Namespace: nagcpp

Namespace: fit

Function: dim1_spline_eval

NAG CPP Interface
nagcpp::fit::dim1_spline_eval (e02bb)

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NAG Library Manual, Mark 30.2

Interfaces: FL CL CPP AD PY MB

NAG CPP Interface Introduction

E02 (Fit) Chapter Contents

e02bb: FL CL CPP AD PY MB

Namespace: nagcpp

Namespace: fit

Function: dim1_spline_eval

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Exceptions and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

▸▿ 10 Example

10.1 Example Program

10.2 Plot

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1 Purpose

dim1_spline_eval evaluates a cubic spline from its B-spline representation.

2 Specification

#include "e02/nagcpp_e02bb.hpp"

template <typename LAMDA, typename C>

void function dim1_spline_eval(const LAMDA &lamda, const C &c, const double x, double &s, OptionalE02BB opt)

template <typename LAMDA, typename C>

void function dim1_spline_eval(const LAMDA &lamda, const C &c, const double x, double &s)

3 Description

dim1_spline_eval evaluates the cubic spline

s (x)

at a prescribed argument

x

from its augmented knot set

λ_{i}

, for

i = 1, 2, \dots, n + 7

, (see e02baf (no CPP interface)) and from the coefficients

c_{i}

, for

i = 1, 2, \dots, q

in its B-spline representation

s (x) = \sum_{i = 1}^{q} c_{i} N_{i} (x) .

Here

q = \bar{n} + 3

, where

\bar{n}

is the number of intervals of the spline, and

N_{i} (x)

denotes the normalized B-spline of degree

3

defined upon the knots

λ_{i}, λ_{i + 1}, \dots, λ_{i + 4}

. The prescribed argument

x

must satisfy

λ_{4} \leq x \leq λ_{\bar{n} + 4}

It is assumed that

λ_{j} \geq λ_{j - 1}

, for

j = 2, 3, \dots, \bar{n} + 7

, and

λ_{\bar{n} + 4} > λ_{4}

x

is a point at which

4

knots coincide,

s (x)

is discontinuous at

x

; in this case, s contains the value defined as

x

is approached from the right.

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 (1972) and Cox and Hayes (1973).

It is expected that a common use of dim1_spline_eval will be the evaluation of the cubic spline approximations produced by e02baf (no CPP interface). A generalization of dim1_spline_eval which also forms the derivative of

s (x)

is e02bcf (no CPP interface). e02bcf (no CPP interface) takes about

50 %

longer than dim1_spline_eval.

4 References

Cox M G (1972) The numerical evaluation of B-splines J. Inst. Math. Appl. 10 134–149

Cox M G (1978) The numerical evaluation of a spline from its B-spline representation J. Inst. Math. Appl. 21 135–143

Cox M G and Hayes J G (1973) Curve fitting: a guide and suite of algorithms for the non-specialist user NPL Report NAC26 National Physical Laboratory

de Boor C (1972) On calculating with B-splines J. Approx. Theory 6 50–62

5 Arguments

1: $lamda (ncap7)$ – double array Input: On entry: $lamda (j - 1)$ must be set to the value of the $j$ th member of the complete set of knots, $λ_{j}$ , for $j = 1, 2, \dots, \bar{n} + 7$ .

Constraint: the $lamda (j - 1)$ must be in nondecreasing order with $lamda (ncap7 - 4) > lamda (3)$ .
2: $c (ncap7)$ – double array Input: On entry: the coefficient $c_{i}$ of the B-spline $N_{i} (x)$ , for $i = 1, 2, \dots, \bar{n} + 3$ . The remaining elements of the array are not referenced.
3: $x$ – double Input: On entry: the argument $x$ at which the cubic spline is to be evaluated.

Constraint: $lamda (3) \leq x \leq lamda (ncap7 - 4)$ .
4: $s$ – double Output: On exit: the value of the spline, $s (x)$ .
5: $opt$ – OptionalE02BB Input/Output: Optional parameter container, derived from Optional.

5.1Additional Quantities

1: $ncap7$: $\bar{n} + 7$ , where $\bar{n}$ is the number of intervals (one greater than the number of interior knots, i.e., the knots strictly within the range $λ_{4}$ to $λ_{\bar{n} + 4}$ ) over which the spline is defined.

6 Exceptions and Warnings

Errors or warnings detected by the function:

All errors and warnings have an associated numeric error code field, errorid, stored either as a member of the thrown exception object (see errorid), or as a member of opt.ifail, depending on how errors and warnings are being handled (see Error Handling for more details).

Raises: ErrorException

$errorid = 1$: On entry, $x = ⟨ value ⟩$ , $ncap7 = ⟨ value ⟩$
and $lamda [ncap7 - 4] = ⟨ value ⟩$ .
Constraint: $x \leq lamda [ncap7 - 4]$ .

$errorid = 1$: On entry, $x = ⟨ value ⟩$ and
$lamda [3] = ⟨ value ⟩$ .
Constraint: $x \geq lamda [3]$ .

$errorid = 2$: On entry, $ncap7 = ⟨ value ⟩$ .
Constraint: $ncap7 \geq 8$ .

$errorid = 10601$: On entry, argument $⟨ value ⟩$ must be a vector of size $⟨ value ⟩$ array.
Supplied argument has $⟨ value ⟩$ dimensions.

$errorid = 10601$: On entry, argument $⟨ value ⟩$ must be a vector of size $⟨ value ⟩$ array.
Supplied argument was a vector of size $⟨ value ⟩$ .

$errorid = 10601$: On entry, argument $⟨ value ⟩$ must be a vector of size $⟨ value ⟩$ array.
The size for the supplied array could not be ascertained.

$errorid = 10602$: On entry, the raw data component of $⟨ value ⟩$ is null.

$errorid = 10603$: On entry, unable to ascertain a value for $⟨ value ⟩$ .

$errorid = −99$: An unexpected error has been triggered by this routine.

$errorid = −399$: Your licence key may have expired or may not have been installed correctly.

$errorid = −999$: Dynamic memory allocation failed.

7 Accuracy

The computed value of

s (x)

has negligible error in most practical situations. Specifically, this value has an absolute error bounded in modulus by

18 \times c_{\max} \times machine precision

, where

c_{\max}

is the largest in modulus of

c_{j}, c_{j + 1}, c_{j + 2}

and

c_{j + 3}

, and

j

is an integer such that

λ_{j + 3} \leq x \leq λ_{j + 4}

. If

c_{j}, c_{j + 1}, c_{j + 2}

and

c_{j + 3}

are all of the same sign, then the computed value of

s (x)

has a relative error not exceeding

20 \times machine precision

in modulus. For further details see Cox (1978).

8 Parallelism and Performance

Please see the description for the underlying computational routine in this section of the FL Interface documentation.

9 Further Comments

The time taken is approximately

c \times (1 + 0.1 \times \log (\bar{n} + 7))

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 5, since to do this would result in a computation time approximately linear in

\bar{n} + 7

instead of

\log (\bar{n} + 7)

. All the conditions are tested in e02baf (no CPP interface), however.

10 Example

Evaluate at nine equally-spaced points in the interval

1.0 \leq x \leq 9.0

the cubic spline with (augmented) knots

1.0

1.0

1.0

1.0

3.0

6.0

8.0

9.0

9.0

9.0

9.0

and normalized cubic B-spline coefficients

1.0

2.0

4.0

7.0

6.0

4.0

3.0

The example program is written in a general form that will enable a cubic spline with

\bar{n}

intervals, in its normalized cubic B-spline form, to be evaluated at

m

equally-spaced points in the interval

lamda (3) \leq x \leq lamda (\bar{n} + 3)

. The program is self-starting in that any number of datasets may be supplied.

10.1 Example Program

Source File	Data	Results
ex_e02bb.cpp	None	ex_e02bb.r

10.2 Plot

NAG Library Manual, Mark 30.2

Interfaces: FL CL CPP AD PY MB

NAG CPP Interface Introduction

E02 (Fit) Chapter Contents

e02bb: FL CL CPP AD PY MB

Namespace: nagcpp

Namespace: fit

Function: dim1_spline_eval

NAG CPP Interfacenagcpp::fit::dim1_spline_eval (e02bb)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

5.1Additional Quantities

6 Exceptions and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Example Program

10.2 Plot

NAG CPP Interface
nagcpp::fit::dim1_spline_eval (e02bb)