naginterfaces.library.fit.dim2_​spline_​derivm

naginterfaces.library.fit.dim2_spline_derivm(x, y, lamda, mu, c, nux, nuy)

dim2_spline_derivm 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 interp.dim2_spline_grid, dim2_spline_panel(), dim2_spline_grid() and dim2_spline_sctr().

For full information please refer to the NAG Library document for e02dh

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/e02/e02dhf.html

Parameters

xfloat, array-like, shape $\left(\text{mx}\right)$

$x[q-1]$ must be set to $x_{\text{q}}$, the $x$ coordinate of the $\text{q}$th grid point along the $x$ axis, for $\text{q}=1,2,\dots ,m_x$, on which values of the partial derivative are sought.

yfloat, array-like, shape $\left(\text{my}\right)$

$y[\text{r}-1]$ must be set to $y_{\text{r}}$, the $y$ coordinate of the $\text{r}$th grid point along the $y$ axis, for $\text{r}=1,2,\dots ,m_y$ on which values of the partial derivative are sought.

lamdafloat, array-like, shape $\left(\text{px}\right)$

Contains the position of the knots in the $x$-direction of the bicubic spline approximation to be differentiated, e.g., $\text{lamda}$ as returned by dim2_spline_grid().

mufloat, array-like, shape $\left(\text{py}\right)$

Contains the position of the knots in the $y$-direction of the bicubic spline approximation to be differentiated, e.g., $\text{mu}$ as returned by dim2_spline_grid().

cfloat, array-like, shape $((\text{px}-4)\times (\text{py}-4))$

The coefficients of the bicubic spline approximation to be differentiated, e.g., $\text{c}$ as returned by dim2_spline_grid().

nuxint

Specifies the order, ${\nu }_x$ of the partial derivative in the $x$-direction.

nuyint

Specifies the order, ${\nu }_y$ of the partial derivative in the $y$-direction.

Returns

zfloat, ndarray, shape $(\text{mx}\times \text{my})$

$z[m_y\times (\text{q}-1)+\text{r}-1]$ contains the derivative $\frac{{\partial }^{{\nu }_x+{\nu }_y}}{{\partial x}^{{\nu }_x}{\partial y}^{{\nu }_y}}s(x_q,y_r)$, for $\text{r}=1,2,\dots ,m_y$, for $\text{q}=1,2,\dots ,m_x$.

Raises

NagValueError

(errno $1$)

On entry, $nux=⟨value⟩$.

Constraint: $0\le nux\le 2$.

(errno $2$)

On entry, $nuy=⟨value⟩$.

Constraint: $0\le nuy\le 2$.

(errno $3$)

On entry, $\text{mx}=⟨value⟩$.

Constraint: $\text{mx}\ge 1$.

(errno $4$)

On entry, $\text{my}=⟨value⟩$.

Constraint: $\text{my}\ge 1$.

(errno $5$)

On entry, for $i=⟨value⟩$, $x[i-2]=⟨value⟩$ and $x[i-1]=⟨value⟩$.

Constraint: $x[\text{i}-2]\le x[\text{i}-1]$, for $\text{i}=2,3,\dots ,\text{mx}$.

(errno $6$)

On entry, for $i=⟨value⟩$, $y[i-2]=⟨value⟩$ and $y[i-1]=⟨value⟩$.

Constraint: $y[\text{i}-2]\le y[\text{i}-1]$, for $\text{i}=2,3,\dots ,\text{my}$.

Notes

dim2_spline_derivm determines the partial derivative $\frac{{\partial }^{{\nu }_x+{\nu }_y}}{\partial x^{{\nu }_x}\partial y^{{\nu }_y}}$ of a smooth bicubic spline approximation $s(x,y)$ at the set of data points $(x_q,y_r)$.

The spline is given in the B-spline representation

$$s(x,y)=\sum _{i=1}^{n_x-4}\sum _{j=1}^{n_y-4}{c}_{ij}{M}_i\left(x\right)N_j\left(y\right)\text{,}$$

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 interp.dim2_spline_grid, dim2_spline_panel(), dim2_spline_grid() and dim2_spline_sctr().

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 $x_{\text{q}}$, for $\text{q}=1,2,\dots ,m_x$, along the $x$ axis, and coordinates $y_{\text{r}}$, for $\text{r}=1,2,\dots ,m_y$, along the $y$ axis.

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