Evaluation of fitted cubic spline, definite integral

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NAG[e02bdc] NAG[nag_1d_spline_intg] - Evaluation of fitted cubic spline, definite integral

Calling Sequence

e02bdc(spline_data, integral, 'fail'=fail)

nag_1d_spline_intg(. . .)

Parameters

spline_data - table;

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

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

n - integer

On entry: , where is the number of intervals of the spline (which is one greater than the number of interior knots, i.e., the knots strictly within the range to ) over which the spline is defined.

Constraint: . .

lamda - Vector(datatype=float[8])

On entry: a pointer to which memory of size n must be allocated. must be set to the value of the th member of the complete set of knots, for .

Constraint: the must be in non-decreasing order with and satisfy

and

c - Vector(datatype=float[8])

On entry: a pointer to which memory of size must be allocated. c holds the coefficient of the B-spline , for .

integral - assignable;

Note: On exit the variable integral will have a value of type float.

On exit: the value of the definite integral of between the limits and , where and .

'fail'=fail - table; (optional)

The NAG error argument, see the documentation for NagError.

Description

	Purpose
	nag_1d_spline_intg (e02bdc) computes the definite integral of a cubic spline from its B-spline representation.

Description

nag_1d_spline_intg (e02bdc) computes the definite integral of the cubic spline between the limits and , where and are respectively the lower and upper limits of the range over which is defined. It is assumed that is represented in terms of its B-spline coefficients , for and (augmented) ordered knot set , for , with , for ,2,3,4 and , for , (see e02bac (nag_1d_spline_fit_knots)), i.e.,

Here , is the number of intervals of the spline and denotes the normalized B-spline of degree 3 (order 4) defined upon the knots .

The method employed uses the formula given in Section 3 of Cox (1975a).

nag_1d_spline_intg (e02bdc) can be used to determine the definite integrals of cubic spline fits and interpolants produced by e02bac (nag_1d_spline_fit_knots), e01bac (nag_1d_spline_interpolant) and e02bec (nag_1d_spline_fit).

	Error Indicators and Warnings
	"NE_INT_ARG_LT" On entry, n must not be less than 8: . "NE_KNOTS_CONS" On entry, the knots must satisfy the following constraints: , , for , with equality in the cases , , and .

	Accuracy
	The rounding errors are such that the computed value of the integral is exact for a slightly perturbed set of B-spline coefficients differing in a relative sense from those supplied by no more than machine precision.

	Further Comments
	Under normal usage, the call to nag_1d_spline_intg (e02bdc) will follow a call to e02bac (nag_1d_spline_fit_knots), e01bac (nag_1d_spline_interpolant) or e02bec (nag_1d_spline_fit). In that case, the structure spline_data will have been set up correctly for input to nag_1d_spline_intg (e02bdc). The time taken by nag_1d_spline_intg (e02bdc) is approximately proportional to .

Examples

>	spline_data := NAG:-Nag_Spline(): spline_data['n'] := 14: spline_data['lamda'] := Vector([ 0, 0, 0, 0, 1, 3, 3, 3, 4, 4, 6, 6, 6, 6 ], datatype=float[8]): spline_data['c'] := Vector([ 10, 12, 13, 15, 22, 26, 24, 18, 14, 12 ], datatype=float[8]): NAG:-e02bdc(spline_data, integral):

integral;

	See Also
	Cox M G (1975a) An algorithm for spline interpolation J. Inst. Math. Appl. 15 95–108 e02 Chapter Introduction. NAG Toolbox Overview. NAG Web Site.