#include <AMA.h>

Functions
long int	AMA_UnvApprox (AMA_OPTIONS options, long int n, double x, double z, double epsilon, long int degree, AMA_SPLINE **spline)
	Approximation of Univariate Data More...

Function Documentation

long int AMA_UnvApprox	(	AMA_OPTIONS *	options,
		long int	n,
		double *	x,
		double *	z,
		double *	epsilon,
		long int	degree,
		AMA_SPLINE **	spline
	)

Approximation of Univariate Data

For a given set of data $(x_\ell,z_\ell)$ and approximation tolerances $\epsilon_\ell$ , for $\ell=1,\ldots,N$ , this function employs cnspla to compute the spline

$s(x) = \sum_{j=1}^{m}\alpha_jB_{d,j}(x|{\bf\Lambda})$

that minimizes

$F^{(2)}(s(x)) = \int_{x_1}^{x_N}\left( s^{(2)}(x) \right)^2 dx$

subject to the approximation constraints

$z_\ell - \epsilon_\ell \le s(x_\ell) \le z_\ell + \epsilon_\ell$

for $\ell=1,\ldots,N$ .

In the above definition of $s(x)$ the $\alpha_j$ , for $j=1,\ldots,m$ , are the coefficients of the $m$ univariate B-splines $B_{d,j}(x|{\bf\Lambda})$ of degree $1\le d\le 5$ which are based on the knot vector ${\bf\Lambda}$ . The knot vector ${\bf\Lambda}$ depends on the independent variable data and the degree and is defined by AMA_LamdaInterp().

This function does the following:

Checks the input parameters for validity, see AMA_UnvInputCheck().
Defines the spline based on a knot vector defined by AMA_LamdaInterp().
Defines CNSPLA_CONPNT constraints to represent the approximation constraints, see AMA_UnvConpnt().
Defines CNSPLA_PNLTRM condition to represent the function $F^{(2)}(s(x))$ , see AMA_UnvPnltrm().
Invokes cnspla to compute a spline which minimizes $F^{(2)}(s(x))$ subject to the approximation constraints.
Stores approximation into spline.

Note: By default the spline coefficients are initialized to zero but a different initial value for the spline coefficients can be set with AMA_OptionsSetCoefficients().

Parameters

options	[in] Pointer to AMA_OPTIONS. Must be initialized with AMA_Options() prior to calling AMA_UnvApprox().
n	[in] The number of data points . Must satisfy n $\ge 2$ .
x	[in] Array of size n containing the independent variable data $x_\ell$ , for $\ell=1,\ldots,N$ . Must be in strictly ascending order.
z	[in] Array of size n containing the dependent variable data $z_\ell$ , for $\ell=1,\ldots,N$ .
epsilon	[in] Array of size n containing the approximation tolerances $\epsilon_\ell$ , for $\ell=1,\ldots,N$ . Must satisfy $\epsilon_\ell\ge 0.0$ for all $\ell=1,\ldots,N$ . If epsilon = NULL, then this function uses $\epsilon_\ell = 0.0$ for all $\ell=1,\ldots,N$ .
degree	[in] The degree . Must satisfy $1\le$ degree $\le 5$ .
spline	[out] Pointer to AMA_SPLINE pointer containing the spline approximation. Must satisfy spline $\ne$ NULL.

Returns: Success/Error Code.

User Callable Function - Documented 101715

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