nag_interp_1d_monotonic_intg (e01bh) evaluates the definite integral of a piecewise cubic Hermite interpolant, as computed by nag_interp_1d_monotonic (e01be), over the interval

[a, b]

If either

a

b

lies outside the interval from

x (1)

x (n)

computation of the integral involves extrapolation and a warning is returned.

The function is derived from function PCHIA in Fritsch (1982).

References

Fritsch F N (1982) PCHIP final specifications Report UCID-30194 Lawrence Livermore National Laboratory

Parameters

Compulsory Input Parameters

1: $x (n)$ – double array
2: $f (n)$ – double array
3: $d (n)$ – double array: n, x, f and d must be unchanged from the previous call of nag_interp_1d_monotonic (e01be).
4: $a$ – double scalar
5: $b$ – double scalar: The interval $[a, b]$ over which integration is to be performed.

Optional Input Parameters

1: $n$ – int64int32nag_int scalar: Default: the dimension of the arrays x, f, d. (An error is raised if these dimensions are not equal.)
n, x, f and d must be unchanged from the previous call of nag_interp_1d_monotonic (e01be).

Output Parameters

1: $pint$ – double scalar: The value of the definite integral of the interpolant over the interval $[a, b]$ .
2: $ifail$ – 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:

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

$ifail = 1$

On entry,

n < 2

$ifail = 2$: The values of $x (r)$ , for $r = 1, 2, \dots, n$ , are not in strictly increasing order.

W $ifail = 3$: On entry, at least one of a or b lies outside the interval [ $x (1), x (n)$ ], and extrapolation was performed to compute the integral. The value returned is therefore unreliable.

$ifail = - 99$: 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.

Accuracy

The computational error in the value returned for pint should be negligible in most practical situations.

Further Comments

The time taken by nag_interp_1d_monotonic_intg (e01bh) is approximately proportional to the number of data points included within the interval

[a, b]

Example

This example reads in values of n, x, f and d. It then reads in pairs of values for a and b, and evaluates the definite integral of the interpolant over the interval

[a, b]

until end-of-file is reached.

Open in the MATLAB editor: e01bh_example

function e01bh_example


fprintf('e01bh example results\n\n');

x = [7.99 8.09    8.19    8.7     9.2     10      12      15      20];
f = [0 2.7643e-05 0.04375 0.16918 0.46943 0.94374 0.99864 0.99992 0.99999];

% Theses are as returned by e01be(x,f)
d = [0;
     0.00055251;
     0.33587;
     0.34944;
     0.59696;
     0.060326;
     0.000898335;
     2.93954e-05;
     0];

m = 11;
dx = (x(end)-x(1))/(m-1);
px = [x(1):dx:x(end)];

ia = [1 6 7 8];
ib = [9 7 6 8];
fprintf('                               Integral\n');
fprintf('          a            b     over (a,b)\n');

for i = 1:4
  a = x(ia(i));
  b = x(ib(i));
  [pint, ifail] = e01bh( ...
                         x, f, d, a, b);
  fprintf('%13.4f%13.4f%13.4f\n', a, b, pint);
end

e01bh example results

                               Integral
          a            b     over (a,b)
       7.9900      20.0000      10.7648
      10.0000      12.0000       1.9622
      12.0000      10.0000      -1.9622
      15.0000      15.0000       0.0000

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

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