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.
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.
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)