e02dhc 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 e01dac,e02dac,e02dccande02ddc.
The function may be called by the names: e02dhc, nag_fit_dim2_spline_derivm or nag_2d_spline_deriv_rect.
3Description
e02dhc 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
(1)
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 e01dac,e02dac,e02dccande02ddc.
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.
4References
de Boor C (1972) On calculating with B-splines J. Approx. Theory6 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: – IntegerInput
On entry: , the number of grid points along the axis.
Constraint:
.
2: – IntegerInput
On entry: , the number of grid points along the axis.
Constraint:
.
3: – const doubleInput
On entry: must be set to
, the coordinate of the th grid point along the axis, for , on which values of the partial derivative are sought.
Constraint:
.
4: – const doubleInput
On entry: must be set to , the coordinate of the th grid point along the axis, for on which values of the partial derivative are sought.
Constraint:
.
5: – IntegerInput
On entry: specifies the order, of the partial derivative in the -direction.
Constraint:
.
6: – IntegerInput
On entry: specifies the order, of the partial derivative in the -direction.
Constraint:
.
7: – doubleOutput
On exit: contains the derivative , for and .
8: – 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: – NagError *Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).
6Error 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 had an illegal value.
NE_INT
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
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 , and .
Constraint: , for .
On entry, for , and .
Constraint: , for .
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 e01dac,e02dac,e02dccande02ddc of the function document for the respective function which calculated the spline approximant for details on the accuracy of that approximation.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
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.
9Further Comments
None.
10Example
This example reads in values of , , , for , and , for , followed by values of the ordinates defined at the grid points . It then calls e02dcc to compute a bicubic spline approximation for one specified value of . Finally it evaluates the spline and its first derivative at a small sample of points on a rectangular grid by calling e02dhc.