Evaluation of fitted cubic spline, function only

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NAG[e02bbc] NAG[nag_1d_spline_evaluate] - Evaluation of fitted cubic spline, function only

Calling Sequence

e02bbc(x, s, spline_data, 'fail'=fail)

nag_1d_spline_evaluate(. . .)

Parameters

x - float;

On entry: the argument at which the cubic spline is to be evaluated.

Constraint: . .

s - assignable;

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

On exit: the value of the spline, .

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 (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 . .

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 .

'fail'=fail - table; (optional)

The NAG error argument, see the documentation for NagError.

Description

	Purpose
	nag_1d_spline_evaluate (e02bbc) evaluates a cubic spline from its B-spline representation.

Description

nag_1d_spline_evaluate (e02bbc) evaluates the cubic spline at a prescribed argument from its augmented knot set , for , (see e02bac (nag_1d_spline_fit_knots)) and from the coefficients , for in its B-spline representation

Here , where is the number of intervals of the spline, and denotes the normalized B-spline of degree 3 defined upon the knots . The prescribed argument must satisfy .

It is assumed that , for , and .

The method employed is that of evaluation by taking convex combinations due to De Boor (1972). For further details of the algorithm and its use see Cox (1972a) and Cox (1978).

It is expected that a common use of nag_1d_spline_evaluate (e02bbc) will be the evaluation of the cubic spline approximations produced by e02bac (nag_1d_spline_fit_knots). A generalization of nag_1d_spline_evaluate (e02bbc) which also forms the derivative of is e02bcc (nag_1d_spline_deriv). e02bcc (nag_1d_spline_deriv) takes about 50% longer than nag_1d_spline_evaluate (e02bbc).

	Error Indicators and Warnings
	"NE_ABSCI_OUTSIDE_KNOT_INTVL" On entry, x must satisfy : , , . In this case s is set arbitrarily to zero. "NE_INT_ARG_LT" On entry, n must not be less than 8: .

	Accuracy
	The computed value of has negligible error in most practical situations. Specifically, this value has an absolute error bounded in modulus by machine precision, where is the largest in modulus of and , and is an integer such that . If and are all of the same sign, then the computed value of has a relative error not exceeding machine precision in modulus. For further details see Cox (1978).

Further Comments

The time taken by nag_1d_spline_evaluate (e02bbc) is approximately C seconds, where C is a machine-dependent constant.

Note: the function does not test all the conditions on the knots given in the description of lamda in Section [Parameters], since to do this would result in a computation time approximately linear in instead of . All the conditions are tested in e02bac (nag_1d_spline_fit_knots), however, and the knots returned by e01bac (nag_1d_spline_interpolant) or e02bec (nag_1d_spline_fit) will satisfy the conditions.

Examples

m := 9:
ncap := 4:
ncap7 := ncap+7:
spline_data := NAG:-Nag_Spline():
spline_data['n'] := ncap7:
spline_data['lamda'] := Vector(ncap7,[1, 1, 1, 1, 3, 6, 8, 9, 9, 9, 9 ],
                             datatype=float[8]):
spline_data['c'] := Vector(ncap7,[ 1, 2, 4, 7, 6, 4, 3 ],
                         datatype=float[8]):
a := spline_data['lamda'][4]:
b := spline_data['lamda'][ncap+4]:
for r from 1 to m do
   x := ((m-r)*a+(r-1)*b)/(m-1):
   NAG:-e02bbc(x, s, spline_data):
   print(s);

end do: