Internally d04aac calculates the odd order derivatives and the even order derivatives separately. There is an option you can use for restricting the calculation to only odd (or even) order derivatives. For each derivative the function employs an extension of the Neville Algorithm (see Lyness and Moler (1969)) to obtain a selection of approximations.

For example, for odd derivatives, based on

20

function values, d04aac calculates a set of numbers:

T_{k, p, s}, p = s, s + 1, \dots, 6, k = 0, 1, \dots, 9 - p

each of which is an approximation to

f^{(2 s + 1)} (x_{0}) / (2 s + 1)!

. A specific approximation

T_{k, p, s}

is of polynomial degree

2 p + 2

and is based on polynomial interpolation using function values

f (x_{0} \pm (2 i - 1) h)

, for

k = k, \dots, k + p

. In the absence of round-off error, the better approximations would be associated with the larger values of

p

and of

k

. However, round-off error in function values has an increasingly contaminating effect for successively larger values of

p

. This function proceeds to make a judicious choice between all the approximations in the following way.

For a specified value of

s

, let:

R_{p} = U_{p} - L_{p}, p = s, s + 1, \dots, 6

where

U_{p} = \max_{k} (T_{k, p, s})

and

L_{p} = \min_{k} (T_{k, p, s})

, for

k = 0, 1, \dots, 9 - p

, and let

\bar{p}

be such that

R_{\bar{p}} = \min_{p} (R_{p})

, for

p = s, \dots, 6

The function returns:

der [2 s] = \frac{1}{8 - \bar{p}} \times {\sum_{k = 0}^{9 - \bar{p}} T_{k, \bar{p}, s} - U_{\bar{p}} - L_{\bar{p}}} (2 s + 1)!

and

erest [2 s] = R_{\bar{p}} \times (2 s + 1)! \times K_{2 s + 1}

where

K_{j}

is a safety factor which has been assigned the values:

$K_{j} = 1$ ,	$j \leq 9$
$K_{j} = 1.5$ ,	$j = 10, 11$
$K_{j} = 2$	$j \geq 12$ ,

on the basis of performance statistics.

The even order derivatives are calculated in a precisely analogous manner.

4 References

Lyness J N and Moler C B (1966) van der Monde systems and numerical differentiation Numer. Math. 8 458–464

Lyness J N and Moler C B (1969) Generalised Romberg methods for integrals of derivatives Numer. Math. 14 1–14

5 Arguments

1: $xval$ – double Input

On entry: the point at which the derivatives are required,

x_{0}

2: $nder$ – Integer Input

On entry: must be set so that its absolute value is the highest order derivative required.

$nder > 0$: All derivatives up to order $\min (nder, 14)$ are calculated.
$nder < 0$ and nder is even: Only even order derivatives up to order $\min (- nder, 14)$ are calculated.
$nder < 0$ and nder is odd: Only odd order derivatives up to order $\min (- nder, 13)$ are calculated.

3: $hbase$ – double Input

On entry: the initial step length which may be positive or negative. For advice on the choice of hbase see Section 9.

Constraint:

hbase \neq 0.0

4: $der [14]$ – double Output

On exit:

der [j - 1]

contains an approximation to the

j

th derivative of

f (x)

x = xval

, so long as the

j

th derivative is one of those requested by you when specifying nder. For other values of

j

der [j - 1]

is unused.

5: $erest [14]$ – double Output

On exit: an estimate of the absolute error in the corresponding result

der [j - 1]

so long as the

j

th derivative is one of those requested by you when specifying nder. The sign of

erest [j - 1]

is positive unless the result

der [j - 1]

is questionable. It is set negative when

| der [j - 1] | < | erest [j - 1] |

or when for some other reason there is doubt about the validity of the result

der [j - 1]

(see Section 6). For other values of

j

erest [j - 1]

is unused.

6: $f$ – function, supplied by the user External Function

f must evaluate the function

f (x)

at a specified point.

The specification of f is:

double

f (double x, Nag_Comm *comm)

1: $x$ – double Input

On entry: the value of the argument

x

If you have equally spaced tabular data, the following information may be useful:

(i)in any call of d04aac the only values of $x$ for which $f (x)$ will be required are $x = xval$ and $x = xval \pm (2 j - 1) hbase$ , for $j = 1, 2, \dots, 10$ ; and
(ii) $f (x_{0})$ is always computed, but it is disregarded when only odd order derivatives are required.

2: $comm$ – Nag_Comm *

Pointer to structure of type Nag_Comm; the following members are relevant to f.

user – double *
iuser – Integer *
p – Pointer: The type Pointer will be void *. Before calling d04aac you may allocate memory and initialize these pointers with various quantities for use by f when called from d04aac (see Section 3.1.1 in the Introduction to the NAG Library CL Interface).

Note: f should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d04aac. If your code inadvertently does return any NaNs or infinities, d04aac is likely to produce unexpected results.

7: $comm$ – Nag_Comm *

The NAG communication argument (see Section 3.1.1 in the Introduction to the NAG Library CL Interface).

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

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_INT: On entry, $nder = 0$ .
Constraint: $nder \neq 0$ .
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, $hbase = 0.0$ .
Constraint: $hbase \neq 0.0$ .

7 Accuracy

The accuracy of the results is problem dependent. An estimate of the accuracy of each result

der [j - 1]

is returned in

erest [j - 1]

(see Sections 3, 5 and 9).

A basic feature of any floating-point function for numerical differentiation based on real function values on the real axis is that successively higher order derivative approximations are successively less accurate. It is expected that in most cases

der [13]

will be unusable. As an aid to this process, the sign of

erest [j - 1]

is set negative when the estimated absolute error is greater than the approximate derivative itself, i.e., when the approximate derivative may be so inaccurate that it may even have the wrong sign. It is also set negative in some other cases when information available to the function indicates that the corresponding value of

der [j - 1]

is questionable.

The actual values in erest depend on the accuracy of the function values, the properties of the machine arithmetic, the analytic properties of the function being differentiated and the user-supplied step length hbase (see Section 9). The only hard and fast rule is that for a given

f (xval)

and hbase, the values of

erest [j - 1]

increase with increasing

j

. The limit of

14

is dictated by experience. Only very rarely can one obtain meaningful approximations for higher order derivatives on conventional machines.

8 Parallelism and Performance

d04aac is not threaded in any implementation.

9 Further Comments

The time taken by d04aac depends on the time spent for function evaluations. Otherwise the time is roughly equivalent to that required to evaluate the function

21

times and calculate a finite difference table having about

200

entries in total.

The results depend very critically on the choice of the user-supplied step length hbase. The overall accuracy is diminished as hbase becomes small (because of the effect of round-off error) and as hbase becomes large (because the discretization error also becomes large). If the function is used four or five times with different values of hbase one can find a reasonably good value. A process in which the value of hbase is successively halved (or doubled) is usually quite effective. Experience has shown that in cases in which the Taylor series for

f (x)

about xval has a finite radius of convergence

R

, the choices of

hbase > R / 19

are not likely to lead to good results. In this case some function values lie outside the circle of convergence.

10 Example

This example evaluates the odd-order derivatives of the function:

f (x) = \frac{1}{2} e^{2 x - 1}

up to order

7

at the point

x = \frac{1}{2}

. Several different values of hbase are used, to illustrate that:

(i)extreme choices of hbase, either too large or too small, yield poor results;
(ii)the quality of these results is adequately indicated by the values of erest.