# NAG C Library Function Document

## 1Purpose

nag_2d_spline_deriv_rect (e02dhc) computes the partial derivative (of order ${\nu }_{x}$, ${\nu }_{y}$), of a bicubic spline approximation to a set of data values, from its B-spline representation, at points on a rectangular grid in the $x$-$y$ plane. This function may be used to calculate derivatives of a bicubic spline given in the form produced 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).

## 2Specification

 #include #include
 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)

## 3Description

nag_2d_spline_deriv_rect (e02dhc) determines the partial derivative $\frac{{\partial }^{{\nu }_{x}+{\nu }_{y}}}{\partial {x}^{{\nu }_{x}}\partial {y}^{{\nu }_{y}}}$ of a smooth bicubic spline approximation $s\left(x,y\right)$ at the set of data points $\left({x}_{q},{y}_{r}\right)$.
The spline is given in the B-spline representation
 $sx,y = ∑ i=1 nx-4 ∑ j=1 ny-4 cij Mix Njy ,$ (1)
where ${M}_{i}\left(x\right)$ and ${N}_{j}\left(y\right)$ denote normalized cubic B-splines, the former defined on the knots ${\lambda }_{i}$ to ${\lambda }_{i+4}$ and the latter on the knots ${\mu }_{j}$ to ${\mu }_{j+4}$, 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 ${x}_{\mathit{q}}$, for $\mathit{q}=1,2,\dots ,{m}_{x}$, along the $x$ axis, and coordinates ${y}_{\mathit{r}}$, for $\mathit{r}=1,2,\dots ,{m}_{y}$, along the $y$ axis.

## 4References

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

## 5Arguments

1:    $\mathbf{mx}$IntegerInput
On entry: ${m}_{x}$, the number of grid points along the $x$ axis.
Constraint: ${\mathbf{mx}}\ge 1$.
2:    $\mathbf{my}$IntegerInput
On entry: ${m}_{y}$, the number of grid points along the $y$ axis.
Constraint: ${\mathbf{my}}\ge 1$.
3:    $\mathbf{x}\left[{\mathbf{mx}}\right]$const doubleInput
On entry: ${\mathbf{x}}\left[q-1\right]$ must be set to ${x}_{\mathit{q}}$, the $x$ coordinate of the $\mathit{q}$th grid point along the $x$ axis, for $\mathit{q}=1,2,\dots ,{m}_{x}$, on which values of the partial derivative are sought.
Constraint: ${x}_{1}<{x}_{2}<\cdots <{x}_{{m}_{x}}$.
4:    $\mathbf{y}\left[{\mathbf{my}}\right]$const doubleInput
On entry: ${\mathbf{y}}\left[\mathit{r}-1\right]$ must be set to ${y}_{\mathit{r}}$, the $y$ coordinate of the $\mathit{r}$th grid point along the $y$ axis, for $\mathit{r}=1,2,\dots ,{m}_{y}$ on which values of the partial derivative are sought.
Constraint: ${y}_{1}<{y}_{2}<\cdots <{y}_{{m}_{y}}$.
5:    $\mathbf{nux}$IntegerInput
On entry: specifies the order, ${\nu }_{x}$ of the partial derivative in the $x$-direction.
Constraint: $0\le {\mathbf{nux}}\le 2$.
6:    $\mathbf{nuy}$IntegerInput
On entry: specifies the order, ${\nu }_{y}$ of the partial derivative in the $y$-direction.
Constraint: $0\le {\mathbf{nuy}}\le 2$.
7:    $\mathbf{z}\left[{\mathbf{mx}}×{\mathbf{my}}\right]$doubleOutput
On exit: ${\mathbf{z}}\left[{m}_{y}×\left(\mathit{q}-1\right)+\mathit{r}-1\right]$ contains the derivative $\frac{{\partial }^{{\nu }_{x}+{\nu }_{y}}}{{\partial x}^{{\nu }_{x}}{\partial y}^{{\nu }_{y}}}s\left({x}_{q},{y}_{r}\right)$, for $\mathit{q}=1,2,\dots ,{m}_{x}$ and $\mathit{r}=1,2,\dots ,{m}_{y}$.
8:    $\mathbf{spline}$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:    $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT
On entry, ${\mathbf{mx}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{mx}}\ge 1$.
On entry, ${\mathbf{my}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{my}}\ge 1$.
On entry, ${\mathbf{nux}}=〈\mathit{\text{value}}〉$.
Constraint: $0\le {\mathbf{nux}}\le 2$.
On entry, ${\mathbf{nuy}}=〈\mathit{\text{value}}〉$.
Constraint: $0\le {\mathbf{nuy}}\le 2$.
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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NE_NOT_STRICTLY_INCREASING
On entry, for $i=〈\mathit{\text{value}}〉$, ${\mathbf{x}}\left[i-2\right]=〈\mathit{\text{value}}〉$ and ${\mathbf{x}}\left[i-1\right]=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{x}}\left[\mathit{i}-2\right]\le {\mathbf{x}}\left[\mathit{i}-1\right]$, for $\mathit{i}=2,3,\dots ,{\mathbf{mx}}$.
On entry, for $i=〈\mathit{\text{value}}〉$, ${\mathbf{y}}\left[i-2\right]=〈\mathit{\text{value}}〉$ and ${\mathbf{y}}\left[i-1\right]=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{y}}\left[\mathit{i}-2\right]\le {\mathbf{y}}\left[\mathit{i}-1\right]$, for $\mathit{i}=2,3,\dots ,{\mathbf{my}}$.

## 7Accuracy

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.

## 8Parallelism and Performance

nag_2d_spline_deriv_rect (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.

None.

## 10Example

This example reads in values of ${m}_{x}$, ${m}_{y}$, ${x}_{\mathit{q}}$, for $\mathit{q}=1,2,\dots ,{m}_{x}$, and ${y}_{\mathit{r}}$, for $\mathit{r}=1,2,\dots ,{m}_{y}$, followed by values of the ordinates ${f}_{q,r}$ defined at the grid points $\left({x}_{q},{y}_{r}\right)$. 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).

### 10.1Program Text

Program Text (e02dhce.c)

### 10.2Program Data

Program Data (e02dhce.d)

### 10.3Program Results

Program Results (e02dhce.r)