Question

Suppose you are asked to construct a clamped cubic spline interpolating the following set of data: $$ \begin{array}{|c|c|c|c|c|c|c|c|c|} \hline i & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ \hline x & 1.2 & 1.4 & 1.6 & 1.67 & 1.8 & 2.0 & 2.1 & 2.2 \\ \hline f(x) & 4.561 & 5.217 & 5.634 & 5.935 & 6.562 & 6.242 & 5.812 & 5.367 \\\ \hline \end{array} $$ The function underlying these data points is unknown, but clamped cubic splines require interpolation of the first derivative of the underlying function at the end points \(x_{0}=1.2\) and \(x_{7}=2.2 .\) Select the formula in the following list that would best assist you to construct the clamped cubic spline interpolating this set of data: $$ \begin{aligned} &\text { i. } f^{\prime}\left(x_{0}\right)=\frac{1}{2 \hbar}\left(-f\left(x_{0}-h\right)+f\left(x_{0}+h\right)\right)-\frac{h^{2}}{6} f^{(3)}(\xi) \\ &\text { ii. } f^{\prime}\left(x_{n}\right)=\frac{1}{2 h}\left(f\left(x_{n}-2 h\right)-4 f\left(x_{n}-h\right)+3 f\left(x_{n}\right)\right)+\frac{h^{2}}{3} f^{(3)}(\xi) \\ &\text { iii. } f^{\prime}\left(x_{0}\right)=\frac{1}{12 h}\left(f\left(x_{0}-2 h\right)-8 f\left(x_{0}-h\right)+8 f\left(x_{0}+h\right)-f\left(x_{0}+2 h\right)\right)+\frac{h^{4}}{30} f^{(5)}(\xi) \\ &\text { iv. } f^{\prime}\left(x_{n}\right)=\frac{1}{12 h}\left(3 f\left(x_{n}-4 h\right)-16 f\left(x_{n}-3 h\right)+36 f\left(x_{n}-2 h\right)-48 f\left(x_{n}-h\right)\right. \\ &\left.\quad+25 f\left(x_{n}\right)\right)+\frac{h^{4}}{5} f^{(5)}(\xi) \\ &\text { v. } f^{\prime}\left(x_{0}\right)=\frac{1}{180 h}\left(-441 f\left(x_{0}\right)+1080 f\left(x_{0}+h\right)-1350 f\left(x_{0}+2 h\right)\right. \\ &\quad+1200 f\left(x_{0}+3 h\right)-675 f\left(x_{0}+4 h\right)+216 f\left(x_{0}+5 h\right) \\ &\left.\quad-30 f\left(x_{0}+6 h\right)\right)+\frac{h^{6}}{7} f^{(7)}(\xi) \end{aligned} $$ Provide an explanation supporting your choice of formula.

Short answer

Answer: The best formulas for constructing the clamped cubic spline at the endpoints are: For x_0: $$ f^{\prime}\left(x_{0}\right)=\frac{1}{12 h}\left(f\left(x_{0}-2 h\right)-8 f\left(x_{0}-h\right)+8 f\left(x_{0}+h\right)-f\left(x_{0}+2 h\right)\right)+\frac{h^{4}}{30} f^{(5)}(\xi) $$ For x_n: $$ f^{\prime}\left(x_{n}\right)=\frac{1}{2 h}\left(f\left(x_{n}-2 h\right)-4 f\left(x_{n}-h\right)+3 f\left(x_{n}\right)\right)+\frac{h^{2}}{3} f^{(3)}(\xi) $$

Step-by-step solution

Choose the Formula to Calculate First Derivative at x_0

We will be using the formula for \(f'(x_0)\) that involves less error term and uses either central or one-sided difference. In this case, we can see from the list that option iii is most appropriate. This is a central finite difference formula of order \(O(h^4)\): $$ f^{\prime}\left(x_{0}\right)=\frac{1}{12 h}\left(f\left(x_{0}-2 h\right)-8 f\left(x_{0}-h\right)+8 f\left(x_{0}+h\right)-f\left(x_{0}+2 h\right)\right)+\frac{h^{4}}{30} f^{(5)}(\xi) $$

Choose the Formula to Calculate First Derivative at x_n

For calculating the first derivative at the rightmost endpoint (\(x_n\)), we can use a one-sided backward finite difference formula. According to the list given, Formula ii meets the requirements, which is a one-sided backward difference formula of order \(O(h^2)\): $$ f^{\prime}\left(x_{n}\right)=\frac{1}{2 h}\left(f\left(x_{n}-2 h\right)-4 f\left(x_{n}-h\right)+3 f\left(x_{n}\right)\right)+\frac{h^{2}}{3} f^{(3)}(\xi) $$

Explanation for Choosing Formulas

The selected formulas iii and ii are ideal for calculating the first derivative at the left and right endpoints, respectively. Formula iii is a central finite difference formula that utilizes more neighboring points and has smaller error terms which give a precise estimation for \(f'(x_0)\). On the other hand, Formula ii is a one-sided backward finite difference formula that takes into consideration a few of the neighboring points and lesser error terms which is suitable for estimating the first derivative at the right endpoint \(f'(x_n)\). Hence, the best formulas for constructing the clamped cubic spline from the given list are: $$ f^{\prime}\left(x_{0}\right)=\frac{1}{12 h}\left(f\left(x_{0}-2 h\right)-8 f\left(x_{0}-h\right)+8 f\left(x_{0}+h\right)-f\left(x_{0}+2 h\right)\right)+\frac{h^{4}}{30} f^{(5)}(\xi) $$ $$ f^{\prime}\left(x_{n}\right)=\frac{1}{2 h}\left(f\left(x_{n}-2 h\right)-4 f\left(x_{n}-h\right)+3 f\left(x_{n}\right)\right)+\frac{h^{2}}{3} f^{(3)}(\xi) $$

Key concepts

Cubic Spline Interpolation

Cubic spline interpolation is a method used to create smooth curves through a set of data points. It's a type of polynomial interpolation, which means it uses polynomials to approximate the function. In cubic spline interpolation, the data is divided into intervals, and a cubic polynomial is fitted to each interval.
A key feature of cubic splines is their continuity. The resulting curve not only passes through each data point, but also ensures that the first and second derivatives are continuous across each interval. This gives the curve a smooth and natural look.
Clamped cubic splines go a step further by specifying the first derivatives at the end points. This helps to minimize boundary errors and keeps the spline neat. With clamped splines, the curve can better reflect the behavior of the unknown underlying function. Particularly in this exercise, determining the first derivatives at initial and final points is crucial to formulating the spline correctly.

Finite Difference Method

The finite difference method is a numerical technique often used to estimate derivatives. In the context of cubic spline interpolation, finite differences help calculate the derivatives of the underlying function at specific data points.
There are different types of finite difference approaches:

• Central difference: Uses point values before and after the target point, providing a higher accuracy.
• Forward difference: Estimates based on the point and subsequent values, often used at the beginning of an interval.
• Backward difference: Calculates derivatives using prior values, suitable for endpoints of data sets.

In our original exercise, different finite difference formulas were chosen based on their accuracy and suitability for endpoints. Central finite difference provides smaller error and is well-suited for the left boundary, while backward finite difference fits the right end.

Numerical Derivatives

Numerical derivatives are approximations of the derivative of a function using numerical methods. This is particularly useful when the function itself is unknown or complex.
In practice, derivatives can be inaccurately estimated due to discretization errors. To minimize these errors, numerical derivatives can be calculated using finite difference methods, which offer effective approximations in practical scenarios. For example, in the original exercise, deriving the first derivative at each endpoint was achieved by employing formulas adopting finite differences.
Furthermore, the choice of formula can influence accuracy. Higher-order methods involve more terms and refined formulas, which results in reduced error and increased precision. Hence, selecting appropriate formulas is crucial, especially when constructing precise interpolations like clamped cubic splines.