-1, fx must contain fx for the initial value of x.
On intermediate re-entry: must contain fx for the current value of x.
3: tol double 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 ( θm=0 in Section 3), and tol is used for the problem defined by θr , 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: tol>0.0 .
4: ir Nag_ErrorControl Input
On initial entry: indicates the type of error test required, as follows. Solving the problem defined by θr , 1rm , involves computing a sequence of secant iterates xr0,xr1, . This sequence will be considered to have converged only if:
for ir=Nag_Mixed,
xr i+1 -xri eps×max1.0,xri ,  
for ir=Nag_Absolute,
xr i+1 -xri eps,  
for ir=Nag_Relative,
xr i+1 -xri eps×xri ,  
for some i>1 ; here eps is either tol or 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: ir=Nag_Mixed, Nag_Absolute or Nag_Relative.
5: scal double Input
On initial entry: a factor for use in determining a significant approximation to the derivative of fx at x=x0 , the initial value. A number of difference approximations to fx0 are calculated using
where h<scal 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: ε, where ε is the machine precision returned by X02AJC.
Constraint: scal must be sufficiently large that x+scalx on the computer.
6: c[26] double Communication Array
( c[4] contains the current θr , this value may be useful in the event of an error exit.)
7: ind Integer * Input/Output
On initial entry: must be set to 1 or -1 .
fx need not be set.
fx must contain fx .
On intermediate exit: contains 2, 3 or 4. The calling program must evaluate f at x, storing the result in fx, and re-enter c05axc with all other arguments unchanged.
On final exit: contains 0.
Constraint: on entry ind=-1, 1, 2, 3 or 4.
Note: any values you return to c05axc 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 c05axc. If your code inadvertently does return any NaNs or infinities, c05axc is likely to produce unexpected results.
8: fail NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument value had an illegal value.
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.
Current problem in the continuation sequence cannot be solved. Perhaps the original problem had no solution or the continuation path passes through a set of insoluble problems: consider refining the initial approximation to the zero. Alternatively, tol is too small, and the accuracy requirement is too stringent, or too large and the initial approximation too poor.
Final problem (with θm=0) cannot be solved. It is likely that too much accuracy has been requested, or that the zero is at α=0 and ir=Nag_Relative.
On initial entry, ind=value.
Constraint: ind=-1 or 1.
On intermediate entry, ind=value.
Constraint: ind=2, 3 or 4.
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.
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.
On entry, scal=value.
Constraint: x+scalx (to machine accuracy).
On entry, tol=value.
Constraint: tol>0.0.
Significant derivatives of f cannot be computed. This can happen when f is almost constant and nonzero, for any value of scal.

7 Accuracy

The accuracy of the approximation to the zero depends on tol and ir. In general decreasing tol will give more accurate results. Care must be exercised when using the relative error criterion ( ir=2 ).
If the zero is at x=0 , or if the initial value of x and the zero bracket the point x=0 , it is likely that an error exit with fail.code= NE_CONTIN_AWAY_NOT_POSS, NE_CONTIN_PROB_NOT_SOLVED or NE_FINAL_PROB_NOT_SOLVED will occur.
It is possible to request too much or too little accuracy. Since it is not possible to achieve more than machine accuracy, a value of tolmachine precision should not be input and may lead to an error exit with fail.code= NE_CONTIN_AWAY_NOT_POSS, NE_CONTIN_PROB_NOT_SOLVED or NE_FINAL_PROB_NOT_SOLVED. For the reasons discussed under fail.code= NE_CONTIN_PROB_NOT_SOLVED in Section 6, tol should not be taken too large, say no larger than tol=1.0e−3 .

8 Parallelism and Performance

c05axc is not threaded in any implementation.

9 Further Comments

For most problems, the time taken on each call to c05axc will be negligible compared with the time spent evaluating fx between calls to c05axc. However, the initial value of x and the choice of tol will clearly affect the timing. The closer that x is to the root, the less evaluations of f required. The effect of the choice of tol will not be large, in general, unless tol is very small, in which case the timing will increase.

10 Example

This example calculates a zero of x - e-x with initial approximation x0=1.0 , and tol=1.0e−3 and 1.0e−4 .

10.1 Program Text

Program Text (c05axce.c)

10.2 Program Data


10.3 Program Results

Program Results (c05axce.r)