NAG Library Function Document

nag_2d_spline_deriv_rect (e02dhc)

void	nag_2d_spline_deriv_rect (Integer mx, Integer my, const double x[], const double y[], Integer nux, Integer nuy, double z[], Nag_2dSpline spline, NagError fail)

3 Description

nag_2d_spline_deriv_rect (e02dhc) determines the partial derivative

\frac{\partial^{ν_{x} + ν_{y}}}{\partial x^{ν_{x}} \partial y^{ν_{y}}}

of a smooth bicubic spline approximation

s (x, y)

at the set of data points

(x_{q}, y_{r})

The spline is given in the B-spline representation

s (x, y) = \sum_{i = 1}^{n_{x} - 4} \sum_{j = 1}^{n_{y} - 4} c_{i j} M_{i} (x) N_{j} (y),

(1)

where

M_{i} (x)

and

N_{j} (y)

denote normalized cubic B-splines, the former defined on the knots

λ_{i}

λ_{i + 4}

and the latter on the knots

μ_{j}

μ_{j + 4}

, with

n_{x}

and

n_{y}

the total numbers of knots of the computed spline with respect to the

x

and

y

variables respectively. For further details, see Hayes and Halliday (1974) for bicubic splines and de Boor (1972) for normalized B-splines. This function is suitable for B-spline representations returned by nag_2d_spline_interpolant (e01dac), nag_2d_spline_fit_panel (e02dac), nag_2d_spline_fit_grid (e02dcc) and nag_2d_spline_fit_scat (e02ddc).

The partial derivatives can be up to order

2

in each direction; thus the highest mixed derivative available is

\frac{\partial^{4}}{\partial x^{2} \partial y^{2}}

The points in the grid are defined by coordinates

x_{q}

, for

q = 1, 2, \dots, m_{x}

, along the

x

axis, and coordinates

y_{r}

, for

r = 1, 2, \dots, m_{y}

, along the

y

axis.

4 References

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

Dierckx P (1981) An improved algorithm for curve fitting with spline functions Report TW54 Department of Computer Science, Katholieke Univerciteit Leuven

Dierckx P (1982) A fast algorithm for smoothing data on a rectangular grid while using spline functions SIAM J. Numer. Anal. 19 1286–1304

Hayes J G and Halliday J (1974) The least squares fitting of cubic spline surfaces to general data sets J. Inst. Math. Appl. 14 89–103

Reinsch C H (1967) Smoothing by spline functions Numer. Math. 10 177–183

5 Arguments

1: $mx$ – IntegerInput: On entry: $m_{x}$ , the number of grid points along the $x$ axis.

Constraint: $mx \geq 1$ .
2: $my$ – IntegerInput: On entry: $m_{y}$ , the number of grid points along the $y$ axis.

Constraint: $my \geq 1$ .
3: $x [mx]$ – const doubleInput: On entry: $x [q - 1]$ must be set to $x_{q}$ , the $x$ coordinate of the $q$ th grid point along the $x$ axis, for $q = 1, 2, \dots, m_{x}$ , on which values of the partial derivative are sought.

Constraint: $x_{1} < x_{2} < \dots < x_{m_{x}}$ .
4: $y [my]$ – const doubleInput: On entry: $y [r - 1]$ must be set to $y_{r}$ , the $y$ coordinate of the $r$ th grid point along the $y$ axis, for $r = 1, 2, \dots, m_{y}$ on which values of the partial derivative are sought.

Constraint: $y_{1} < y_{2} < \dots < y_{m_{y}}$ .
5: $nux$ – IntegerInput: On entry: specifies the order, $ν_{x}$ of the partial derivative in the $x$ -direction.

Constraint: $0 \leq nux \leq 2$ .
6: $nuy$ – IntegerInput: On entry: specifies the order, $ν_{y}$ of the partial derivative in the $y$ -direction.

Constraint: $0 \leq nuy \leq 2$ .
7: $z [mx \times my]$ – doubleOutput: On exit: $z [m_{y} \times (q - 1) + r - 1]$ contains the derivative $\frac{\partial^{ν_{x} + ν_{y}}}{{\partial x}^{ν_{x}} {\partial y}^{ν_{y}}} s (x_{q}, y_{r})$ , for $q = 1, 2, \dots, m_{x}$ and $r = 1, 2, \dots, m_{y}$ .
8: $spline$ – Nag_2dSpline *Input: Pointer to structure of type Nag_2dSpline describing the bicubic spline approximation to be differentiated.

In normal usage, the call to nag_2d_spline_deriv_rect (e02dhc) follows a call to nag_2d_spline_interpolant (e01dac), nag_2d_spline_fit_panel (e02dac), nag_2d_spline_fit_grid (e02dcc) or nag_2d_spline_fit_scat (e02ddc), in which case, members of the structure spline will have been set up correctly for input to nag_2d_spline_deriv_rect (e02dhc).
9: $fail$ – NagError *Input/Output: The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.2.1.2 in the Essential Introduction for further information.
NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
NE_INT: On entry, $mx = 〈value〉$ .
Constraint: $mx \geq 1$ .

On entry, $my = 〈value〉$ .
Constraint: $my \geq 1$ .

On entry, $nux = 〈value〉$ .
Constraint: $0 \leq nux \leq 2$ .

On entry, $nuy = 〈value〉$ .
Constraint: $0 \leq nuy \leq 2$ .
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.

An unexpected error has been triggered by this function. Please contact NAG.
See Section 3.6.6 in the Essential Introduction for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 3.6.5 in the Essential Introduction for further information.
NE_NOT_STRICTLY_INCREASING: On entry, for $i = 〈value〉$ , $x [i - 2] = 〈value〉$ and $x [i - 1] = 〈value〉$ .
Constraint: $x [i - 2] \leq x [i - 1]$ , for $i = 2, 3, \dots, mx$ .

On entry, for $i = 〈value〉$ , $y [i - 2] = 〈value〉$ and $y [i - 1] = 〈value〉$ .
Constraint: $y [i - 2] \leq y [i - 1]$ , for $i = 2, 3, \dots, my$ .

7 Accuracy

On successful exit, the partial derivatives on the given mesh are accurate to machine precision with respect to the supplied bicubic spline. Please refer to Section 7 in nag_2d_spline_interpolant (e01dac), nag_2d_spline_fit_panel (e02dac), nag_2d_spline_fit_grid (e02dcc) and nag_2d_spline_fit_scat (e02ddc) of the function document for the respective function which calculated the spline approximant for details on the accuracy of that approximation.

8 Parallelism and Performance

Not applicable.

9 Further Comments

None.

10 Example

This example reads in values of

m_{x}

m_{y}

x_{q}

, for

q = 1, 2, \dots, m_{x}

, and

y_{r}

, for

r = 1, 2, \dots, m_{y}

, followed by values of the ordinates

f_{q, r}

defined at the grid points

(x_{q}, y_{r})

. It then calls nag_2d_spline_fit_grid (e02dcc) to compute a bicubic spline approximation for one specified value of

S

. Finally it evaluates the spline and its first

x

derivative at a small sample of points on a rectangular grid by calling nag_2d_spline_deriv_rect (e02dhc).

NAG Library Function Documentnag_2d_spline_deriv_rect (e02dhc)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG Library Function Document

nag_2d_spline_deriv_rect (e02dhc)