Evaluation of bicubic spline, at a set of points

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NAG[e02dec] NAG[nag_2d_spline_eval] - Evaluation of bicubic spline, at a set of points

Calling Sequence

e02dec(x, y, ff, spline_data, 'm'=m, 'fail'=fail)

nag_2d_spline_eval(. . .)

Parameters

x - Vector(1..m, datatype=float[8]);
y - Vector(1..m, datatype=float[8]);

On entry: x and y must contain and , for , respectively. These are the co-ordinates of the points at which values of the spline are required. The order of the points is immaterial.

Constraint: x and y must satisfy and , for . The spline representation is not valid outside these intervals .

ff - Vector(1..m, datatype=float[8]);

On exit: contains the value of the spline at the point , for .

spline_data - table;

A Maple table, which should be generated using NAG[Nag_2dSpline], corresponding to the Nag_2dSpline structure.

The following fields may be accessed via NAG[GetOptions] and NAG[SetOptions]:

nx - integer

On entry: nx must specify the total number of knots associated with the variables . It is such that is the number of interior knots.

Constraint: . .

lamda - Vector(datatype=float[8])

On entry: a pointer to which memory of size nx must be allocated. lamda must contain the complete sets of knots associated with the variable.

Constraint: the knots must be in non-decreasing order, with . .

ny - integer

On entry: ny must specify the total number of knots associated with the variable .

It is such that is the number of interior knots.

Constraint: . .

mu - Vector(datatype=float[8])

On entry: a pointer to which memory of size ny must be allocated. mu must contain the complete sets of knots associated with the variable.

Constraint: the knots must be in non-decreasing order, with . .

c - Vector(datatype=float[8])

On entry: a pointer to which memory of size must be allocated. must contain the coefficient described in Section [Description], for ; .

'm'=m - integer; (optional)

Default value: the first dimension of the arrays x, y, ff.

On entry: , the number of points at which values of the spline are required.

Constraint: . .

'fail'=fail - table; (optional)

The NAG error argument, see the documentation for NagError.

Description

	Purpose
	nag_2d_spline_eval (e02dec) calculates values of a bicubic spline from its B-spline representation.

Description

nag_2d_spline_eval (e02dec) calculates values of the bicubic spline at prescribed points , for , from its augmented knot sets and and from the coefficients , for ; , in its B-spline representation

Here and denote normalized cubic B-splines, the former defined on the knots to and the latter on the knots to .

This function may be used to calculate values of a bicubic spline given in the form produced by e01dac (nag_2d_spline_interpolant), e02dcc (nag_2d_spline_fit_grid) and e02ddc (nag_2d_spline_fit_scat). It is derived from the routine B2VRE in Anthony et al. (1982).

	Error Indicators and Warnings
	"NE_ALLOC_FAIL" Dynamic memory allocation failed. "NE_END_KNOTS_CONS" On entry, the end knots must satisfy , , . "NE_INT_ARG_LT" On entry, m must not be less than 1: . "NE_NOT_INCREASING" The sequence lamda is not increasing: , . The sequence mu is not increasing: , . "NE_POINT_OUTSIDE_RECT" On entry, point lies outside the rectangle bounded by , , , .

	Accuracy
	The method used to evaluate the B-splines is numerically stable, in the sense that each computed value of 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.

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

Examples

m := 7:
spline_data := NAG:-Nag_2dSpline():
spline_data['nx'] := 11:
spline_data['ny'] := 10:
spline_data['lamda'] := Vector([ 1, 1, 1, 1, 1.3, 1.5, 1.6, 2, 2, 2, 2 ],
                        datatype=float[8]):
spline_data['mu'] := Vector([ 0, 0, 0, 0, 0.4, 0.7, 1, 1, 1, 1 ],
                     datatype=float[8]):
spline_data['c'] := Vector([ 1, 1.1333, 1.3667, 1.7, 1.9, 2, 1.2, 1.3333,
                      1.5667, 1.9, 2.1, 2.2, 1.5833, 1.7167, 1.95,
                      2.2833, 2.4833, 2.5833, 2.1433, 2.2767, 2.51,
                      2.8433, 3.0433, 3.1433, 2.8667, 3, 3.2333, 3.5667,
                      3.7667, 3.8667, 3.4667, 3.6, 3.8333, 4.1667, 4.3667,
                      4.4667, 4, 4.1333, 4.3667, 4.7, 4.9, 5 ],
                     datatype=float[8]):
x := Vector([1, 1.1, 1.5, 1.6, 1.9, 1.9, 2], datatype=float[8]):
y := Vector([0, 0.1, 0.7, 0.4, 0.3, 0.8, 1], datatype=float[8]):
ff := Vector(7, datatype=float[8]):
NAG:-e02dec(x, y, ff, spline_data, 'm' = m):

ff;

	See Also
	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 e02 Chapter Introduction. NAG Toolbox Overview. NAG Web Site.