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NAG Toolbox: nag_interp_1d_monotonic_deriv (e01bg)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_interp_1d_monotonic_deriv (e01bg) evaluates a piecewise cubic Hermite interpolant and its first derivative at a set of points.

Syntax

[pf, pd, ifail] = e01bg(x, f, d, px, 'n', n, 'm', m)
[pf, pd, ifail] = nag_interp_1d_monotonic_deriv(x, f, d, px, 'n', n, 'm', m)

Description

nag_interp_1d_monotonic_deriv (e01bg) evaluates a piecewise cubic Hermite interpolant, as computed by nag_interp_1d_monotonic (e01be), at the points pxi, for i=1,2,,m. The first derivatives at the points are also computed. If any point lies outside the interval from x1 to xn, values of the interpolant and its derivative are extrapolated from the nearest extreme cubic, and a warning is returned.
If values of the interpolant only, and not of its derivative, are required, nag_interp_1d_monotonic_eval (e01bf) should be used.
The function is derived from function PCHFD in Fritsch (1982).

References

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

Parameters

Compulsory Input Parameters

1:     xn – double array
2:     fn – double array
3:     dn – double array
n, x, f and d must be unchanged from the previous call of nag_interp_1d_monotonic (e01be).
4:     pxm – double array
The m values of x at which the interpolant is to be evaluated.

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).
2:     m int64int32nag_int scalar
Default: the dimension of the array px.
m, the number of points at which the interpolant is to be evaluated.
Constraint: m1.

Output Parameters

1:     pfm – double array
pfi contains the value of the interpolant evaluated at the point pxi, for i=1,2,,m.
2:     pdm – double array
pdi contains the first derivative of the interpolant evaluated at the point pxi, for i=1,2,,m.
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:

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 xr, for r=1,2,,n, are not in strictly increasing order.
   ifail=3
On entry,m<1.
W  ifail=4
At least one of the points pxi, for i=1,2,,m, lies outside the interval [x1,xn], and extrapolation was performed at all such points. Values computed at these points may be very 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 errors in the arrays pf and pd should be negligible in most practical situations.

Further Comments

The time taken by nag_interp_1d_monotonic_deriv (e01bg) is approximately proportional to the number of evaluation points, m. The evaluation will be most efficient if the elements of px are in nondecreasing order (or, more generally, if they are grouped in increasing order of the intervals xr-1,xr). A single call of nag_interp_1d_monotonic_deriv (e01bg) with m>1 is more efficient than several calls with m=1.

Example

This example reads in values of n, x, f and d, and calls nag_interp_1d_monotonic_deriv (e01bg) to compute the values of the interpolant and its derivative at equally spaced points.
function e01bg_example


fprintf('e01bg 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)];

[pf, pd, ifail] = e01bg(x, f, d, px);

fprintf('\n                Interpolated   Interpolated\n');
fprintf('     Abscissa          Value     Derivative\n');
fprintf('%13.4f%15.4f%15.3e\n', [px' pf pd]');

% Recalculate on finer mesh for plotting
m = 61;
dx = (x(end)-x(1))/(m-1);
px = [x(1):dx:x(end)];

[pf, pd, ifail] = e01bg(x, f, d, px);

fig1 = figure;
plot(px,pf,px,pd,x,f,'*');
legend('Function','Derivative','Data points','Location','East');
title('Monotonic Hermite interpolant');
xlabel('x');
axis([8 20 -0.1 1.2]);


e01bg example results


                Interpolated   Interpolated
     Abscissa          Value     Derivative
       7.9900         0.0000      0.000e+00
       9.1910         0.4640      6.060e-01
      10.3920         0.9645      4.569e-02
      11.5930         0.9965      9.917e-03
      12.7940         0.9992      6.249e-04
      13.9950         0.9998      2.708e-04
      15.1960         0.9999      2.809e-05
      16.3970         1.0000      2.034e-05
      17.5980         1.0000      1.308e-05
      18.7990         1.0000      6.297e-06
      20.0000         1.0000     -3.388e-21
e01bg_fig1.png

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