Internally nag_numdiff (d04aa) 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, nag_numdiff (d04aa) 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 + 1) = \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 + 1) = 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.

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

Parameters

Compulsory Input Parameters

1: $xval$ – double scalar

The point at which the derivatives are required,

x_{0}

2: $nder$ – int64int32nag_int scalar

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 scalar

The initial step length which may be positive or negative. For advice on the choice of hbase see Further Comments.

Constraint:

hbase \neq 0.0

4: $fun$ – function handle or string containing name of m-file

fun must evaluate the function

f (x)

at a specified point.

[result] = fun(x)

Input Parameters

1: $x$ – double scalar

The value of the argument

x

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

(i)	in any call of nag_numdiff (d04aa) 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.

Output Parameters

1: $result$ – double scalar: The value of $f (x)$ at the specified point.

Optional Input Parameters

None.

Output Parameters

1: $der (14)$ – double array: $der (j)$ contains an approximation to the $j$ th derivative of $f (x)$ at $x = xval$ , so long as the $j$ th derivative is one of those requested by you when specifying nder. For other values of $j$ , $der (j)$ is unused.
2: $erest (14)$ – double array: An estimate of the absolute error in the corresponding result $der (j)$ so long as the $j$ th derivative is one of those requested by you when specifying nder. The sign of $erest (j)$ is positive unless the result $der (j)$ is questionable. It is set negative when $|der (j)| < |erest (j)|$ or when for some other reason there is doubt about the validity of the result $der (j)$ (see Error Indicators and Warnings). For other values of $j$ , $erest (j)$ is unused.
3: $ifail$ – int64int32nag_int scalar: $ifail = 0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:

$ifail = 1$

On entry,	$nder = 0$ ,
or	$hbase = 0.0$ .

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: Dynamic memory allocation failed.

If ifail has a value zero on exit then nag_numdiff (d04aa) has terminated successfully, but before any use is made of a derivative

der (j)

the value of

erest (j)

must be checked.

Accuracy

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

der (j)

is returned in

erest (j)

(see Description, Arguments and Further Comments).

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 (14)

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

erest (j)

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)

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 Further Comments). The only hard and fast rule is that for a given

fun (xval)

and hbase, the values of

erest (j)

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.

Further Comments

The time taken by nag_numdiff (d04aa) 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

fun (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.

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.

Open in the MATLAB editor: d04aa_example

function d04aa_example


fprintf('d04aa example results\n\n');

fprintf(['Four separate runs to calculate the first four odd order ',...
	 'derivatives of\n    fun(x) = exp(2*x-1)/2 at x = 0.5.\n']);
fprintf('The exact results are 1, 4, 16 and 64\n\n');

xval = 0.5;
nder = int64(-7);
hbase = 0.5;
fun = @(x) exp(2*x-1)/2;

for k = 1:4
  hbase = 5*10^(-k);
  [der, erest, ifail] = d04aa( ...
			       xval, nder, hbase, fun);

  fprintf('\nwith step length %9.4f the results are:\n',hbase);
  fprintf('Order        Derivative       Error estimate\n');

  for i = 1:2:abs(nder)
    fprintf('%d%21.3e%21.4e\n',i, der(i), erest(i));
  end
end

d04aa example results

Four separate runs to calculate the first four odd order derivatives of
    fun(x) = exp(2*x-1)/2 at x = 0.5.
The exact results are 1, 4, 16 and 64


with step length    0.5000 the results are:
Order        Derivative       Error estimate
1            1.392e+03          -1.0734e+05
3           -3.139e+03          -1.4378e+05
5            8.762e+03          -2.4790e+05
7           -2.475e+04          -4.4838e+05

with step length    0.0500 the results are:
Order        Derivative       Error estimate
1            1.000e+00           1.5294e-11
3            4.000e+00           2.1127e-09
5            1.600e+01           3.8162e-07
7            6.400e+01           7.3906e-05

with step length    0.0050 the results are:
Order        Derivative       Error estimate
1            1.000e+00           3.5527e-14
3            4.000e+00           4.9597e-10
5            1.600e+01           1.4335e-05
7            6.404e+01           2.8513e-01

with step length    0.0005 the results are:
Order        Derivative       Error estimate
1            1.000e+00           1.4289e-13
3            4.000e+00           3.0894e-07
5            1.599e+01           6.3314e-01
7            3.826e+04          -1.9644e+06

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox