NAG CL Interface
e02dhc (dim2_spline_derivm)
1
Purpose
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,
e02dcc and
e02ddc.
2
Specification
void |
e02dhc (Integer mx,
Integer my,
const double x[],
const double y[],
Integer nux,
Integer nuy,
double z[],
Nag_2dSpline *spline,
NagError *fail) |
|
The function may be called by the names: e02dhc, nag_fit_dim2_spline_derivm or nag_2d_spline_deriv_rect.
3
Description
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
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,
e02dcc and
e02ddc.
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.
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:
– Integer
Input
-
On entry: , the number of grid points along the axis.
Constraint:
.
-
2:
– Integer
Input
-
On entry: , the number of grid points along the axis.
Constraint:
.
-
3:
– const double
Input
-
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 double
Input
-
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:
– Integer
Input
-
On entry: specifies the order, of the partial derivative in the -direction.
Constraint:
.
-
6:
– Integer
Input
-
On entry: specifies the order, of the partial derivative in the -direction.
Constraint:
.
-
7:
– double
Output
-
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).
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 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 .
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.
None.
10
Example
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.
10.1
Program Text
10.2
Program Data
10.3
Program Results