# NAG FL Interfacee01ebf (dim2_​triang_​bary_​eval)

## 1Purpose

e01ebf 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 e01eaf.

## 2Specification

Fortran Interface
 Subroutine e01ebf ( m, n, x, y, f, px, py, pf,
 Integer, Intent (In) :: m, n, triang(7*n) Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: x(n), y(n), f(n), px(m), py(m) Real (Kind=nag_wp), Intent (Out) :: pf(m)
#include <nag.h>
 void e01ebf_ (const Integer *m, const Integer *n, const double x[], const double y[], const double f[], const Integer triang[], const double px[], const double py[], double pf[], Integer *ifail)
The routine may be called by the names e01ebf or nagf_interp_dim2_triang_bary_eval.

## 3Description

e01ebf takes as input a set of scattered data points $\left({x}_{\mathit{r}},{y}_{\mathit{r}},{f}_{\mathit{r}}\right)$, for $\mathit{r}=1,2,\dots ,n$, and a Thiessen triangulation of the $\left({x}_{r},{y}_{r}\right)$ computed by e01eaf, and interpolates at a set of points $\left({\mathit{px}}_{i},{\mathit{py}}_{i}\right)$, for $\mathit{i}=1,2,\dots ,m$.
If the $i$th interpolation point $\left({\mathit{px}}_{i},{\mathit{py}}_{i}\right)$ is equal to $\left({x}_{r},{y}_{r}\right)$ 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 $\left({\mathit{px}}_{i},{\mathit{py}}_{i}\right)$, 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 $F\left({\mathit{px}}_{i},{\mathit{py}}_{i}\right)$.
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 $\left({x}_{s},{y}_{s}\right)$.
e01ebf must only be called after a call to e01eaf.
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

## 5Arguments

1: $\mathbf{m}$Integer Input
On entry: $m$, the number of points to interpolate.
Constraint: ${\mathbf{m}}\ge 1$.
2: $\mathbf{n}$Integer Input
On entry: $n$, the number of data points. n must be unchanged from the previous call of e01eaf.
Constraint: ${\mathbf{n}}\ge 3$.
3: $\mathbf{x}\left({\mathbf{n}}\right)$Real (Kind=nag_wp) array Input
4: $\mathbf{y}\left({\mathbf{n}}\right)$Real (Kind=nag_wp) array Input
On entry: the coordinates of the $\mathit{r}$th data point, $\left({x}_{r},{y}_{r}\right)$, for $\mathit{r}=1,2,\dots ,n$. x and y must be unchanged from the previous call of e01eaf.
5: $\mathbf{f}\left({\mathbf{n}}\right)$Real (Kind=nag_wp) array Input
On entry: the function values ${f}_{\mathit{r}}$ at $\left({x}_{\mathit{r}},{y}_{\mathit{r}}\right)$, for $\mathit{r}=1,2,\dots ,n$.
6: $\mathbf{triang}\left(7×{\mathbf{n}}\right)$Integer array Input
On entry: the triangulation computed by the previous call of e01eaf. See Section 9 in e01eaf for details of how the triangulation used is encoded in triang.
7: $\mathbf{px}\left({\mathbf{m}}\right)$Real (Kind=nag_wp) array Input
8: $\mathbf{py}\left({\mathbf{m}}\right)$Real (Kind=nag_wp) array Input
On entry: the coordinates $\left({\mathit{px}}_{\mathit{i}},{\mathit{py}}_{\mathit{i}}\right)$, for $\mathit{i}=1,2,\dots ,m$, at which interpolated function values are sought.
9: $\mathbf{pf}\left({\mathbf{m}}\right)$Real (Kind=nag_wp) array Output
On exit: the interpolated values $F\left({\mathit{px}}_{\mathit{i}},{\mathit{py}}_{\mathit{i}}\right)$, for $\mathit{i}=1,2,\dots ,m$.
10: $\mathbf{ifail}$Integer Input/Output
On entry: ifail must be set to $0$, $-1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $-1$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $-1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $-\mathbf{1}$ or $\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 3$.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{m}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{m}}\ge 1$.
${\mathbf{ifail}}=3$
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 e01eaf.
${\mathbf{ifail}}=4$
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.
${\mathbf{ifail}}=-99$
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

Not applicable.

## 8Parallelism and Performance

e01ebf is not threaded in any implementation.

The time taken for a call of e01ebf is approximately proportional to the number of interpolation points, $m$.