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 e01dac, e02dac, e02dcc and 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: $mx$ – Integer Input: On entry: $m_{x}$ , the number of grid points along the $x$ axis.

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

Constraint: $my \geq 1$ .
3: $x [mx]$ – const double Input: 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 double Input: 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: $nux$ – Integer Input: On entry: specifies the order, $ν_{x}$ of the partial derivative in the $x$ -direction.

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

Constraint: $0 \leq nuy \leq 2$ .
7: $z [mx \times my]$ – double Output: 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 e02dhc follows a call to e01dac, e02dac, e02dcc or e02ddc, in which case, members of the structure spline will have been set up correctly for input to e02dhc.
9: $fail$ – NagError * Input/Output: The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface 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.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface 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 e01dac, e02dac, e02dcc and 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

e02dhc makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.

Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

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 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 e02dhc.

e02dh: FL CL CPP AD

NAG CL Interfacee02dhc (dim2_​spline_​derivm)

▸▿ 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 CL Interface
e02dhc (dim2_spline_derivm)