naginterfaces.library.fit.dim2_​spline_​derivm

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

dim2_spline_derivm computes the partial derivative (of order , ), of a bicubic spline approximation to a set of data values, from its B-spline representation, at points on a rectangular grid in the - 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://www.nag.com/numeric/nl/nagdoc_28.6/flhtml/e02/e02dhf.html

Parameters
xfloat, array-like, shape

must be set to , the coordinate of the th grid point along the axis, for , on which values of the partial derivative are sought.

yfloat, array-like, shape

must be set to , the coordinate of the th grid point along the axis, for on which values of the partial derivative are sought.

lamdafloat, array-like, shape

Contains the position of the knots in the -direction of the bicubic spline approximation to be differentiated, e.g., as returned by dim2_spline_grid().

mufloat, array-like, shape

Contains the position of the knots in the -direction of the bicubic spline approximation to be differentiated, e.g., as returned by dim2_spline_grid().

cfloat, array-like, shape

The coefficients of the bicubic spline approximation to be differentiated, e.g., as returned by dim2_spline_grid().

nuxint

Specifies the order, of the partial derivative in the -direction.

nuyint

Specifies the order, of the partial derivative in the -direction.

Returns
zfloat, ndarray, shape

contains the derivative , for , for .

Raises
NagValueError
(errno )

On entry, .

Constraint: .

(errno )

On entry, .

Constraint: .

(errno )

On entry, .

Constraint: .

(errno )

On entry, .

Constraint: .

(errno )

On entry, for , and .

Constraint: , for .

(errno )

On entry, for , and .

Constraint: , for .

Notes

dim2_spline_derivm determines the partial derivative of a smooth bicubic spline approximation at the set of data points .

The spline is given in the B-spline representation

where and denote normalized cubic B-splines, the former defined on the knots to and the latter on the knots to , with and the total numbers of knots of the computed spline with respect to the and 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 in each direction; thus the highest mixed derivative available is .

The points in the grid are defined by coordinates , for , along the axis, and coordinates , for , along the 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