#include <AMA.h>

Functions
long int	AMA_SplineCheck (AMA_OPTIONS options, AMA_SPLINE spline)
	Check Validity of Spline Structure More...

long int	AMA_SplineFree (AMA_OPTIONS options, AMA_SPLINE *spline)
	Free a Spline Structure More...

long int	AMA_SplineIntegral (AMA_OPTIONS options, long int porder, AMA_SPLINE spline, AMA_SPLINE *integral)
	Analytic Integral of a Spline More...

long int	AMA_SplinePartial (AMA_OPTIONS options, long int porder, AMA_SPLINE spline, AMA_SPLINE *partial)
	Analytic Partial of a Spline More...

long int	AMA_SplinePartialValue (AMA_OPTIONS options, double x, long int porder, AMA_SPLINE spline, double *sval)
	Evaluate Partial Derivative of a Spline More...

long int	AMA_SplineRead (AMA_OPTIONS options, const char splfile, const char type, AMA_SPLINE *spline)
	Read a Spline More...

long int	AMA_SplineValue (AMA_OPTIONS options, double x, AMA_SPLINE spline, double sval)
	Evaluate a Spline More...

long int	AMA_SplineWrite (AMA_OPTIONS options, const char splfile, const char type, AMA_SPLINE spline)
	Write a Spline More...

double	AMA_SplineInfbnd (void)
	Define a finite number which represents an infinite bound. More...

Function Documentation

long int AMA_SplineCheck	(	AMA_OPTIONS *	options,
		AMA_SPLINE *	spline
	)

Check Validity of Spline Structure

This function determines if the AMA_SPLINE structure which represents the spline $s({\bf X}):{\rm R}^n\rightarrow {\rm R}^1$ given as

$s({\bf X}) = \sum_{j_n=1}^{m_n}\cdots\sum_{j_1=1}^{m_1}\alpha_{j_1,\ldots,j_n}B_{d_1,j_1}(x_1|{\bf\Lambda}_1)\cdots B_{d_n,j_n}(x_n|{\bf\Lambda_n})$

is properly defined. It insures the following conditions are satisfied:

$1\le n\le$ AMA_MXNIND;
$d_k\ge 0$ , for $1\le k\le n$ ;
$m_k\ge 2(d_k+1)$ , for $1\le k\le n$ .

Additionally, for each knot vector ${\bf\Lambda}_k$ , for $1\le k\le n$ , defined as

${\bf\Lambda}_k = ( \lambda_{k,1},\cdots,\lambda_{k,d_k+1}, \lambda_{k,d_k+2}, \cdots, \lambda_{k,m_k}, \lambda_{k,m_k+1},\cdots,\lambda_{k,m_k+d_k+1} )^T$

this function insures the following conditions are satisfied:

$\lambda_{k,1} = \ldots = \lambda_{k,d_k+1}$ ;
$\lambda_{k,i} \le \lambda_{k,i+1}$ , for $1\le i\le m_k+d_k$ ;
$\lambda_{k,i} < \lambda_{i+d_k+1}$ , for $1\le i\le m_k$ ; and
$\lambda_{k,m_k+1} = \ldots = \lambda_{k,m_k+d_k+1}$ .

Parameters

options	[in] Pointer to AMA_OPTIONS. Must be initialized with AMA_Options() prior to calling AMA_SplineCheck().
spline	[in] Pointer to AMA_SPLINE structure to check. Must satisfy spline $\ne$ NULL.

Returns: Success/Error Code.

User Callable Function - Documented 112514

long int AMA_SplineFree	(	AMA_OPTIONS *	options,
		AMA_SPLINE **	spline
	)

Free a Spline Structure

This function frees an AMA_SPLINE structure which represents the spline $s({\bf X}):{\rm R}^n\rightarrow {\rm R}^1$ given as

$s({\bf X}) = \sum_{j_n=1}^{m_n}\cdots\sum_{j_1=1}^{m_1}\alpha_{j_1,\ldots,j_n}B_{d_1,j_1}(x_1|{\bf\Lambda}_1)\cdots B_{d_n,j_n}(x_n|{\bf\Lambda_n}).$

This function does the following:

Frees AMA_SPLINE, see cnsplaSplineFree().
Frees AMA_SPLINE pointer.

Parameters

options	[in] Pointer to AMA_OPTIONS. Must be initialized with AMA_Options() prior to calling AMA_SplineFree().
spline	[in] Pointer to AMA_SPLINE pointer containing structure to free. Must satisfy spline $\ne$ NULL and be a valid AMA_SPLINE structure, see AMA_SplineCheck().

Returns: Success/Error Code.

User Callable Function - Documented 112514

double AMA_SplineInfbnd ( void )

Define a finite number which represents an infinite bound.

This function defines the finite number $\alpha_\infty = 0.01\times {\rm DBL\_EPSILON}^{-1}$ which represents an infinite bound. By default the univariate splines are subject to the bounds

$\alpha_l \le s(x) \le \alpha_u$

and the multivariate splines are subject to the bounds

$\alpha_l \le s({\bf X}) \le \alpha_u,$

where $\alpha_l = -\alpha_\infty$ and $\alpha_u = \alpha_\infty$ . Therefore, the splines are unbounded but finite bounds can be set with AMA_OptionsSetBounds().

Returns: $\alpha_\infty$ .

User Support Function - Documented 100415 - !!!THIS IS NOT A USER CALLABLE FUNCTION - DOCUMENT IS INCLUDED FOR COMPLETENESS!!!

long int AMA_SplineIntegral	(	AMA_OPTIONS *	options,
		long int *	porder,
		AMA_SPLINE *	spline,
		AMA_SPLINE **	integral
	)

Analytic Integral of a Spline

For a spline $s({\bf X}):{\rm R}^n\rightarrow {\rm R}^1$ given as

$s({\bf X}) = \sum_{j_n=1}^{m_n}\cdots\sum_{j_1=1}^{m_1}\alpha_{j_1,\ldots,j_n}B_{d_1,j_1}(x_1|{\bf\Lambda}_1)\cdots B_{d_n,j_n}(x_n|{\bf\Lambda_n})$

this function computes the indefinite integral

$\int\cdots\int s({\bf X}) dx_1\cdots dx_n.$

The integral with respect to $x_k$ is computed $p_k$ times, for $1\le k\le n$ ; where $p_k = 0$ or $p_k = 1$ .

Parameter Note: In the parameter descriptions given below the limits on $k$ are $1\le k\le n$ and k $= k-1$ .

Parameters

options	[in] Pointer to AMA_OPTIONS. Must be initialized with AMA_Options() prior to calling AMA_SplineIntegral().
porder	[in] Array of size containing the integral order where porder[k] . Must satisfy porder[k] or porder[k] .
spline	[in] Pointer to AMA_SPLINE containing the spline to integrate. Must be a valid AMA_SPLINE structure, see AMA_SplineCheck().
integral	[out] Pointer to AMA_SPLINE pointer containing the spline's indefinite integral. Must satisfy integral $\ne$ NULL.

Returns: Success/Error Code.

User Callable Function - Documented 021116

long int AMA_SplinePartial	(	AMA_OPTIONS *	options,
		long int *	porder,
		AMA_SPLINE *	spline,
		AMA_SPLINE **	partial
	)

Analytic Partial of a Spline

For a spline $s({\bf X}):{\rm R}^n\rightarrow {\rm R}^1$ given as

$s({\bf X}) = \sum_{j_n=1}^{m_n}\cdots\sum_{j_1=1}^{m_1}\alpha_{j_1,\ldots,j_n}B_{d_1,j_1}(x_1|{\bf\Lambda}_1)\cdots B_{d_n,j_n}(x_n|{\bf\Lambda_n})$

this function computes the partial derivative

${\partial^{q} s({\bf X})\over \partial x^{p_1}_1\cdots\partial x^{p_n}_n}$

where the partial order $q=\sum_{k=1}^np_k$ and $p_k$ is the partial order in $x_k$ , for $1\le k\le n$ .

Parameter Note: In the parameter descriptions given below the limits on $k$ are $1\le k\le n$ and k $= k-1$ .

Parameters

options	[in] Pointer to AMA_OPTIONS. Must be initialized with AMA_Options() prior to calling AMA_SplinePartial().
porder	[in] Array of size containing the partial order where porder[k] . Must satisfy porder[k] $\ge 0$ .
spline	[in] Pointer to AMA_SPLINE containing the spline to differentiate. Must be a valid AMA_SPLINE structure, see AMA_SplineCheck().
partial	[out] Pointer to AMA_SPLINE pointer containing the spline's partial derivative. Must satisfy partial $\ne$ NULL.

Returns: Success/Error Code.

User Callable Function - Documented 021116

long int AMA_SplinePartialValue	(	AMA_OPTIONS *	options,
		double *	x,
		long int *	porder,
		AMA_SPLINE *	spline,
		double *	sval
	)

Evaluate Partial Derivative of a Spline

For a spline $s({\bf X}):{\rm R}^n\rightarrow {\rm R}^1$ given as

$s({\bf X}) = \sum_{j_n=1}^{m_n}\cdots\sum_{j_1=1}^{m_1}\alpha_{j_1,\ldots,j_n}B_{d_1,j_1}(x_1|{\bf\Lambda}_1)\cdots B_{d_n,j_n}(x_n|{\bf\Lambda_n})$

this function evaluates the partial derivative

${\partial^{q}s({\bf X}) \over \partial x_1^{p_1}\cdots\partial x_n^{p_n}}$

of the spline at a point ${\bf X}^o = (x^o_1,\cdots,x^o_n)$ . The partial order $q$ is $q = \sum_{k=1}^np_k$ where $p_k$ is the partial order in $u_k$ , for $1\le k\le n$ . The value of ${\bf X}^o$ must lie within the spline's domain; that is, if the knot vector ${\bf\Lambda}_k$ is defined as

${\bf\Lambda}_k = ( \lambda_{k,1},\cdots,\lambda_{k,d_k+1}, \lambda_{k,d_k+2}, \cdots, \lambda_{k,m_k}, \lambda_{k,m_k+1},\cdots,\lambda_{k,m_k+d_k+1} )^T,$

then $x^o_k$ must satisfy $\lambda_{k,1}\le x^o_k \le \lambda_{k,m_k+d_k+1}$ , for all $1\le k\le n$ . However, extrapolation can be enabled with AMA_OptionsSetExtrapolation().

Parameter Note: In the parameter descriptions given below the limits on $k$ are $1\le k\le n$ and k $= k-1$ .

Parameters

options	[in] Pointer to AMA_OPTIONS. Must be initialized with AMA_Options() prior to calling AMA_SplinePartialValue().
x	[in] Array of size containing the evaluation point ${\bf X}^o$ where x[k] .
porder	[in] Array of size containing the partial order where porder[k] . Must satisfy $0\le$ porder[k] $\le d_k$ . However, if the spline is linear in , ; then porder[k] can satisfy $0\le$ porder[k] $\le 2$ .
spline	[in] Pointer to AMA_SPLINE containing the spline whose partial derivative is evaluated. Must satisfy spline $\ne$ NULL.
sval	[out] The value of the partial derivative of $s({\bf X})$ at ${\bf X}^o$ . Must satisfy sval $\ne$ NULL.

Returns: Success/Error Code.

Note: For efficiency purposes this function does not check if spline is a valid AMA_SPLINE structure. Also, except for insuring the options parameter is properly initialized, all input error checking can be disabled with AMA_OptionsSetInputCheck().; If ${\bf X}^o$ lies outside of the spline's domain; then this function returns AMA_EXTRAPOLATION if extrapolation is enabled and AMA_INPUT_ERROR, otherwise.; If an input error occurs, then this function returns sval equal to DBL_MAX.

User Callable Function - Documented 021116

long int AMA_SplineRead	(	AMA_OPTIONS *	options,
		const char *	splfile,
		const char *	type,
		AMA_SPLINE **	spline
	)

Read a Spline

This function allocates and reads a spline from either an ascii or binary file.

Parameters

options	[in] Pointer to AMA_OPTIONS. Must be initialized with AMA_Options() prior to calling AMA_SplineRead().
splfile	[in] Name of file from which spline is read.
type	[in] Format of splfile. Must equal either ascii or binary. The value of type is not case sensitive.
spline	[out] Pointer to AMA_SPLINE pointer containing the spline read from splfile. Must satisfy spline $\ne$ NULL.

Returns: Success/Error Code.

User Callable Function - Documented 021116

long int AMA_SplineValue	(	AMA_OPTIONS *	options,
		double *	x,
		AMA_SPLINE *	spline,
		double *	sval
	)

Evaluate a Spline

For a spline $s({\bf X}):{\rm R}^n\rightarrow {\rm R}^1$ given as

$s({\bf X}) = \sum_{j_n=1}^{m_n}\cdots\sum_{j_1=1}^{m_1}\alpha_{j_1,\ldots,j_n}B_{d_1,j_1}(x_1|{\bf\Lambda}_1)\cdots B_{d_n,j_n}(x_n|{\bf\Lambda_n})$

this function evaluates the spline at a point ${\bf X}^o = (x^o_1,\cdots,x^o_n)$ . The value of ${\bf X}^o$ must lie within the spline's domain; that is, if the knot vector ${\bf\Lambda}_k$ is defined as

${\bf\Lambda}_k = ( \lambda_{k,1},\cdots,\lambda_{k,d_k+1}, \lambda_{k,d_k+2}, \cdots, \lambda_{k,m_k}, \lambda_{k,m_k+1},\cdots,\lambda_{k,m_k+d_k+1} )^T,$

then $x^o_k$ must satisfy $\lambda_{k,1}\le x^o_k \le \lambda_{k,m_k+d_k+1}$ , for all $1\le k\le n$ . However, extrapolation can be enabled with AMA_OptionsSetExtrapolation().

Parameters

options	[in] Pointer to AMA_OPTIONS. Must be initialized with AMA_Options() prior to calling AMA_SplineValue().
x	[in] Array of size containing the evaluation point ${\bf X}^o$ where x[k] , for $1\le k\le n$ and k .
spline	[in] Pointer to AMA_SPLINE containing the spline to evaluate. Must satisfy spline $\ne$ NULL.
sval	[out] The value of $s({\bf X}^o)$ . Must satisfy sval $\ne$ NULL.

Returns: Success/Error Code.

Note: For efficiency purposes this function does not check if spline is a valid AMA_SPLINE structure. Also, except for insuring the options parameter is properly initialized, all input error checking can be disabled with AMA_OptionsSetInputCheck().; If ${\bf X}^o$ lies outside of the spline's domain; then this function returns AMA_EXTRAPOLATION if extrapolation is enabled and AMA_INPUT_ERROR, otherwise.; If an input error occurs, then this function returns sval equal to DBL_MAX.

User Callable Function - Documented 021116

long int AMA_SplineWrite	(	AMA_OPTIONS *	options,
		const char *	splfile,
		const char *	type,
		AMA_SPLINE *	spline
	)

Write a Spline

This function writes a spline to either an ascii or binary file.

Parameters

options	[in] Pointer to AMA_OPTIONS. Must be initialized with AMA_Options() prior to calling AMA_SplineWrite().
splfile	[in] Name of file to which spline is written.
type	[in] Format of splfile. Must equal either ascii or binary. The value of type is not case sensitive.
spline	[out] Pointer to AMA_SPLINE containing the spline to write to splfile. Must satisfy spline $\ne$ NULL and be a valid AMA_SPLINE structure, see AMA_SplineCheck().

Returns: Success/Error Code.

User Callable Function - Documented 021116

Functions

Function Documentation