NAG Library Function Document

nag_zero_cont_func_cntin_rcomm (c05axc)

1  Purpose

2  Specification

3  Description

4  References

5  Arguments

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 – doubleInput

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.

3:     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:     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 $x_r^0,x_r^1,\dots$. This sequence will be considered to have converged only if:

for $\mathbf{ir}=\mathrm{Nag\_Mixed}$,

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

for $\mathbf{ir}=\mathrm{Nag\_Absolute}$,

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

for $\mathbf{ir}=\mathrm{Nag\_Relative}$,

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

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:     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

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

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 nag_machine_precision (X02AJC).

Constraint: $\mathbf{scal}$ must be sufficiently large that $\mathbf{x}+\mathbf{scal}\ne \mathbf{x}$ on the computer.

6:     c[$26$] – doubleCommunication Array

($\mathbf{c}\left[4\right]$ contains the current ${\theta }_r$, 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 $-1$.

$\mathbf{ind}=1$

fx need not be set.

$\mathbf{ind}=-1$

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

On intermediate exit: contains $2$, $3$ or $4$. The calling program must evaluate $f$ at x, storing the result in fx, and re-enter nag_zero_cont_func_cntin_rcomm (c05axc) with all other arguments unchanged.

On final exit: contains $0$.

Constraint: on entry $\mathbf{ind}=-1$, $1$, $2$, $3$ or $4$.

8:     fail – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

NE_BAD_PARAM

On entry, argument $⟨\mathit{\text{value}}⟩$ 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, $\mathbf{ind}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{ind}=-1$ or $1$.

On intermediate entry, $\mathbf{ind}=⟨\mathit{\text{value}}⟩$.
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.

NE_REAL

On entry, $\mathbf{scal}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{x}+\mathbf{scal}\ne \mathbf{x}$ (to machine accuracy).

On entry, $\mathbf{tol}=⟨\mathit{\text{value}}⟩$.
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.

7  Accuracy

8  Parallelism and Performance

9  Further Comments

10  Example

10.1  Program Text

Program Text (c05axce.c)

10.2  Program Data

10.3  Program Results

Program Results (c05axce.r)