d04ba:: Numerical Differentiation (NAG Toolbox)

nag_numdiff_rcomm (d04ba) calculates a set of derivatives (up to order

14

) of a function at a point with respect to a single variable. A corresponding set of error estimates is also returned. Derivatives are calculated using an extension of the Neville algorithm. This function differs from nag_numdiff (d04aa), in that the abscissae and corresponding function values must be calculated before this function is called. The abscissae may be generated using nag_numdiff_sample (d04bb).

Syntax

Description

nag_numdiff_rcomm (d04ba) provides a set of approximations:

der (j), j = 1, 2, \dots, 14

to the derivatives:

f^{(j)} (x_{0}), j = 1, 2, \dots, 14

of a real valued function

f (x)

at a real abscissa

x_{0}

, together with a set of error estimates:

erest (j), j = 1, 2, \dots, 14

which hopefully satisfy:

|der (j) - f^{(j)} (x_{0})| < erest (j), j = 1, 2, \dots, 14 .

The abscissae

x

and the corresponding function values

f (x)

should be passed into nag_numdiff_rcomm (d04ba) as the vectors xval and fval respectively. The step size

h

is derived from the abscissae in xval. See Further Comments for a discussion of how the derived value of

h

may affect the results of nag_numdiff_rcomm (d04ba). The order in which the abscissae and function values are stored in xval and fval is irrelevant, provided that the function value at any given index corresponds to the value of the abscissa at the same index. Abscissae may be automatically generated using nag_numdiff_sample (d04bb) if desired. For each derivative nag_numdiff_rcomm (d04ba) employs an extension of the Neville Algorithm (see Lyness and Moler (1969)) to obtain a selection of approximations.

For example, for odd derivatives, this function 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.

This 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.

References

Parameters

Compulsory Input Parameters

Optional Input Parameters

Output Parameters

Error Indicators and Warnings

Accuracy

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 nag_numdiff_rcomm (d04ba) 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 step length

h

(see Further Comments). The only hard and fast rule is that for a given objective function and

h

, 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 results depend very critically on the choice of the step length

h

. The overall accuracy is diminished as

h

becomes small (because of the effect of round-off error) and as

h

becomes large (because the discretization error also becomes large). If this function is used four or five times with different values of

h

one can find a reasonably good value. A process in which the value of

h

is successively halved (or doubled) is usually quite effective. Experience has shown that in cases in which the Taylor series for the objective function about

x_{0}

has a finite radius of convergence

R

, the choices of

h > R / 19

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

As mentioned, the order of the abscissae in xval does not matter, provided the corresponding values of fval are ordered identically. If the abscissae are generated by nag_numdiff_sample (d04bb), then they will be in ascending order. In particular, the target abscissa

x_{0}

will be stored in

xval (11)

Example

function d04ba_example


fprintf('d04ba example results\n\n');

fprintf('Find the derivatives of the polygamma (psi) function\n');
fprintf('using function values generated by s14ae.\n\n');
fprintf('Demonstrate the effect of successively reducing hbase.\n\n');

% Set the target location and calculate the objective value
x_0 = 0.05;
[f_0, ifail] = s14ae(x_0, int64(0));

% Compute the actual derivatives using s14ae for comparison
actder = zeros(3, 1);
for j=1:3
  [actder(j), ifail] = s14ae(x_0, int64(j));
end

der_comp = zeros(14, 3, 4);
% Attempt 4 applications, reducing hbase by factor 0.1 each time
for j=1:4
  % Generate the abscissa xval using d04bb
  hbase = 0.025*10^(-j);
  [xval] = d04bb(x_0, hbase);

  % Calculate the corresponding objective function values
  fval(11) = f_0;
  for k=[1:10,12:21]
    [fval(k), ifail] = s14ae(xval(k), int64(0));
  end

  % Call d04ba to calculate the derivative estimates
  [der, erest, ifail] = d04ba(xval, fval);

  % Store results in der_comp
  der_comp(:, 1, j) = hbase;
  der_comp(:, 2, j) = der;
  der_comp(:, 3, j) = erest;

end

% Display results for first 3 derivatives
for i=1:3
  fprintf('\nderivative %d calculated using s14ae: %11.4e\n', i, actder(i));
  fprintf('Derivative and error estimates for derivative %d\n', i);
  fprintf('      hbase        der(%d)      erest(%d)\n', i, i);
  for j=1:4
    fprintf('  %12.4e %12.4e %12.4e\n', der_comp(i,:,j));
  end
end

d04ba example results

Find the derivatives of the polygamma (psi) function
using function values generated by s14ae.

Demonstrate the effect of successively reducing hbase.


derivative 1 calculated using s14ae:  4.0153e+02
Derivative and error estimates for derivative 1
      hbase        der(1)      erest(1)
    2.5000e-03   4.0204e+02   1.3940e+02
    2.5000e-04   4.0153e+02   4.9170e-11
    2.5000e-05   4.0153e+02   2.1799e-10
    2.5000e-06   4.0153e+02   1.1826e-09

derivative 2 calculated using s14ae: -1.6002e+04
Derivative and error estimates for derivative 2
      hbase        der(2)      erest(2)
    2.5000e-03  -1.6022e+04   5.5760e+03
    2.5000e-04  -1.6002e+04   1.2831e-07
    2.5000e-05  -1.6002e+04   6.0543e-06
    2.5000e-06  -1.6002e+04   9.5762e-04

derivative 3 calculated using s14ae:  9.6001e+05
Derivative and error estimates for derivative 3
      hbase        der(3)      erest(3)
    2.5000e-03   9.1465e+05  -7.3750e+06
    2.5000e-04   9.6001e+05   2.3718e-04
    2.5000e-05   9.6001e+05   4.2253e-02
    2.5000e-06   9.6001e+05   5.9679e+01

NAG Toolbox: nag_numdiff_rcomm (d04ba)

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Purpose