NAG FL Interface
e02def (dim2_​spline_​evalv)

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1 Purpose

e02def calculates values of a bicubic spline from its B-spline representation.

2 Specification

Fortran Interface
Subroutine e02def ( m, px, py, x, y, lamda, mu, c, ff, wrk, iwrk, ifail)
Integer, Intent (In) :: m, px, py
Integer, Intent (Inout) :: ifail
Integer, Intent (Out) :: iwrk(py-4)
Real (Kind=nag_wp), Intent (In) :: x(m), y(m), lamda(px), mu(py), c((px-4)*(py-4))
Real (Kind=nag_wp), Intent (Out) :: ff(m), wrk(py-4)
C Header Interface
#include <nag.h>
void  e02def_ (const Integer *m, const Integer *px, const Integer *py, const double x[], const double y[], const double lamda[], const double mu[], const double c[], double ff[], double wrk[], Integer iwrk[], Integer *ifail)
The routine may be called by the names e02def or nagf_fit_dim2_spline_evalv.

3 Description

e02def calculates values of the bicubic spline s(x,y) at prescribed points (xr,yr), for r=1,2,,m, from its augmented knot sets {λ} and {μ} and from the coefficients cij, for i=1,2,,px-4 and j=1,2,,py-4, in its B-spline representation
Here Mi(x) and Nj(y) denote normalized cubic B-splines, the former defined on the knots λi to λi+4 and the latter on the knots μj to μj+4.
This routine may be used to calculate values of a bicubic spline given in the form produced by e01daf, e02daf, e02dcf and e02ddf. It is derived from the routine B2VRE in Anthony et al. (1982).

4 References

Anthony G T, Cox M G and Hayes J G (1982) DASL – Data Approximation Subroutine Library National Physical Laboratory
Cox M G (1978) The numerical evaluation of a spline from its B-spline representation J. Inst. Math. Appl. 21 135–143

5 Arguments

1: m Integer Input
On entry: m, the number of points at which values of the spline are required.
Constraint: m1.
2: px Integer Input
3: py Integer Input
On entry: px and py must specify the total number of knots associated with the variables x and y respectively. They are such that px-8 and py-8 are the corresponding numbers of interior knots.
Constraint: px8 and py8.
4: x(m) Real (Kind=nag_wp) array Input
5: y(m) Real (Kind=nag_wp) array Input
On entry: x and y must contain xr and yr, for r=1,2,,m, respectively. These are the coordinates of the points at which values of the spline are required. The order of the points is immaterial.
Constraint: x and y must satisfy
mu(4)y(r)mu(py- 3),   r= 1,2,,m.  
The spline representation is not valid outside these intervals
6: lamda(px) Real (Kind=nag_wp) array Input
7: mu(py) Real (Kind=nag_wp) array Input
On entry: lamda and mu must contain the complete sets of knots {λ} and {μ} associated with the x and y variables respectively.
Constraint: the knots in each set must be in nondecreasing order, with lamda(px-3)>lamda(4) and mu(py-3)>mu(4).
8: c((px-4)×(py-4)) Real (Kind=nag_wp) array Input
On entry: c( (py-4) × (i-1) +j ) must contain the coefficient cij described in Section 3, for i=1,2,,px-4 and j=1,2,,py-4.
9: ff(m) Real (Kind=nag_wp) array Output
On exit: ff(r) contains the value of the spline at the point (xr,yr), for r=1,2,,m.
10: wrk(py-4) Real (Kind=nag_wp) array Workspace
11: iwrk(py-4) Integer array Workspace
12: 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 -1 or 1 is used it is essential to test the value of ifail on exit.
On exit: ifail=0 unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry 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:
On entry, m=value.
Constraint: m1.
On entry, px=value.
Constraint: px8.
On entry, py=value.
Constraint: py8.
On entry, the knots in lamda are not in nondecreasing order.
On entry, the knots in mu are not in nondecreasing order.
On entry, point (x(K),y(K)) lies outside the rectangle bounded by lamda(4), lamda(px-3), mu(4), mu(py-3): K=value, x(K)=value and y(K)=value.
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
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.
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

The method used to evaluate the B-splines is numerically stable, in the sense that each computed value of s(xr,yr) can be regarded as the value that would have been obtained in exact arithmetic from slightly perturbed B-spline coefficients. See Cox (1978) for details.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
e02def is not threaded in any implementation.

9 Further Comments

Computation time is approximately proportional to the number of points, m, at which the evaluation is required.

10 Example

This program reads in knot sets lamda(1),,lamda(px) and mu(1),,mu(py), and a set of bicubic spline coefficients cij. Following these are a value for m and the coordinates (xr,yr), for r=1,2,,m, at which the spline is to be evaluated.

10.1 Program Text

Program Text (e02defe.f90)

10.2 Program Data

Program Data (e02defe.d)

10.3 Program Results

Program Results (e02defe.r)
GnuplotProduced by GNUPLOT 5.4 patchlevel 6 1 1.2 1.4 1.6 1.8 2 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 gnuplot_plot_1 Evaluatuation points gnuplot_plot_2 gnuplot_plot_3 Bicubic Spline Surface x y Example Program Evaluation of Least-squares Bicubic Spline Fit at Scattered Points