naginterfaces.library.roots.contfn_​cntin_​rcomm

naginterfaces.library.roots.contfn_cntin_rcomm(x, fx, tol, ir, comm, ind, scal=1.0536712127723509e-08)

contfn_cntin_rcomm 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.

For full information please refer to the NAG Library document for c05ax

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/c05/c05axf.html

Parameters

xfloat

On initial entry: an initial approximation to the zero.

fxfloat

On initial entry: if $ind=1$, $fx$ need not be set.

If $ind=-1$, $fx$ must contain $f\left(x\right)$ for the initial value of $x$.

On intermediate entry: must contain $f\left(x\right)$ for the current value of $x$.

tolfloat

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 Notes), and $\sqrt{tol}$ 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.

irint

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 $ir=0$,

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

for $ir=1$,

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

for $ir=2$,

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

for some $i>1$; here $\text{eps}$ is either $tol$ or $\sqrt{tol}$ 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.

commdict, communication object, modified in place

Communication structure.

On initial entry: need not be set.

indint

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

$ind=1$

$fx$ need not be set.

$ind=-1$

$fx$ must contain $f\left(x\right)$.

scalfloat, optional

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

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

where $\left|h\right|<\left|scal\right|$ 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.

Returns

xfloat

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.

indint

On intermediate exit: contains $2$, $3$ or $4$. The calling program must evaluate $f$ at $x$, storing the result in $fx$, and re-enter contfn_cntin_rcomm with all other arguments unchanged.

On final exit: contains $0$.

Raises

NagValueError

(errno $1$)

On entry, $tol=⟨value⟩$.

Constraint: $tol>0.0$.

(errno $1$)

On entry, $ir=⟨value⟩$.

Constraint: $ir=0$, $1$ or $2$.

(errno $2$)

On initial entry, $ind=⟨value⟩$.

Constraint: $ind=-1$ or $1$.

(errno $2$)

On intermediate entry, $ind=⟨value⟩$.

Constraint: $ind=2$, $3$ or $4$.

(errno $3$)

On entry, $scal=⟨value⟩$.

Constraint: $x+scal\ne x$ (to machine accuracy).

(errno $3$)

Significant derivatives of $f$ cannot be computed. This can happen when $f$ is almost constant and nonzero, for any value of $scal$.

(errno $4$)

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.

(errno $5$)

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.

(errno $6$)

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 $ir=2$.

Notes

contfn_cntin_rcomm 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

$$f\left(x\right)-{\theta }_{r}f\left(x_0\right)\text{,}r=0,1,\dots ,m$$

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 $\sqrt{tol}$ is used to control the accuracy in the intermediate problems (${\theta }_1,{\theta }_2,\dots ,{\theta }_{m-1}$).

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