ama_spline Struct Reference

This structure is used to define a tensor product spline $s({\bf X}):{\rm R}^n\rightarrow{\rm R}^m$. More...

#include <AMA.h>

Detailed Description

This structure is used to define a tensor product spline $s({\bf X}):{\rm R}^n\rightarrow{\rm R}^m$.

A tensor product spline is a function $s({\bf X}):{\rm R}^n\rightarrow {\rm R}^m$ given as

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

where $n$ is the number of independent variables for the spline and ${\bf U}\in{\rm R}^n$ is a vector of independent variables ${\bf X} = (x_1,x_2,\ldots,x_n)$. In the above definition of $s({\bf X})$ the $\alpha^l_{j_1,\ldots,j_n}$, for $j_k=1,\ldots,m_k$ and $1\le k\le n$, are the coefficients in the $l$-th dependent variable, for $1\le l\le m$, of the tensor product B-splines

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

where $B_{d_k,j_k}(x_k|{\bf\Lambda}_k)$ are the $m_k$ univariate B-splines of degree $d_k$ defined by the knot vector

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

From the definition of ${\bf\Lambda}_k$ it can be seen that $m_k+d_k+1$ knots are required to define $m_k$ univariate B-splines of degree $d_k$. The number of tensor product spline coefficients $\alpha^l_{j_1,\ldots,j_n}$ is $M = \prod_{k=1}^n m_k$ for all $1\le l\le m$.

This structure is equivalent to CNSPLA_SPLINE where within the AMA Spline Library the number of dependent variables $m$ equals one.

In the AMA Spline Library a valid tensor product spline $s({\bf X}):{\rm R}^n\rightarrow{\rm R}^m$ must satisfy the following conditions:

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

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

the following conditions must be satisfied:

Input/Output Structure - Documented 110815


The documentation for this struct was generated from the following file: