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.
The function may be called by the names: c05axc, nag_roots_contfn_cntin_rcomm or nag_zero_cont_func_cntin_rcomm.
3Description
c05axc uses a modified version of an algorithm given in Swift and Lindfield (1978) to compute a zero $\alpha $ of a continuous function $f\left(x\right)$. The algorithm used is based on a continuation method in which a sequence of problems
are solved, where $1={\theta}_{0}>{\theta}_{1}>\cdots >{\theta}_{m}=0$
(the value of $m$ is determined as the algorithm proceeds) and where ${x}_{0}$ is your initial estimate for the zero of $f\left(x\right)$. For each ${\theta}_{r}$ 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 (${\theta}_{m}=0$)
in the continuation method, and $\sqrt{{\mathbf{tol}}}$ is used to control the accuracy in the intermediate problems
(${\theta}_{1},{\theta}_{2},\dots ,{\theta}_{m-1}$).
4References
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
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 thanfx must remain unchanged.
1: $\mathbf{x}$ – 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: $\mathbf{fx}$ – doubleInput
On initial entry: if ${\mathbf{ind}}=1$, fx need not be set.
If ${\mathbf{ind}}=\mathrm{-1}$, 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: $\mathbf{tol}$ – doubleInput
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: $\mathbf{ir}$ – Nag_ErrorControlInput
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 ${x}_{r}^{0},{x}_{r}^{1},\dots \text{}$. This sequence will be considered to have converged only if:
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:
${\mathbf{ir}}=\mathrm{Nag\_Mixed}$, $\mathrm{Nag\_Absolute}$ or $\mathrm{Nag\_Relative}$.
5: $\mathbf{scal}$ – doubleInput
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
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 X02AJC.
Constraint:
${\mathbf{scal}}$ must be sufficiently large that ${\mathbf{x}}+{\mathbf{scal}}\ne {\mathbf{x}}$ on the computer.
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 ${\mathbf{ind}}=\mathrm{-1}$, $1$, $2$, $3$ 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: $\mathbf{fail}$ – 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.
NE_BAD_PARAM
On entry, argument $\u27e8\mathit{\text{value}}\u27e9$ had an illegal value.
NE_CONTIN_AWAY_NOT_POSS
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.
NE_CONTIN_PROB_NOT_SOLVED
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.
NE_FINAL_PROB_NOT_SOLVED
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 ${\mathbf{ir}}=\mathrm{Nag\_Relative}$.
NE_INT
On initial entry, ${\mathbf{ind}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{ind}}=\mathrm{-1}$ or $1$.
On intermediate entry, ${\mathbf{ind}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{ind}}=2$, $3$ 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_REAL
On entry, ${\mathbf{scal}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{x}}+{\mathbf{scal}}\ne {\mathbf{x}}$ (to machine accuracy).
On entry, ${\mathbf{tol}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{tol}}>0.0$.
NE_SIGNIF_DERIVS_NOT_COMPUT
Significant derivatives of $f$ cannot be computed. This can happen when $f$ is almost constant and nonzero, for any value of scal.
7Accuracy
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 (${\mathbf{ir}}=2$).
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 ${\mathbf{tol}}\ll \mathit{machineprecision}$ should not be input and may lead to an error exit with ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$NE_CONTIN_AWAY_NOT_POSS, NE_CONTIN_PROB_NOT_SOLVED or NE_FINAL_PROB_NOT_SOLVED. For the reasons discussed under ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$NE_CONTIN_PROB_NOT_SOLVED in Section 6,
tol should not be taken too large, say no larger than ${\mathbf{tol}}=\text{1.0e\u22123}$.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
c05axc is not threaded in any implementation.
9Further Comments
For most problems, the time taken on each call to c05axc will be negligible compared with the time spent evaluating $f\left(x\right)$ 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.
10Example
This example calculates a zero of $x-{e}^{-x}$ with initial approximation ${x}_{0}=1.0$, and ${\mathbf{tol}}=\text{1.0e\u22123}$ and $\text{1.0e\u22124}$.