Functions
AMA_Mltv.c File Reference
#include <AMA.h>

Functions

long int AMA_MltvBoundaryConditions (AMA_OPTIONS *options, enum AMA_Boolean interp, CNSPLA_SPLFUN *splfun)
 Define boundary conditions for multivariate approximation and interpolation functions. More...
 
long int AMA_MltvData (AMA_OPTIONS *options, const char *datfile, long int *nind, long int *degree, long int *n, double ***x, double **z, double **epsilon, double **wht, double *theta, long int *mlamda, double ***lamda)
 Read data and approximation options for AMA Spline Library Multivariate Random Data Functions. More...
 
long int AMA_MltvDataFree (AMA_OPTIONS *options, long int nind, double ***x, double **z, double **epsilon, double **wht, double ***lamda)
 Free Multivariate Data Functions data arrays. More...
 
long int AMA_MltvGrdData (AMA_OPTIONS *options, const char *datfile, long int *nind, long int *degree, long int *ng, double ***x, double **z, double **epsilon, double **wht, double *theta, long int *mlamda, double ***lamda)
 Read data and approximation options for AMA Spline Library Multivariate Gridded Data Functions. More...
 
long int AMA_MltvGrdInputCheck (AMA_OPTIONS *options, long int nind, long int *ng, double **x, double *z, double *wht, double *epsilon, long int *degree, long int *mlamda, double **lamda, enum AMA_Boolean lstsqr, enum AMA_Boolean *mltknt)
 Perform input check for multivariate gridded data. More...
 
long int AMA_MltvInputCheck (AMA_OPTIONS *options, long int nind, long int n, double **x, double *z, double *wht, double *epsilon, long int *degree, long int *mlamda, double **lamda, enum AMA_Boolean lstsqr)
 Perform input check for multivariate data approximation functions. More...
 
long int AMA_MltvPnltrm (AMA_OPTIONS *options, double theta, long int porder, CNSPLA_SPLFUN *splfun)
 Define penalty term on a multivariate spline. More...
 
long int AMA_MltvConpnt (AMA_OPTIONS *options, long int nind, long int n, double **x, double *z, double *epsilon, CNSPLA_SPLFUN *splfun)
 Define approximation or interpolation constraints on a multivariate spline. More...
 
long int AMA_MltvConreg (AMA_OPTIONS *options, long int nind, long int n, double **x, double *z, double *epsilon, long int *degree, CNSPLA_SPLFUN *splfun)
 Define monotonicity constraints on a multivariate spline. More...
 

Function Documentation

long int AMA_MltvBoundaryConditions ( AMA_OPTIONS *  options,
enum AMA_Boolean  interp,
CNSPLA_SPLFUN *  splfun 
)

Define boundary conditions for multivariate approximation and interpolation functions.

Parameters
options[in] Pointer to AMA_OPTIONS. Should be initialized with AMA_Options() prior to calling AMA_MltvBoundaryConditions().
interp[in] Interpolation function flag. It has one of the following two values:
splfun[in] Pointer to CNSPLA_SPLFUN containing the spline $s(x)$ upon which the penalty term is imposed. Should satisfy splfun $\ne$ NULL.
Returns
Success/Error Code.

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

long int AMA_MltvConpnt ( AMA_OPTIONS *  options,
long int  nind,
long int  n,
double **  x,
double *  z,
double *  epsilon,
CNSPLA_SPLFUN *  splfun 
)

Define approximation or interpolation constraints on a multivariate spline.

For a given set of independent variable data ${\bf X}_\ell=(x_{1,\ell},\cdots,x_{n,\ell})$, dependent variable data $z_\ell$ and approximation tolerances $\epsilon_\ell$, for $\ell=1,\ldots,N$, this function defines either the approximation constraints

\[ z_\ell - \epsilon_\ell \le s({\bf X}_\ell) \le z_\ell + \epsilon_\ell \]

or the interpolation constraints

\[ s({\bf X}_\ell) = z_\ell, \]

for $\ell=1,\ldots,N$, on a multivariate spline $s({\bf X}):{\rm R}^n\rightarrow {\rm R}^1$ given as

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

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

Parameters
options[in] Pointer to AMA_OPTIONS. Should be initialized with AMA_Options() prior to calling AMA_MltvConpnt().
nind[in] The number of independent variables $n$. Should satisfy nind $\ge 2$.
n[in] The number of data points $N$. Should satisfy n $\ge 2$.
x[in] Array of size nind containing arrays of size n where x[k] contains the independent variable data $x_{k,\ell}$, for $\ell=1,\ldots,N$.
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 $\ell=1,\ldots,N$. Should satisfy $\epsilon_\ell\ge 0.0$ for all $\ell=1,\ldots,N$. If epsilon = NULL, then the interpolation constraints are defined. Otherwise, the approximation constraints are defined.
splfun[in] Pointer to CNSPLA_SPLFUN containing spline $s({\bf X})$ upon which approximation or interpolation constraints are imposed. Should satisfy splfun $\ne$ NULL.
Returns
Success/Error Code.

User Support Function - Documented 110914 - !!!THIS IS NOT A USER CALLABLE FUNCTION - DOCUMENT IS INCLUDED FOR COMPLETENESS!!!

long int AMA_MltvConreg ( AMA_OPTIONS *  options,
long int  nind,
long int  n,
double **  x,
double *  z,
double *  epsilon,
long int *  degree,
CNSPLA_SPLFUN *  splfun 
)

Define monotonicity constraints on a multivariate spline.

For a given set of independent variable data ${\bf X}_\ell=(x_{1,\ell},\cdots,x_{n,\ell})$, dependent variable data $z_\ell$ and approximation tolerances $\epsilon_\ell$, for $\ell=1,\ldots,N$, this function defines local monotonicity constraints on a multivariate 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 first defines the rectilinear grid $(x^g_{1,1},x^g_{1,2},\cdots,x^g_{1,N^g_1-1},x^g_{1,N^g_1})\times\cdots\times(x^g_{n,1},x^g_{n,2},\cdots,x^g_{n,N^g_n-1},x^g_{n,N^g_n})$ upon which the independent variable data lies and then it defines the gridded dependent variable data $z^g_{l_1,\cdots,l_n}$ and the gridded approximation tolerances $\epsilon^g_{l_1,\cdots,l_n}$ for $1\le l_k\le N^g_k$ and $1\le k\le n$. If the independent variable data defines a gridded data distribution, then $N=\prod_{k=1}^n N^g_k$. If it defines a gridded data with holes distribution, then $N<\prod_{k=1}^n N^g_k$; that is, dependent variable data does not lie everywhere on the grid. In the latter case the local monotonicity constraints are not defined in regions of missing data.

After determining the data's underlying rectilinear grid this function imposes constraints on the first order partials ${\partial s({\bf X})\over \partial x_k}$, for all $1\le k\le n$, which insure the spline satisfies the local monotonicity constraints. The form of the constraints depends on the approximation tolerances.

If $\max_\ell\lbrace{\epsilon_\ell\rbrace}=0.0$, then for some $1\le k_o\le n$ a local monotonicity constraint is defined for all $1\le l_k\le N^g_k$ with $k \ne k_o$ and it is of the form

\[ {\partial s(x^g_{l_1},\cdots,x^g_{l_{k_o-1}},x_{k_o},x^g_{l_{k_o}+1},\cdots,x^g_{l_n})\over\partial x_{k_o}} \hspace{5pt} \cases { \ge 0.0 {\rm \hspace{5pt}for\hspace{5pt}all\hspace{5pt}} x_{k_o}\in[x^g_{l_{k_o}},x^g_{l_{k_o}+1}], & if $z^g_{l_{k_o}} < z^g_{l_{k_o}+1}$; \cr \le 0.0 {\rm \hspace{5pt}for\hspace{5pt}all\hspace{5pt}} x_{k_o}\in[x^g_{l_{k_o}},x^g_{l_{k_o}+1}], & if $z^g_{l_{k_o}} > z^g_{l_{k_o}+1}$; \cr = 0.0 {\rm \hspace{5pt}for\hspace{5pt}all\hspace{5pt}} x_{k_o}\in[x^g_{l_{k_o}},x^g_{l_{k_o}+1}], & otherwise.\cr } \]

The above constraint is defined for all $l_{k_o} = 1,\ldots, N^g_{k_o}-1$. The notation $z^g_{l_{k_o}}$ and $z^g_{l_{k_o}+1}$ refers to the dependent variable data associated with the grid points $(x^g_{l_1},\cdots,x^g_{l_{k_o-1}},x^g_{l_{k_o}},x^g_{l_{k_o}+1},\cdots,x^g_{l_n})$ and $(x^g_{l_1},\cdots,x^g_{l_{k_o-1}},x^g_{l_{k_o}},x^g_{l_{k_o}+1},\cdots,x^g_{l_n})$, respectively. If data does not lie at both grid points, then the constraint is not defined.

Alternatively, if there exists an $1\le \ell\le N$ for which $\epsilon_\ell\ne 0.0$, then for some $1\le k_o\le n$ a local monotonicity constraint is defined for all $1\le l_k\le N^g_k$ with $k \ne k_o$ and it is of the form

\[ {\partial s(x^g_{l_1},\cdots,x^g_{l_{k_o}-1},x_{k_o},x^g_{l_{k_o}+1},\cdots,x^g_{l_n})\over\partial x_{k_o}} \hspace{5pt} \cases { \ge 0.0 {\rm \hspace{5pt}for\hspace{5pt}all\hspace{5pt}} x_{k_o}\in[x^g_{l_{k_o}},x^g_{l_{k_o}+1}], & if $z^g_{l_{k_o}} + \epsilon^g_{l_{k_o}} < z^g_{l_{k_o}+1} - \epsilon^g_{l_{k_o}+1}$; \cr \le 0.0 {\rm \hspace{5pt}for\hspace{5pt}all\hspace{5pt}} x_{k_o}\in[x^g_{l_{k_o}},x^g_{l_{k_o}+1}], & if $z^g_{l_{k_o}} - \epsilon^g_{l_{k_o}} > z^g_{l_{k_o}+1} + \epsilon^g_{l_{k_o}+1}$. \cr } \]

The above constraint is defined for all $l_{k_o} = 1,\ldots, N^g_{k_o}-1$. The notation $\epsilon^g_{l_{k_o}}$ and $\epsilon^g_{l_{k_o}+1}$ refers to the approximation tolerances associated with the grid points $(x^g_{l_1},\cdots,x^g_{l_{k_o}-1},x^g_{l_{k_o}},x^g_{l_{k_o}+1},\cdots,x^g_{l_n})$ and $(x^g_{l_1},\cdots,x^g_{l_{k_o}-1},x^g_{l_{k_o}+1},x^g_{l_{k_o}+1},\cdots,x^g_{l_n})$, respectively. If data does not lie at both grid points, then the constraint is not defined.

If for some $1\le l_{k_o}\le N^g_{k_o}-1$ the above conditions are not met, then the intervals $[z^g_{l_{k_o}} - \epsilon^g_{l_{k_o}}, z^g_{l_{k_o}} + \epsilon^g_{l_{k_o}}]$ and $[z^g_{l_{k_o}+1} - \epsilon^g_{l_{k_o}+1}, z^g_{l_{k_o}+1} + \epsilon^g_{l_{k_o}+1}]$ intersect and the equality constraint

\[ {\partial s(x^g_{l_1},\cdots,x^g_{l_{k_o}-1},x_{k_o},x^g_{l_{k_o}+1},\cdots,x^g_{l_n})\over\partial x_{k_o}} = 0.0 \]

can be imposed over the interval $[x^g_{l_{k_o}},x^g_{l_{k_o}+1}]$. However, it is possible for the intervals $[z^g_q - \epsilon^g_q, z^g_q + \epsilon^g_q]$ and $[z^g_{q+1} - \epsilon^g_{q+1}, z^g_{q+1} + \epsilon^g_{q+1}]$ to intersect, for each $l_{k_o}\le q\le l_{k_o}+r$ and $r\ge 1$; without an intersection existing between all the intervals. In this case the aforementioned equality constraint is inconsistent with the approximation constraints

\[ z^g_q - \epsilon^g_q \le s(x^g_{l_1},\cdots,x^g_{l_{k_o}-1},x_q,x^g_{l_{k_o}+1},\cdots,x^g_{l_n}) \le z^g_q + \epsilon^g_q, \]

for $l_{k_o}\le q\le l_{k_o}+r+1$; and, therefore, can not be imposed over the interval $[x^g_{l_{k_o}},x^g_{l_{k_o}+r+1}]$. Hence, instead of imposing an equality constraint this function minimizes

\[ \Biggr\Vert {\partial s(x^g_{l_1},\cdots,x^g_{l_{k_o}-1},x_{k_o},x^g_{l_{k_o}+1},\cdots,x^g_{l_n})\over\partial x_{k_o}} \Biggr\Vert_\infty \]

over all intervals $[x^g_{l_{k_o}},x^g_{l_{k_o}+1}]$ for which the intervals $[z^g_{l_{k_o}} - \epsilon^g_{l_{k_o}}, z^g_{l_{k_o}}+ \epsilon^g_{l_{k_o}}]$ and $[z^g_{l_{k_o}+1} - \epsilon^g_{l_{k_o}+1}, z^g_{l_{k_o}+1} + \epsilon^g_{l_{k_o}+1}]$ intersect.

Note
By default the local monotonicity constraints are imposed on the spline in all of its independent variables but the constraints can be disabled in one or more of its independent variables with AMA_OptionsSetMonotonicity().

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

Parameters
options[in] Pointer to AMA_OPTIONS. Should be initialized with AMA_Options() prior to calling AMA_MltvConreg().
nind[in] The number of independent variables $n$. Should satisfy nind $\ge 2$.
n[in] The number of data points $N$. Should satisfy n $\ge 2$.
x[in] Array of size nind containing arrays of size n where x[k] contains the independent variable data $x_{k,\ell}$, for $\ell=1,\ldots,N$.
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 $\ell=1,\ldots,N$. Should 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] Array of size nind containing the degree $d_k$ where degree[k] $= d_k$. Should satisfy $1\le$ degree[k] $\le 5$.
splfun[in] Pointer to CNSPLA_SPLFUN containing spline $s({\bf X})$ upon which monotonicity constraints are imposed. Should satisfy splfun $\ne$ NULL.
Returns
Success/Error Code.

User Support Function - Documented 121114 - !!!THIS IS NOT A USER CALLABLE FUNCTION - DOCUMENT IS INCLUDED FOR COMPLETENESS!!!

long int AMA_MltvData ( AMA_OPTIONS *  options,
const char *  datfile,
long int *  nind,
long int *  degree,
long int *  n,
double ***  x,
double **  z,
double **  epsilon,
double **  wht,
double *  theta,
long int *  mlamda,
double ***  lamda 
)

Read data and approximation options for AMA Spline Library Multivariate Random Data Functions.

This function reads data and approximation options for AMA Spline Library functions which compute spline approximations of independent variable data ${\bf X}\in{\rm R}^n$ and dependent variable data ${\bf Z}\in{\rm R}^1$.

The argument datfile should reference a readable file which consists of a data section and several, optional, approximation options sections. The Data section must preceed the approximation options sections and has the following structure:

\[ \begin{array}{llllll} {\bf Data} & & & & & \cr {\bf Nind:}n &{\bf N:}N & & & & \cr {\bf Degree:}d_1 & & & & & \cr \vdots & & & & & \cr {\bf Degree:}d_n & & & & & \cr {\bf X_1} &\cdots &{\bf X}_n &{\bf Z} &{\bf Epsilon} &{\bf Wht} \cr x_{1,1} &\cdots &x_{n,1} &z_1 &\epsilon_1 &w_1 \cr x_{1,2} &\cdots &x_{n,2} &z_2 &\epsilon_2 &w_2 \cr \vdots &\vdots &\vdots &\vdots &\vdots &\vdots \cr x_{1,N} &\cdots &x_{n,N} &z_N &\epsilon_N &w_N \cr {\bf End\_Data} & & & & & \cr \end{array} \]

where the Epsilon and Wht columns are optional. The data must satisfy the following conditions:

  • $2\le n\le$ AMA_MXNIND;
  • $1\le d_k \le 5$, for $1\le k\le n$;
  • $N\ge 2$;
  • the approximation tolerances must satisfy $\epsilon_\ell\ge 0.0$, for $\ell=1,\cdots,N$;
  • the weights must satisfy $w_\ell\ge 0.0$, for $\ell=1,\cdots,N$.

Following the Data section may be one or more of the Bounds, Least_Squares, Monotonicity or Penalty_Term sections. If an approximation options section is not defined, then this function sets the options to their default values. See Table Approximation Options Defaults for a list of Multivariate Random Data Functions approximation options default values.

The Bounds section specifies the lower and upper bounds. It has the following structure:

\[ \begin{array}{ll} {\bf Bounds} & \cr {\bf Lwrbnd:}\alpha_l &{\bf Uprbnd:}\alpha_u \cr {\bf End\_Bounds} & \cr \end{array} \]

and its options must satisfy the following conditions:

The lower bound is read as a string lwrstr and based on the value of lwrstr the value of $\alpha_l$ is set as follows:

  • If lwrstr equals infbnd, then $\alpha_l = -\alpha_\infty$ where $\alpha_\infty$ = AMA_SplineInfbnd(); or,
  • If lwrstr equals zmin , then $\alpha_l = \min_\ell\lbrace{z_\ell\rbrace}$.
  • Otherwise, $\alpha_l =$ atof(lwrstr).

Similarly, the upper bound is read as a string uprstr and based on the value of uprstr the value of $\alpha_u$ is set as follows:

  • If uprstr equals infbnd, then $\alpha_u = \alpha_\infty$ where $\alpha_\infty$ = AMA_SplineInfbnd(); or,
  • If uprstr equals zmax , then $\alpha_u = \max_\ell\lbrace{z_\ell\rbrace}$.
  • Otherwise, $\alpha_u =$ atof(uprstr).

The Least_Squares section specifies the approximation options and knots employed by AMA_MltvLstsqr(). It has the following structure:

\[ \begin{array}{ll} {\bf Least\_Squares} & \cr {\bf Theta:}\theta &{\bf Minorm:}\rm minorm \cr {\bf Mlamda:}m_1 & \cr \lambda^o_{1,1} & \cr \lambda^o_{1,2} & \cr \vdots & \cr \lambda^o_{1,m_1} & \cr \vdots & \cr {\bf Mlamda:}m_n & \cr \lambda^o_{n,1} & \cr \lambda^o_{n,2} & \cr \vdots & \cr \lambda^o_{n,m_n} & \cr {\bf End\_Least\_Squares} & \cr \end{array} \]

and its options must satisfy the following conditions:

  • $0.0\le\theta< 1.0$;
  • ${\rm minorm}$ must be Enabled or Disabled;
  • $m_k\ge 2$, for $1\le k\le n$;
  • the knots $\lambda^o_{k,i}$, for $i=1,\cdots,m_k$ and $1\le k\le n$, must be in increasing order;
  • $\lambda^o_{k,1} \le x_{k,1}$ and $\lambda^o_{k,m_k} \ge x_{k,N^g_k}$, for all $1\le k\le n$.

The Monotonicity section specifies the monotonicity constraints and continuity conditions employed by AMA_MltvMonoApprox() and AMA_MltvMonoInterp(). It has the the following structure:

\[ \begin{array}{llll} {\bf Monotonicity} & & & \cr {\bf Monpos:\rm monpos[1]} &{\bf Monneg:\rm monneg[1]} &{\bf Monzer:\rm monzer[1]} &{\bf Concnd:\rm concnd[1]} \cr \vdots &\vdots &\vdots &\vdots \cr {\bf Monpos:\rm monpos[n]} &{\bf Monneg:\rm monneg[n]} &{\bf Monzer:\rm monzer[n]} &{\bf Concnd:\rm concnd[n]} \cr {\bf End\_Monotonicity} & & & \cr \end{array} \]

and its options must satisfy the following conditions:

  • the positive monotonicity constraint flag ${\rm monpos}[k]$, negative monotonicity constraint flag ${\rm monneg}[k]$ and zero monotonicity constraint flag ${\rm monzer}[k]$ must be Enabled or Disabled, for $1\le k\le n$;
  • the continuity condition ${\rm concnd}[k]$ must be Full or Reduced, for $1\le k\le n$;

The Penalty_Term section specifies which cross partial terms are used in the penalty term. It has the following structure:

\[ \begin{array}{l} {\bf Penalty\_Term} \cr {\bf Partial\_1\_2:\rm pnltrm[1][2]} \cr \vdots \cr {\bf Partial\_1\_n:\rm pnltrm[1][n]} \cr \vdots \cr {\bf Partial\_n-1\_n:\rm pnltrm[n-1][n]} \cr {\bf End\_Penalty\_Term} \cr \end{array} \]

and its options must satisfy the following condition:

  • ${\rm pnltrm[k][l]}$ must be Include or Exclude; for $1\le k\le n-1$ and $k+1\le l\le n$.

The bold keywords are case sensitive and the string values for the approximation options are case insensitive.

This function performs the following tasks:

Parameter Note: In the parameter definitions 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_MltvData().
datfile[in] The data file name. Must satisfy datfile $\ne$ NULL.
nind[out] The number of independent variables $n$. Must satisfy nind $\ne$ NULL.
degree[out] Array of size nind containing the degree $d_k$ where degree[k] $= d_k$. Must satisfy degree $\ne$ NULL.
n[out] The number of data points $N$. Must satisfy n $\ne$ NULL.
x[out] Array of size nind containing arrays of size n where x[k] contains the independent variable data $x_{k,\ell}$, for $\ell=1,\ldots,N$. Must satisfy x $\ne$ NULL.
z[out] Array of size n containing the dependent variable data $z_\ell$, for $\ell=1,\ldots,N$. Must satisfy z $\ne$ NULL.
epsilon[out] Array of size n containing the approximation tolerances $\epsilon_\ell$, for $\ell=1,\ldots,N$. Must satisfy epsilon $\ne$ NULL.
wht[out] Array of size n containing the weights $w_\ell$, for $\ell=1,\ldots,N$. Must satisfy wht $\ne$ NULL.
theta[out] The penalty term weight $\theta$. Must satisfy theta $\ne$ NULL.
mlamda[out] Array of size nind containing the number of knots $m_k$ where mlamda[k] $= m_k$. Must satisfy mlamda $\ne$ NULL.
lamda[out] Array of size nind containing arrays of size mlamda[k] where lamda[k] contains the knots $\lambda^o_{i,k}$, for $i=1,\ldots,m_k$. Must satisfy lambda $\ne$ NULL.
Returns
Success/Error Code.

User Support Function - Documented 110315 - !!!THIS IS NOT A USER CALLABLE FUNCTION - DOCUMENT IS INCLUDED FOR COMPLETENESS!!!

long int AMA_MltvDataFree ( AMA_OPTIONS *  options,
long int  nind,
double ***  x,
double **  z,
double **  epsilon,
double **  wht,
double ***  lamda 
)

Free Multivariate Data Functions data arrays.

This function frees the arrays allocated by AMA_MltvData() and AMA_MltvGrdData().

Parameters
options[in] Pointer to AMA_OPTIONS. Must be initialized with AMA_Options() prior to calling AMA_MltvDataFree().
nind[out] The number of independent variables $n$. Must satisfy nind $\ge 2$.
x[out] Pointer to array of size nind containing the independent variable data arrays.
z[out] Pointer to array containing the dependent variable data.
epsilon[out] Pointer to array containing the approximation tolerances.
wht[out] Pointer to array containing the weights.
lamda[out] Pointer to array of size nind containing the knot arrays.
Returns
Success/Error Code.

User Support Function - Documented 110515 - !!!THIS IS NOT A USER CALLABLE FUNCTION - DOCUMENT IS INCLUDED FOR COMPLETENESS!!!

long int AMA_MltvGrdData ( AMA_OPTIONS *  options,
const char *  datfile,
long int *  nind,
long int *  degree,
long int *  ng,
double ***  x,
double **  z,
double **  epsilon,
double **  wht,
double *  theta,
long int *  mlamda,
double ***  lamda 
)

Read data and approximation options for AMA Spline Library Multivariate Gridded Data Functions.

This function reads data for AMA Spline Library functions which compute spline approximations of independent variable data ${\bf X}\in{\rm R}^n$ and dependent variable data ${\bf Z}\in{\rm R}^1$. The independent variable data defines the rectilinear grid

\[ (x_{1,1},x_{1,2},\cdots,x_{1,N^g_1-1},x_{1,N^g_1})\times\cdots\times(x_{n,1},x_{n,2},\cdots,x_{n,N^g_n-1},x_{n,N^g_n}) \]

where $N^g_k\ge 2$ and $1\le k\le n$. The dependent variable data lies on the grid and is given as $z_{\ell_1,\cdots,\ell_n}$ for $\ell_1=1,\ldots,N^g_1,\cdots,\ell_n=1,\ldots,N^g_n$; that is, there are $N=\prod_{k=1}^nN^g_k$ dependent variable values. Similarly, the approximation tolerances $\epsilon_{\ell_1,\cdots,\ell_n}$ for $\ell_1=1,\ldots,N^g_1,\cdots,\ell_n=1,\ldots,N^g_n$ lie on the grid.

The argument datfile should reference a readable file which consists of a data section and several, optional, approximation options sections. The Data section must preceed the approximation options sections and has the following structure:

\[ \begin{array}{lll} {\bf Data} & & \cr {\bf Nind:}n & & \cr {\bf Degree:}d_1 & & \cr \vdots & & \cr {\bf Degree:}d_n & & \cr {\bf Ng:}N^g_1 & & \cr x_{1,1} & & \cr \vdots & & \cr x_{1,N^g_1} & & \cr {\bf Ng:}N^g_n & & \cr x_{n,1} & & \cr \vdots & & \cr x_{n,N^g_n} & & \cr {\bf Z} &{\bf Epsilon} &{\bf Wht} \cr z_{1,\cdots,1} &\epsilon_{1,\cdots,1} &w_{1,\cdots,1} \cr z_{2,\cdots,1} &\epsilon_{2,\cdots,1} &w_{2,\cdots,1} \cr \vdots &\vdots &\vdots \cr z_{N^g_1,\cdots,1} &\epsilon_{N^g_1,\cdots,1} &w_{N^g_1,\cdots,1} \cr \vdots &\vdots &\vdots \cr z_{1,\cdots,N^g_n} &\epsilon_{1,\cdots,N^g_n} &w_{1,\cdots,N^g_n} \cr z_{2,\cdots,N^g_n} &\epsilon_{2,\cdots,N^g_n} &w_{2,\cdots,N^g_n} \cr \vdots &\vdots &\vdots \cr z_{N^g_1,\cdots,N^g_n} &\epsilon_{N^g_1,\cdots,N^g_n} &w_{N^g_1,\cdots,N^g_n} \cr {\bf End\_Data} & & \cr \end{array} \]

where the Epsilon and Wht columns are optional. The data must satisfy the following conditions:

  • $2\le n\le$ AMA_MXNIND;
  • $1\le d_k \le 5$, for $1\le k\le n$;
  • $N^g_k\ge 2$, for $1\le k\le n$;
  • the independent variable data $x_{k,\ell_k}$, for $\ell_k=1,\cdots,N^g_k$ and $1\le k\le n$, must be in increasing order;
  • the approximation tolerances must satisfy $\epsilon_{\ell_1,\cdots,\ell_n}\ge 0.0$, for $\ell_1=1,\ldots,N^g_1,\cdots,\ell_n=1,\ldots,N^g_n$;
  • the weights must satisfy $w_{\ell_1,\cdots,\ell_n}\ge 0.0$, for $\ell_1=1,\ldots,N^g_1,\cdots,\ell_n=1,\ldots,N^g_n$.

Following the Data section may be one or more of the Bounds, Least_Squares or Monotonicity sections. If an approximation options section is not defined, then this function sets the options to their default values. See Table Approximation Options Defaults for a list of Multivariate Gridded Data Functions approximation options default values.

The Bounds section specifies the lower and upper bounds employed by AMA_MltvGrdLstsqr(). It has the following structure:

\[ \begin{array}{ll} {\bf Bounds} & \cr {\bf Lwrbnd:}\alpha_l &{\bf Uprbnd:}\alpha_u \cr {\bf End\_Bounds} & \cr \end{array} \]

and its options must satisfy the following conditions:

The lower bound is read as a string lwrstr and based on the value of lwrstr the value of $\alpha_l$ is set as follows:

  • If lwrstr equals infbnd, then $\alpha_l = -\alpha_\infty$ where $\alpha_\infty$ = AMA_SplineInfbnd(); or,
  • If lwrstr equals zmin , then $\alpha_l = \min_{\ell_1,\cdots,\ell_n}\lbrace{z_{\ell_1,\cdots,\ell_n}\rbrace}$.
  • Otherwise, $\alpha_l =$ atof(lwrstr).

Similarly, the upper bound is read as a string uprstr and based on the value of uprstr the value of $\alpha_u$ is set as follows:

  • If uprstr equals infbnd, then $\alpha_u = \alpha_\infty$ where $\alpha_\infty$ = AMA_SplineInfbnd(); or,
  • If uprstr equals zmax , then $\alpha_u = \max_{\ell_1,\cdots,\ell_n}\lbrace{z_{\ell_1,\cdots,\ell_n}\rbrace}$.
  • Otherwise, $\alpha_u =$ atof(uprstr).

The Least_Squares section specifies the approximation options and knots employed by AMA_MltvGrdLstsqr(). It has the following structure:

\[ \begin{array}{ll} {\bf Least\_Squares} & \cr {\bf Theta:}\theta &{\bf Minorm:}\rm minorm \cr {\bf Mlamda:}m_1 & \cr \lambda^o_{1,1} & \cr \lambda^o_{1,2} & \cr \vdots & \cr \lambda^o_{1,m_1} & \cr \vdots & \cr {\bf Mlamda:}m_n & \cr \lambda^o_{n,1} & \cr \lambda^o_{n,2} & \cr \vdots & \cr \lambda^o_{n,m_n} & \cr {\bf End\_Least\_Squares} & \cr \end{array} \]

and its options must satisfy the following conditions:

  • $0.0\le\theta< 1.0$;
  • ${\rm minorm}$ must be Enabled or Disabled;
  • $m_k\ge 2$, for $1\le k\le n$;
  • the knots $\lambda^o_{k,i}$, for $i=1,\cdots,m_k$ and $1\le k\le n$, must be in increasing order;
  • $\lambda^o_{k,1} \le x_{k,1}$ and $\lambda^o_{k,m_k} \ge x_{k,N^g_k}$, for all $1\le k\le n$.

The Monotonicity section specifies the monotonicity constraints and continuity conditions employed by AMA_MltvGrdMonoApprox() and AMA_MltvGrdMonoInterp(). It has the the following structure:

\[ \begin{array}{llll} {\bf Monotonicity} & & & \cr {\bf Monpos:\rm monpos[1]} &{\bf Monneg:\rm monneg[1]} &{\bf Monzer:\rm monzer[1]} &{\bf Concnd:\rm concnd[1]} \cr \vdots &\vdots &\vdots &\vdots \cr {\bf Monpos:\rm monpos[n]} &{\bf Monneg:\rm monneg[n]} &{\bf Monzer:\rm monzer[n]} &{\bf Concnd:\rm concnd[n]} \cr {\bf End\_Monotonicity} & & & \cr \end{array} \]

and its options must satisfy the following conditions:

  • the positive monotonicity constraint flag ${\rm monpos}[k]$, negative monotonicity constraint flag ${\rm monneg}[k]$ and zero monotonicity constraint flag ${\rm monzer}[k]$ must be Enabled or Disabled, for $1\le k\le n$;
  • the continuity condition ${\rm concnd}[k]$ must be Full or Reduced, for $1\le k\le n$;

The bold keywords are case sensitive and the string values for the approximation options are case insensitive.

This function performs the following tasks:

Parameter Note: In the parameter definitions 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_MltvGrdData().
datfile[in] The data file name. Must satisfy datfile $\ne$ NULL.
nind[out] The number of independent variables $n$. Must satisfy nind $\ne$ NULL.
degree[out] Array of size nind containing the degree $d_k$ where degree[k] $= d_k$. Must satisfy degree $\ne$ NULL.
ng[out] Array of size nind containing the number of points $N^g_k$ where ng[k] $= N^g_k$. Must satisfy ng $\ne$ NULL.
x[out] Array of size nind containing arrays of size ng[k] where x[k] contains the independent variable data $x_{k,\ell_k}$, $\ell_k=1,\ldots,N^g_k$. Must satisfy x $\ne$ NULL.
z[out] Array of size $N$ containing the dependent variable data $z_{\ell_1,\cdots,\ell_n}$, for $\ell_1=1,\ldots,N^g_1,\cdots,\ell_n=1,\ldots,N^g_n$. Must satisfy z $\ne$ NULL.
epsilon[out] Array of size $N$ containing the approximation tolerances $\epsilon_{\ell_1,\cdots,\ell_n}$, for $\ell_1=1,\ldots,N^g_1,\cdots,\ell_n=1,\ldots,N^g_n$. Must satisfy epsilon $\ne$ NULL.
wht[out] Array of size $N$ containing the weights $w_{\ell_1,\cdots,\ell_n}$, for $\ell_1=1,\ldots,N^g_1,\cdots,\ell_n=1,\ldots,N^g_n$. Must satisfy wht $\ne$ NULL.
theta[out] The penalty term weight $\theta$. Must satisfy theta $\ne$ NULL.
mlamda[out] Array of size nind containing the number of knots $m_k$ where mlamda[k] $= m_k$. Must satisfy mlamda $\ne$ NULL.
lamda[out] Array of size nind containing arrays of size mlamda[k] where lamda[k] contains the knots $\lambda^o_{i,k}$, for $i=1,\ldots,m_k$. Must satisfy lamda $\ne$ NULL.
Returns
Success/Error Code.

User Support Function - Documented 110315 - !!!THIS IS NOT A USER CALLABLE FUNCTION - DOCUMENT IS INCLUDED FOR COMPLETENESS!!!

long int AMA_MltvGrdInputCheck ( AMA_OPTIONS *  options,
long int  nind,
long int *  ng,
double **  x,
double *  z,
double *  wht,
double *  epsilon,
long int *  degree,
long int *  mlamda,
double **  lamda,
enum AMA_Boolean  lstsqr,
enum AMA_Boolean *  mltknt 
)

Perform input check for multivariate gridded data.

This function checks the input parameters for the Multivariate Gridded Data Functions which compute the spline

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

of degree $d_k$ which is based on the knot vectors ${\bf\Lambda_k}$, for $1\le k\le n$.

Along with the degree the Multivariate Gridded Data Functions require the independent variable data $x_{k,\ell}$, for $\ell=1,\ldots,N^g_k$ and $1\le k\le n$, which defines the rectilinear grid

\[ (x_{1,1},x_{1,2},\cdots,x_{1,N^g_1-1},x_{1,N^g_1})\times\cdots\times(x_{n,1},x_{n,2},\cdots,x_{n,N^g_n-1},x_{n,N^g_n}) \]

and either the weights $w_{\ell_1,\cdots,\ell_n}$ or the approximation tolerances $\epsilon_{\ell_1,\cdots,\ell_n}$, for $\ell_1=1,\ldots,N^g_1,\cdots,\ell_n=1,\ldots,N^g_n$. There are a total of N $=\prod_{k=1}^nN^g_k$ points. Also, for least squares approximation functions the knot vectors ${\bf\Lambda}_k\in{\rm R}^{M_k}$, for $1\le k\le n$, are required. This function insures the following conditions are satisfied:

  • $1\le n\le$ AMA_MXNIND;
  • $N^g_k\ge 2$, for all $1\le k\le n$;
  • $1\le d_k\le 5$, for all $1\le k\le n$;
  • $x_{k,\ell}$, for all $1\le k\le n$, is in ascending order and defines at least two distinct points;
  • $w_{\ell_1,\cdots,\ell_n}\ge 0.0$ for all $\ell_1=1,\ldots,N^g_1,\cdots,\ell_n=1,\ldots,N^g_n$ and $w_{\ell_1,\cdots,\ell_n} >0.0$ for at least one $\ell_1=1,\ldots,N^g_1,\cdots,\ell_n=1,\ldots,N^g_n$;
  • $\epsilon_{\ell_1,\cdots,\ell_n}\ge 0.0$ for all $\ell_1=1,\ldots,N^g_1,\cdots,\ell_n=1,\ldots,N^g_n$;
  • $M_k\ge 2$, for all $1\le k\le n$.

Also, if lstsqr equals AMA_Boolean_True and the knot vector ${\bf\Lambda}_k$ is represented by its distinct knot vector

\[ {\bf\rm K}^{{\bf\Lambda}_k} = ( \kappa_{k,1}, \kappa_{k,2}, \cdots, \kappa_{k,m^d_k-1}, \kappa_{k,m^d_k} )^T \]

and its knot multiplicity vector

\[ {\bf\rm H}^{{\bf\Lambda}_k} = ( \eta_{k,1}, \eta_{k,2}, \cdots, \eta_{k,m^d_k-1}, \eta_{k,m^d_k} )^T, \]

then, for all $1\le k\le n$, this function insures either the conditions

  • $\kappa_{k,i} < \kappa_{k,i+1}$, for $1\le i\le m^d_k - 1$;
  • $\eta_{k,i} \le d_k + 1$, for $2\le i\le m^d_k-1$;
  • $\eta_{k,1} = \eta_{k,m^d_k} = d_k+1$;

or

  • $\kappa_{k,i} < \kappa_{k,i+1}$, for $1\le i\le m^d_k - 1$;
  • $\eta_{k,i} \le d_k + 1$, for $2\le i\le m^d_k-1$;
  • $\eta_{k,1} = \eta_{k,m^d_k} = 1$;

are satisfied. It also insures the first and last distinct knots satisfy the conditions $\kappa_{k,1}\le x_{k,1}$ and $\kappa_{k,m^d_k}\ge x_{k,N^g_k}$, respectively.

Parameter Note: In the parameter definitions given below the limits on $k$ are $1\le k\le n$, k = $k-1$ and N $=\prod_{k=1}^nN^g_k$.

Parameters
options[in] Pointer to AMA_OPTIONS. Must be initialized with AMA_Options() prior to calling AMA_MltvGrdInputCheck().
nind[in] The number of independent variables $n$. Must satisfy nind $\ge 2$.
ng[in] Array of size nind containing the number of points $N^g_k$ where ng[k] $= N^g_k$. Must satisfy ng $\ne$ NULL and ng[k] $\ge 2$.
x[in] Array of size nind containing arrays of size ng[k] where x[k] contains the independent variable data $x_{k,\ell_k}$, for $\ell_k=1,\ldots,N^g_k$. Must satisfy x $\ne$ NULL and x[k] $\ne$ NULL. Also, x[k] must be in ascending order.
z[in] Array of size N containing the dependent variable data $z_{\ell_1,\cdots,\ell_n}$, for $\ell_1=1,\ldots,N^g_1,\cdots,\ell_n=1,\ldots,N^g_n$. Must satisfy z $\ne$ NULL.
wht[in] Array of size N containing the weights $w_{\ell_1,\cdots,\ell_n}$, for $\ell_1=1,\ldots,N^g_1,\cdots,\ell_n=1,\ldots,N^g_n$. If wht $\ne$ NULL, then must satisfy $w_\ell\ge 0.0$ for all $\ell=1,\ldots,N$.
epsilon[in] Array of size N containing the approximation tolerances $\epsilon_{\ell_1,\cdots,\ell_n}$, for $\ell_1=1,\ldots,N^g_1,\cdots,\ell_n=1,\ldots,N^g_n$. If epsilon $\ne$ NULL, then must satisfy $\epsilon_\ell\ge 0.0$ for all $\ell=1,\ldots,N$.
degree[in] Array of size nind containing the degree $d_k$ where degree[k] $= d_k$. Must satisfy degree $\ne$ NULL and $1\le$ degree[k] $\le 5$.
mlamda[in] Array of size nind containing the number of knots $M_k$ where mlamda[k] $= M_k$. If lstsqr AMA_Boolean_True, then must satisfy mlamda $\ne$ NULL and mlamda[k] $\ge 2$.
lamda[in] Array of size nind containing arrays of size mlamda[k] where lamda[k] contains the knot vector ${\bf\Lambda}_k$. If lstsqr equals AMA_Boolean_True, then must satisfy lamda $\ne$ NULL and lamda[k] $\ne$ NULL. Also, lamda[k] must be a valid knot vector.
lstsqr[in] Least squares approximation flag. If it equals AMA_Boolean_True, then this function is being called by a least squares approximation function.
mltknt[out] Array of size nind containing the multiple knot flag. Defined only if lstsqr equals AMA_Boolean_True. It has one of the following two values:
  • If mltknt[k] equals AMA_Boolean_True, then ${\bf\Lambda}_k$ has $d_k+1$ order knots at its left and right hand boundaries.
  • If mltknt[k] equals AMA_Boolean_False, then ${\bf\Lambda}_k$ has single knots at its left and right hand boundaries.
Returns
Success/Error Code.

User Support Function - Documented 021116 - !!!THIS IS NOT A USER CALLABLE FUNCTION - DOCUMENT IS INCLUDED FOR COMPLETENESS!!!

long int AMA_MltvInputCheck ( AMA_OPTIONS *  options,
long int  nind,
long int  n,
double **  x,
double *  z,
double *  wht,
double *  epsilon,
long int *  degree,
long int *  mlamda,
double **  lamda,
enum AMA_Boolean  lstsqr 
)

Perform input check for multivariate data approximation functions.

This function checks the input parameters for the Multivariate Random Data Functions which compute the spline

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

of degree $d_k$ which is based on the knot vectors ${\bf\Lambda_k}$, for $1\le k\le n$. Along with the degree the Multivariate Random Data Functions require the independent variable data data ${\bf X}_\ell=(x_{1,\ell},\cdots,x_{n,\ell})$, for $\ell=1,\ldots,N$, and either the weights $w_\ell$ or the approximation tolerances $\epsilon_\ell$, for $\ell=1,\ldots,N$. Also, for least squares approximation functions the knot vectors ${\bf\Lambda}_k\in{\rm R}^{M_k}$, for $1\le k\le n$, are required. This function insures the following conditions are satisfied:

Also, if lstsqr equals AMA_Boolean_True and the knot vector ${\bf\Lambda}_k$ is represented by its distinct knot vector

\[ {\bf\rm K}^{{\bf\Lambda}_k} = ( \kappa_{k,1}, \kappa_{k,2}, \cdots, \kappa_{k,m^d_k-1}, \kappa_{k,m^d_k} )^T \]

and its knot multiplicity vector

\[ {\bf\rm H}^{{\bf\Lambda}_k} = ( \eta_{k,1}, \eta_{k,2}, \cdots, \eta_{k,m^d_k-1}, \eta_{k,m^d_k} )^T, \]

then, for all $1\le k\le n$, this function insures either the conditions

  • $\kappa_{k,i} < \kappa_{k,i+1}$, for $1\le i\le m^d_k - 1$;
  • $\eta_{k,i} \le d_k + 1$, for $2\le i\le m^d_k-1$;
  • $\eta_{k,1} = \eta_{k,m^d_k} = d_k+1$;

or

  • $\kappa_{k,i} < \kappa_{k,i+1}$, for $1\le i\le m^d_k - 1$;
  • $\eta_{k,i} \le d_k + 1$, for $2\le i\le m^d_k-1$;
  • $\eta_{k,1} = \eta_{k,m^d_k} = 1$;

are satisfied. It also insures the first and last distinct knots satisfy the conditions $\kappa_{k,1}\le \min_\ell\lbrace{x_{k,\ell}\rbrace}$ and $\kappa_{k,m^d_k}\ge \max_\ell\lbrace{x_{k,\ell}\rbrace}$, respectively.

Parameter Note: In the parameter definitions 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_MltvInputCheck().
nind[in] The number of independent variables $n$. Must satisfy nind $\ge 2$.
n[in] The number of data points $N$. Must satisfy n $\ge 2$.
x[in] Array of size nind containing arrays of size n where x[k] contains the independent variable data $x_{k,\ell}$, for $\ell=1,\ldots,N$. Must satisfy x $\ne$ NULL and x[k] $\ne$ NULL. Also, x[k] must be in ascending order.
z[in] Array of size n containing the dependent variable data $z_\ell$, for $\ell=1,\ldots,N$. Must satisfy z $\ne$ NULL.
wht[in] Array of size n containing the weights $w_\ell$, for $\ell=1,\ldots,N$. If wht $\ne$ NULL, then must satisfy $w_\ell\ge 0.0$ for all $\ell=1,\ldots,N$.
epsilon[in] Array of size n containing the approximation tolerances $\epsilon_\ell$, for $\ell=1,\ldots,N$. If epsilon $\ne$ NULL, then must satisfy $\epsilon_\ell\ge 0.0$ for all $\ell=1,\ldots,N$.
degree[in] Array of size nind containing the degree $d_k$ where degree[k] $= d_k$. Must satisfy degree $\ne$ NULL and $1\le$ degree[k] $\le 5$.
mlamda[in] Array of size nind containing the number of knots $M_k$ where mlamda[k] $= M_k$. If lstsqr AMA_Boolean_True, then must satisfy mlamda $\ne$ NULL and mlamda[k] $\ge 2$.
lamda[in] Array of size nind containing arrays of size mlamda[k] where lamda[k] contains the knot vector ${\bf\Lambda}_k$. If lstsqr equals AMA_Boolean_True, then must satisfy lamda $\ne$ NULL and lamda[k] $\ne$ NULL. Also, lamda[k] must be a valid knot vector.
lstsqr[in] Least squares approximation flag. If it equals AMA_Boolean_True, then this function is being called by a least squares approximation function.
Returns
Success/Error Code.

User Support Function - Documented 021116 - !!!THIS IS NOT A USER CALLABLE FUNCTION - DOCUMENT IS INCLUDED FOR COMPLETENESS!!!

long int AMA_MltvPnltrm ( AMA_OPTIONS *  options,
double  theta,
long int  porder,
CNSPLA_SPLFUN *  splfun 
)

Define penalty term on a multivariate spline.

This function defines a penalty term $\theta F^{(p)}(s({\bf X}))$ for $p=0$ or $p=2$ on a multivariate spline $s({\bf X}):{\rm R}^n\rightarrow {\rm R}^1$ given as

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

where the $\alpha_{j_1,\ldots,j_n}$s, for $j_k=1,\ldots,m_k$ and $1\le k\le n$, are the coefficients of the tensor product B-splines

\[ \Phi_{j_1,\ldots,j_n}({\bf X}) = B_{d_1,j_1}(x_1|{\bf\Lambda}_1)\cdots B_{d_n,j_n}(x_n|{\bf\Lambda}_n). \]

The $B_{d_k,j_k}(x_k|{\bf\Lambda}_k)$s are the $m_k$ univariate B-splines of degree $1 \le d_k\le 5$ defined by the knot vectors

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

For $p=0$ the penalty term $F^{(0)}(s({\bf X}))$ is

\[ F^{(0)}(s({\bf X})) = \int_R\left(s({\bf X})\right)^2dR \]

where $R = [\lambda_{1,1},\lambda_{1,m_1+1}] \times [\lambda_{2,1},\lambda_{2,m_2+1}] \times\cdots\times [\lambda_{n,1},\lambda_{n,m_n+1}]$. Alternately, for $p=2$ the penalty term $F^{(2)}(s({\bf X}))$ is

\[ F^{(2)}(s({\bf X})) = \sum_{k=1}^n\left[\int_R\left(\partial^2 s({\bf X})\over\partial^2 x_k\right)^2dR + 2.0\sum_{l=k+1}^n\int_R\left(\partial^2 s({\bf X})\over\partial x_k \partial x_l\right)^2dR\right]. \]

The cross partial terms can be excluded from the penalty term $F^{(2)}(s({\bf X}))$ with AMA_OptionsSetPenaltyTerm().

Parameters
options[in] Pointer to AMA_OPTIONS. Should be initialized with AMA_Options() prior to calling AMA_MltvPnltrm().
theta[in] The penalty term weight $\theta$. Should satisfy $0.0\le$ theta $\le 1.0$.
porder[in] The penalty term order $p$. Should satisfy porder $=0$ or porder $=2$.
splfun[in] Pointer to CNSPLA_SPLFUN containing spline $s({\bf X})$ upon which penalty term is imposed. Should satisfy splfun $\ne$ NULL.
Returns
Success/Error Code.

User Support Function - Documented 121114 - !!!THIS IS NOT A USER CALLABLE FUNCTION - DOCUMENT IS INCLUDED FOR COMPLETENESS!!!

AMA Spline Library Document - Version:1.0.0-a Date:02/24/16 - Generated by   doxygen 1.8.5