Find and graph the natural cubic spline interpolant for the...

Find and graph the natural cubic spline interpolant for the following data: (a) \begin{tabular}{c|ccc} $x$ & -1 & 0 & 1 \\ \hline$y$ & -2 & 1 & -1 \end{tabular} (b) \begin{tabular}{l|llll} $x$ & 0 & 1 & 2 & 3 \\ \hline$y$ & 1 & 2 & 0 & 1 \end{tabular} (c) \begin{tabular}{l|lll} $x$ & 1 & 2 & 4 \\ \hline$y$ & 3 & 0 & 2 \end{tabular} (d) \begin{tabular}{c|ccccc} $x$ & -2 & -1 & 0 & 1 & 2 \\ \hline$y$ & 5 & 2 & 3 & -1 & 1 \end{tabular}

Find and graph the natural cubic spline interpolant for the following data:
(a)
\begin{tabular}{c|ccc}
$x$ & -1 & 0 & 1 \\
\hline$y$ & -2 & 1 & -1
\end{tabular}
(b)
\begin{tabular}{l|llll}
$x$ & 0 & 1 & 2 & 3 \\
\hline$y$ & 1 & 2 & 0 & 1
\end{tabular}
(c)
\begin{tabular}{l|lll}
$x$ & 1 & 2 & 4 \\
\hline$y$ & 3 & 0 & 2
\end{tabular}
(d)
\begin{tabular}{c|ccccc}
$x$ & -2 & -1 & 0 & 1 & 2 \\
\hline$y$ & 5 & 2 & 3 & -1 & 1
\end{tabular}

Key Concepts

Spline Interpolation

Spline interpolation is a method of constructing a smooth function that passes through a given set of data points. Instead of using a single high-degree polynomial, the data interval is divided into smaller subintervals and low-degree polynomials are used in each, ensuring that the overall function remains smooth and well-behaved.

Cubic Splines

Cubic splines are a specific type of spline interpolation where the interpolants on each subinterval are cubic polynomials. These cubics are chosen because they provide a good balance between flexibility and smoothness, and they are relatively straightforward to compute and work with in terms of continuity properties.

Natural Cubic Spline

A natural cubic spline is a cubic spline with the additional condition that its second derivative is set to zero at both endpoints of the interpolation interval. These boundary conditions lead to a 'natural' shape of the curve, avoiding overfitting or unnatural bending at the ends of the data range.

Continuity Conditions

When constructing cubic splines, it is essential to enforce continuity conditions at the data points. This includes continuity of the function itself as well as its first and second derivatives across the subinterval boundaries, which ensures a smooth transition between adjacent polynomial pieces.

Piecewise Polynomial Formulation

The process involves determining the coefficients of each cubic polynomial on the individual subintervals. This is achieved by solving a system of equations that arises from the interpolation conditions at the data points and the smoothness conditions (including the natural boundary conditions) at the joins between the subintervals.

Graphical Analysis

Graphing the natural cubic spline interpolant provides a visual confirmation of the interpolation process. It helps in understanding the behavior of the spline over the data interval, verifying the smooth transition at the knots, and appreciating how the natural boundary conditions influence the curvature of the spline at the endpoints.

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