Evaluation of bicubic spline, at a mesh of points

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NAG[e02dfc] NAG[nag_2d_spline_eval_rect] - Evaluation of bicubic spline, at a mesh of points

Calling Sequence

e02dfc(x, y, ff, spline_data, 'mx'=mx, 'my'=my, 'fail'=fail)

nag_2d_spline_eval_rect(. . .)

Parameters

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

On entry: x and y must contain , for , and , for , respectively. These are the and co-ordinates that define the rectangular grid of points at which values of the spline are required.

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

ff - Vector(1.., 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 variable . 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 ; .

'mx'=mx - integer; (optional)
'my'=my - integer; (optional)

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

On entry: mx and my must specify and respectively, the number of points along the and axes that define the rectangular grid.

Constraint: and . .

'fail'=fail - table; (optional)

The NAG error argument, see the documentation for NagError.

Description

	Purpose
	nag_2d_spline_eval_rect (e02dfc) calculates values of a bicubic spline from its B-spline representation. The spline is evaluated at all points on a rectangular grid.

Description

nag_2d_spline_eval_rect (e02dfc) calculates values of the bicubic spline on a rectangular grid of points in the - plane, 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 .

The points in the grid are defined by co-ordinates , for , along the axis, and co-ordinates , for along the axis.

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, mx must not be less than 1: .

"NE_KNOTS_COORD_CONS"

On entry, the end knots and coordinates must satisfy and . , , , .

"NE_NOT_INCREASING"

The sequence lamda is not increasing: , . The sequence mu is not increasing: , .

"NE_NOT_STRICTLY_INCREASING"

The sequence x is not strictly increasing: , . The sequence y is not strictly increasing: , .

	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 .

Examples

mx := 7:
my := 6:
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, 1.3, 1.4, 1.5, 1.7, 2], datatype=float[8]):
y := Vector([0, 0.2, 0.4, 0.6, 0.8, 1], datatype=float[8]):
ff := Vector(42, datatype=float[8]):
NAG:-e02dfc(x, y, ff, spline_data, 'mx' = mx, 'my' = my):

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.