NAG Library Function Document
nag_2d_spline_fit_ts_scat (e02jdc)
Note: this function uses optional arguments to define choices in the problem specification and in the details of the algorithm. If you wish to use default
settings for all of the optional arguments, you need only read Sections 1 to 10 of this document. If, however, you wish to reset some or all of the settings please refer to Section 11 for a detailed description of the specification of the optional arguments produced by the function.
1 Purpose
nag_2d_spline_fit_ts_scat (e02jdc) computes a spline approximation to a set of scattered data using a two-stage approximation method.
The computational complexity of the method grows linearly with the number of data points; hence large datasets are easily accommodated.
2 Specification
#include <nag.h> |
#include <nage02.h> |
void |
nag_2d_spline_fit_ts_scat (Integer n,
const double x[],
const double y[],
const double f[],
Integer lsminp,
Integer lsmaxp,
Integer nxcels,
Integer nycels,
Integer lcoefs,
double coefs[],
Integer iopts[],
double opts[],
NagError *fail) |
|
Before calling nag_2d_spline_fit_ts_scat (e02jdc),
nag_fit_opt_set (e02zkc) must be called with
optstr set to
"Initialize = e02jdc". Settings for optional algorithmic arguments may be specified by calling
nag_fit_opt_set (e02zkc) before a call to nag_2d_spline_fit_ts_scat (e02jdc).
3 Description
nag_2d_spline_fit_ts_scat (e02jdc) determines a smooth bivariate spline approximation to a set of
data points , for . Here, ‘smooth’ means
.
The approximation domain is the bounding box , where (respectively ) and (respectively ) denote the lowest and highest data values of the (respectively ).
The spline is computed by local approximations on a uniform triangulation of the bounding box. These approximations are extended to a smooth spline representation of the surface over the domain. The local approximation scheme is
by least squares polynomials (
Davydov and Zeilfelder (2004)).
The two-stage approximation method employed by nag_2d_spline_fit_ts_scat (e02jdc) is derived from
the TSFIT package of O. Davydov and F. Zeilfelder.
Values of the computed spline can subsequently be computed by calling
nag_2d_spline_ts_eval (e02jec) or
nag_2d_spline_ts_eval_rect (e02jfc).
4 References
Davydov O and Zeilfelder F (2004) Scattered data fitting by direct extension of local polynomials to bivariate splines Advances in Comp. Math. 21 223–271
5 Arguments
- 1:
n – IntegerInput
On entry: , the number of data values to be fitted.
Constraint:
.
- 2:
x[n] – const doubleInput
- 3:
y[n] – const doubleInput
- 4:
f[n] – const doubleInput
On entry: the data values to be fitted.
Constraint:
for some and for some ; i.e., there are at least two distinct and values.
- 5:
lsminp – IntegerInput
- 6:
lsmaxp – IntegerInput
On entry: are control parameters for the local approximations.
Each local approximation is computed on a local domain containing one of the
triangles in the discretization of the bounding box. The size of each local domain will be adaptively chosen such that if it contains fewer than
lsminp sample points it is expanded, else if it contains greater than
lsmaxp sample points a thinning method is applied.
lsmaxp mainly controls computational cost (in that working with a thinned set of points is cheaper and may be appropriate if the input data is densely distributed), while
lsminp allows handling of different types of scattered data.
Setting , and therefore forcing either expansion or thinning, may be useful for computing initial coarse approximations. In general smaller values for these arguments reduces cost.
A calibration procedure (experimenting with a small subset of the data to be fitted and validating the results) may be needed to choose the most appropriate values for
lsminp and
lsmaxp.
- 7:
nxcels – IntegerInput
- 8:
nycels – IntegerInput
On entry:
nxcels (respectively
nycels) is the number of cells in the
(respectively
) direction that will be used to create the triangulation of the bounding box of the domain of the function to be fitted.
Greater efficiency generally comes when
nxcels and
nycels are chosen to be of the same order of magnitude and are such that
n is
. Thus for a ‘square’ triangulation — when
— the quantities
and
nxcels should be of the same order of magnitude. See also
Section 9.
- 9:
lcoefs – IntegerInput
- 10:
coefs[lcoefs] – doubleOutput
On exit: if
NE_NOERROR on exit,
coefs contains the computed spline coefficients.
Constraint:
.
- 11:
iopts[] – IntegerCommunication Array
-
Note: the dimension,
, of this array is dictated by the requirements of associated functions that must have been previously called. This array MUST be the same array passed as argument
iopts in the previous call to
nag_fit_opt_set (e02zkc).
On entry: the contents of
iopts MUST NOT be modified in any way either directly or indirectly, by further calls to
nag_fit_opt_set (e02zkc), before calling either or both of the evaluation routines
nag_2d_spline_ts_eval (e02jec) and
nag_2d_spline_ts_eval_rect (e02jfc).
- 12:
opts[] – doubleCommunication Array
-
Note: the dimension,
, of this array is dictated by the requirements of associated functions that must have been previously called. This array MUST be the same array passed as argument
opts in the previous call to
nag_fit_opt_set (e02zkc).
On entry: the contents of
opts MUST NOT be modified in any way either directly or indirectly, by further calls to
nag_fit_opt_set (e02zkc), before calling either or both of the evaluation routines
nag_2d_spline_ts_eval (e02jec) and
nag_2d_spline_ts_eval_rect (e02jfc).
- 13:
fail – NagError *Input/Output
-
The NAG error argument (see
Section 3.6 in the Essential Introduction).
6 Error Indicators and Warnings
- NE_ALL_ELEMENTS_EQUAL
-
On entry, all elements of
x or of
y are equal.
- NE_ALLOC_FAIL
-
Dynamic memory allocation failed.
- NE_BAD_PARAM
-
On entry, argument had an illegal value.
- NE_INITIALIZATION
-
Option arrays are not initialized or are corrupted.
- NE_INT
-
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
- NE_INT_2
-
On entry, and .
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.
An unexpected algorithmic failure was encountered. Please contact
NAG.
- NE_INVALID_OPTION
-
The value of optional argument was invalid.
7 Accuracy
Technical results on error bounds can be found in
Davydov and Zeilfelder (2004).
Local approximation by polynomials of degree for data points has optimal
approximation order .
The approximation error for global smoothing is .
Whether maximal accuracy is achieved depends on the distribution of the input data and the choices of the algorithmic parameters. The
reference above contains
extensive numerical tests and further technical discussions of how best to configure the method.
8 Parallelism and Performance
nag_2d_spline_fit_ts_scat (e02jdc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_2d_spline_fit_ts_scat (e02jdc) 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
Users' Note for your implementation for any additional implementation-specific information.
-linear complexity and memory usage can be attained for sufficiently dense input data if the triangulation parameters
nxcels and
nycels are chosen as recommended in their descriptions above. For sparse input data on such triangulations, if many expansion steps are required (see
lsminp) the complexity may rise to be loglinear.
10 Example
The Franke function
is widely used for testing surface-fitting methods. The example program randomly generates a number of points on this surface. From these a spline is computed and then evaluated at a vector of points and on a mesh.
10.1 Program Text
Program Text (e02jdce.c)
10.2 Program Data
Program Data (e02jdce.d)
10.3 Program Results
Program Results (e02jdce.r)
11 Optional Arguments
Several optional arguments in nag_2d_spline_fit_ts_scat (e02jdc) control aspects of the algorithm, methodology used, logic or output. Their values are contained in the arrays
iopts and
opts; these must be initialized before calling nag_2d_spline_fit_ts_scat (e02jdc) by first calling
nag_fit_opt_set (e02zkc) with
optstr set to
"Initialize = e02jdc".
Each optional argument has an associated default value; to set any of them to a non-default value, or to reset any of them to the default value, use
nag_fit_opt_set (e02zkc). The current value of an optional argument can be queried using
nag_fit_opt_get (e02zlc).
The remainder of this section can be skipped if you wish to use the default values for all optional arguments.
The following is a list of the optional arguments available. A full description of each optional argument is provided in
Section 11.1.
11.1 Description of the Optional Arguments
For each option, we give a summary line, a description of the optional argument and details of constraints.
The summary line contains:
- the keywords;
- a parameter value,
where the letters , denote options that take character, integer and real values respectively;
- the default value.
Keywords and character values are case insensitive.
For nag_2d_spline_fit_ts_scat (e02jdc) the maximum length of the parameter
cvalue used by
nag_fit_opt_get (e02zlc) is
.
Averaged Spline | | Default |
When the bounding box is triangulated there are 8 equivalent configurations of the mesh. Setting will use the averaged value of the possible local polynomial approximations over each triangle in the mesh. This usually gives better results but at (about 8 times) higher computational cost.
Constraint: or .
Minimum Singular Value LPA | | Default |
A tolerance measure for accepting or rejecting a local polynomial approximation (LPA) as reliable.
The solution of a local least squares problem solved on each triangle subdomain is accepted as reliable if the minimum singular value of the matrix (of Bernstein polynomial values) associated with the least squares problem satisfies .
In general the approximation power will be reduced as is reduced. (A small indicates that the local data has hidden redundancies which prevent it from carrying enough information for a good approximation to be made.) Setting very large may have the detrimental effect that only approximations of low degree are deemed reliable.
will have no effect if , and it will have little effect if the input data is ‘smooth’ (e.g., from a known function).
A calibration procedure (experimenting with a small subset of the data to be fitted and validating the results) may be needed to choose the most appropriate value for this parameter.
Constraint:
.
Polynomial Starting Degree | |
Default |
The degree to be used in the initial step of each local polynomial approximation.
At the initial step the method will attempt to fit with local polynomials of degree . If the approximation is deemed unreliable (according to ), the degree will be decremented by one and a new local approximation computed, ending with a constant approximation if no other is reliable.
is bounded from above by the maximum possible spline
degree, .
The default value gives a good compromise between efficiency and accuracy. In general the best approximation can be obtained by
setting .
Constraint:
.