nag_interp_2d_triang_bary_eval (e01eb) performs barycentric interpolation, at a given set of points, using a set of function values on a scattered grid and a triangulation of that grid computed by nag_interp_2d_triangulate (e01ea).

Syntax

[pf, ifail] = e01eb(x, y, f, triang, px, py, 'm', m, 'n', n)

[pf, ifail] = nag_interp_2d_triang_bary_eval(x, y, f, triang, px, py, 'm', m, 'n', n)

Description

nag_interp_2d_triang_bary_eval (e01eb) takes as input a set of scattered data points

(x_{r}, y_{r}, f_{r})

, for

r = 1, 2, \dots, n

, and a Thiessen triangulation of the

(x_{r}, y_{r})

computed by nag_interp_2d_triangulate (e01ea), and interpolates at a set of points

({px}_{i}, {py}_{i})

, for

i = 1, 2, \dots, m

If the

i

th interpolation point

({px}_{i}, {py}_{i})

is equal to

(x_{r}, y_{r})

for some value of

r

, the returned value will be equal to

f_{r}

; otherwise a barycentric transformation will be used to calculate the interpolant.

For each point

({px}_{i}, {py}_{i})

, a triangle is sought which contains the point; the vertices of the triangle and

f_{r}

values at the vertices are then used to compute the value

F ({px}_{i}, {py}_{i})

If any interpolation point lies outside the triangulation defined by the input arguments, the returned value is the value provided,

f_{s}

, at the closest node

(x_{s}, y_{s})

nag_interp_2d_triang_bary_eval (e01eb) must only be called after a call to nag_interp_2d_triangulate (e01ea).

References

Cline A K and Renka R L (1984) A storage-efficient method for construction of a Thiessen triangulation Rocky Mountain J. Math. 14 119–139

Lawson C L (1977) Software for

C^{1}

surface interpolation Mathematical Software III (ed J R Rice) 161–194 Academic Press

Renka R L (1984) Algorithm 624: triangulation and interpolation of arbitrarily distributed points in the plane ACM Trans. Math. Software 10 440–442

Renka R L and Cline A K (1984) A triangle-based

C^{1}

interpolation method Rocky Mountain J. Math. 14 223–237

Parameters

Compulsory Input Parameters

1: $x (n)$ – double array
2: $y (n)$ – double array: The coordinates of the $r$ th data point, $(x_{r}, y_{r})$ , for $r = 1, 2, \dots, n$ . x and y must be unchanged from the previous call of nag_interp_2d_triangulate (e01ea).
3: $f (n)$ – double array: The function values $f_{r}$ at $(x_{r}, y_{r})$ , for $r = 1, 2, \dots, n$ .
4: $triang (7 \times n)$ – int64int32nag_int array: The triangulation computed by the previous call of nag_interp_2d_triangulate (e01ea). See Further Comments in nag_interp_2d_triangulate (e01ea) for details of how the triangulation used is encoded in triang.
5: $px (m)$ – double array
6: $py (m)$ – double array: The coordinates $({px}_{i}, {py}_{i})$ , for $i = 1, 2, \dots, m$ , at which interpolated function values are sought.

Optional Input Parameters

1: $m$ – int64int32nag_int scalar: Default: the dimension of the arrays px, py. (An error is raised if these dimensions are not equal.)
$m$ , the number of points to interpolate.

Constraint: $m \geq 1$ .
2: $n$ – int64int32nag_int scalar: Default: the dimension of the arrays x, y, f. (An error is raised if these dimensions are not equal.)
$n$ , the number of data points. n must be unchanged from the previous call of nag_interp_2d_triangulate (e01ea).

Constraint: $n \geq 3$ .

Output Parameters

1: $pf (m)$ – double array: The interpolated values $F ({px}_{i}, {py}_{i})$ , for $i = 1, 2, \dots, m$ .
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:

$ifail = 1$: Constraint: $n \geq 3$ .

$ifail = 2$: Constraint: $m \geq 1$ .

$ifail = 3$: On entry, the triangulation information held in the array triang does not specify a valid triangulation of the data points. triang has been corrupted since the call to nag_interp_2d_triangulate (e01ea).

$ifail = 4$: At least one evaluation point lies outside the nodal triangulation. For each such point the value returned in pf is that corresponding to a node on the closest boundary line segment.

$ifail = - 99$: 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

Not applicable.

Further Comments

The time taken for a call of nag_interp_2d_triang_bary_eval (e01eb) is approximately proportional to the number of interpolation points,

m

Example

See Example in nag_interp_2d_triangulate (e01ea).

Open in the MATLAB editor: e01eb_example

function e01eb_example


fprintf('e01eb example results\n\n');

% Scattered Grid Data
x = [11.16; 12.85; 19.85; 19.72; 15.91;  0.00; 20.87;  3.45; 14.26; ...
     17.43; 22.80;  7.58; 25.00;  0.00;  9.66;  5.22; 17.25; 25.00; ...
     12.13; 22.23; 11.52; 15.20;  7.54; 17.32;  2.14;  0.51; 22.69; ...
      5.47; 21.67;  3.31];
y = [ 1.24;  3.06; 10.72;  1.39;  7.74; 20.00; 20.00; 12.78; 17.87; ...
      3.46; 12.39;  1.98; 11.87;  0.00; 20.00; 14.66; 19.57;  3.87; ...
     10.79;  6.21;  8.53;  0.00; 10.69; 13.78; 15.03;  8.37; 19.63; ...
     17.13; 14.36; 0.33];
f = [22.15; 22.11;  7.97; 16.83; 15.30; 34.60;  5.74; 41.24; 10.74; ...
     18.60;  5.47; 29.87;  4.40; 58.20;  4.73; 40.36;  6.43;  8.74; ...
     13.71; 10.25; 15.74; 21.60; 19.31; 12.11; 53.10; 49.43;  3.25; ...
     28.63;  5.52; 44.08];
% Triangulate on (x,y)
[triang,ifail] = e01ea(x,y);
% Perform barycentric interpolation at (3.0,17.0)
px = 3;
py = 17;
[pf, ifail] = e01eb(x, y, f, triang, px, py);

fprintf('Interpolated value for f at (%4.1f,%4.1f) = %7.2f\n',px,py,pf);

e01eb example results

Interpolated value for f at ( 3.0,17.0) =   39.05

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

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