Classes | Namespaces
Spline.hpp File Reference
#include <opm/material/common/TridiagonalMatrix.hpp>
#include <opm/material/common/PolynomialUtils.hpp>
#include <opm/material/common/Exceptions.hpp>
#include <ostream>
#include <vector>
#include <tuple>
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Classes

class  Opm::Spline< Scalar >
 Class implementing cubic splines. More...
 
struct  Opm::Spline< Scalar >::ComparatorX_
 Helper class needed to sort the input sampling points. More...
 

Namespaces

namespace  Opm
 

Detailed Description

Class implementing cubic splines.

This class supports full, natural, periodic and monotonic cubic splines.

Full a splines $s(x)$ are splines which, given $n$ sampling points $x_1, \dots, x_n$, fulfill the following conditions

\begin{align*} s(x_i) & = y_i \quad \forall i \in \{1, \dots, n \} \\ s'(x_1) & = m_1 \\ s'(x_n) & = m_n \end{align*}

for any given boundary slopes $m_1$ and $m_n$.

Natural splines which are defined by

\begin{align*} s(x_i) & = y_i \quad \forall i \in \{1, \dots, n \} \\ s''(x_1) & = 0 \\ s''(x_n) & = 0 *\end{align*}

For periodic splines of splines the slopes at the boundaries are identical:

\begin{align*} s(x_i) & = y_i \quad \forall i \in \{1, \dots, n \} \\ s'(x_1) & = s'(x_n) \\ s''(x_1) & = s''(x_n) \;. *\end{align*}

Finally, there are monotonic splines which guarantee that the curve is confined by its sampling points, i.e.,

\[ y_i \leq s(x) \leq y_{i+1} \quad \text{for} x_i \leq x \leq x_{i+1} \;. \]

For more information on monotonic splines, see http://en.wikipedia.org/wiki/Monotone_cubic_interpolation

Full, natural and periodic splines are continuous in their first and second derivatives, i.e.,

\[ s \in \mathcal{C}^2 \]

holds for such splines. Monotonic splines are only continuous up to their first derivative, i.e., for these only

\[ s \in \mathcal{C}^1 \]

is true.