c05axc (contfn_cntin_rcomm) : NAG Library CL Interface, Mark 28

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

f (x)

. The algorithm used is based on a continuation method in which a sequence of problems

f (x) - θ_{r} f (x_{0}), r = 0, 1, \dots, m

are solved, where

1 = θ_{0} > θ_{1} > \dots > θ_{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 (x)

. For each

θ_{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 (

θ_{m} = 0

) in the continuation method, and

\sqrt{tol}

is used to control the accuracy in the intermediate problems (

θ_{1}, θ_{2}, \dots, θ_{m - 1}

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.

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.

fx

– double Input

On initial entry: if

ind = 1

, fx need not be set.

ind = −1

, fx must contain

f (x)

for the initial value of x.

On intermediate re-entry: must contain

f (x)

for the current value of x.

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 (

θ_{m} = 0

in Section 3), and

\sqrt{tol}

is used for the problem defined by

θ_{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:

tol > 0.0

ir

– Nag_ErrorControl Input

On initial entry: indicates the type of error test required, as follows. Solving the problem defined by

θ_{r}

1 \leq r \leq 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 = Nag_Mixed

| x_{r}^{(i + 1)} - x_{r}^{(i)} | \leq eps \times \max (1.0, | x_{r}^{(i)} |),

for

ir = Nag_Absolute

| x_{r}^{(i + 1)} - x_{r}^{(i)} | \leq eps,

for

ir = Nag_Relative

| x_{r}^{(i + 1)} - x_{r}^{(i)} | \leq eps \times | x_{r}^{(i)} |,

for some

i > 1

; here

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.

Constraint:

ir = Nag_Mixed

Nag_Absolute

Nag_Relative

scal

– double Input

On initial entry: a factor for use in determining a significant approximation to the derivative of

f (x)

x = x_{0}

, the initial value. A number of difference approximations to

f^{'} (x_{0})

are calculated using

f^{'} (x_{0}) \sim (f (x_{0} + h) - f (x_{0})) / h

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:

\sqrt{ε}

, where

ε

is the machine precision returned by X02AJC.

Constraint:

scal

must be sufficiently large that

x + scal \neq x

on the computer.

c [26]

– double Communication Array

(

c [4]

contains the current

θ_{r}

, this value may be useful in the event of an error exit.)

ind

– Integer * Input/Output

On initial entry: must be set to

1

−1

$ind = 1$: fx need not be set.
$ind = −1$: fx must contain $f (x)$ .

On intermediate exit: contains

2

3

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

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.

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

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

⟨ 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

θ_{m} = 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

NE_INT

On initial entry,

ind = ⟨ value ⟩

.
Constraint:

ind = −1

1

On intermediate entry,

ind = ⟨ value ⟩

.
Constraint:

ind = 2

3

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,

scal = ⟨ value ⟩

.
Constraint:

x + scal \neq x

(to machine accuracy).

On entry,

tol = ⟨ value ⟩

.
Constraint:

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

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

tol ≪ machine 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

9 Further Comments

For most problems, the time taken on each call to c05axc will be negligible compared with the time spent evaluating

f (x)

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.

NAG CL Interface
c05axc (contfn_cntin_rcomm)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG CL Interfacec05axc (contfn_​cntin_​rcomm)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG CL Interface
c05axc (contfn_cntin_rcomm)