Question

Given function values \(f\left(t_{0}\right), f\left(t_{1}\right), \ldots, f\left(t_{r}\right)\), as well as those of \(f^{\prime}\left(t_{0}\right)\) and \(f^{\prime}\left(t_{r}\right)\), for some \(r \geq 2\), it is possible to construct the complete interpolating cubic spline. Suppose that we were instead to approximate \(f^{\prime}\left(t_{i}\right)\) by the divided difference \(f\left[t_{i-1}, t_{i+1}\right]\), for \(i=1,2, \ldots, r-1\), and then use these values to construct a Hermite piecewise cubic interpolant. State one advantage and one disadvantage of this procedure over a complete cubic spline interpolation.

Short answer

Answer: One advantage of the Hermite piecewise cubic interpolant with divided differences is that it can give a closer approximation to the original function, especially in cases where the shape of the curve changes significantly between different segments. However, the disadvantage is that it may result in non-continuous second derivatives, leading to a less smooth curve compared to complete cubic spline interpolation.

Step-by-step solution

Understanding Complete Cubic Spline and Hermite Piecewise Cubic Interpolant

The complete cubic spline interpolates a given set of function values by constructing a smooth curve that passes through all of the data. This curve has continuous first and second derivatives, making it a smooth representation of the underlying function. On the other hand, Hermite piecewise cubic interpolant also approximates a function through given data points, but it does so in a piecewise manner, constructing separate cubic polynomials between consecutive data points. Instead of only using function values, it also considers the values of the derivatives at the given points. In this exercise, we approximate the derivatives using divided differences.

Advantage of Hermite Piecewise Cubic Interpolant

One advantage of using Hermite piecewise cubic interpolant with divided differences over complete cubic spline interpolation is that it can give a closer approximation to the original function, especially in cases where the shape of the curve changes significantly between different segments. By taking into account the information about the derivatives at each of the points, it can sometimes produce better local approximations than the cubic spline.

Disadvantage of Hermite Piecewise Cubic Interpolant

The disadvantage of using the Hermite piecewise cubic interpolant approximation with divided differences over complete cubic spline interpolation is that it may result in non-continuous second derivatives. This can lead to a less smooth curve compared to that obtained using the complete cubic spline interpolation. If we emphasize smoothness over precise local approximation, the cubic spline might be a better choice. In conclusion, using Hermite piecewise cubic interpolant approximation with divided differences has the advantage of potentially providing better local approximations of the underlying function curve compared to complete cubic spline interpolation. However, this comes at the cost of potentially having non-continuous second derivatives, which may result in a less smooth overall curve.

Key concepts

Hermite Piecewise Cubic Interpolant

The Hermite piecewise cubic interpolant is a mathematical tool used to construct an approximate function that matches a given set of data points. Unlike a single polynomial that attempts to fit all points, Hermite interpolation works by creating separate cubic polynomials for each interval between points.

These polynomials are defined not only by the function's value at the endpoints but also by the derivatives at these points. Mathematically, for the interval between the points \( t_i \) and \( t_{i+1} \) we have a cubic polynomial \( H_i(x) \) that satisfies the following conditions:

• \( H_i(t_i) = f(t_i) \)
• \( H_i(t_{i+1}) = f(t_{i+1}) \)
• \( H_i'(t_i) = f'(t_i) \)
• \( H_i'(t_{i+1}) = f'(t_{i+1}) \)

Using derivatives provides a 'smoother' interpolation that can capture the nuances of the original function better, especially when the function's behavior changes significantly between data points. However, the challenge emerges when the function's actual derivatives are unknown. This necessitates the use of an approximation technique such as divided differences to estimate these derivatives.

Numerical Differentiation

Numerical differentiation is the process of approximating the derivatives of functions based on known function values. In the context of interpolation, numerical differentiation is critical when exact derivative values are not available—leading one to estimate these values to construct a more accurate interpolant.

For instance, the divided difference \( f[t_i, t_{i+1}] \) that approximates \( f'(t_i) \) is based on function values around \( t_i \) and is defined as \( f[t_i, t_{i+1}] = \frac{f(t_{i+1}) - f(t_i)}{t_{i+1} - t_i} \). This approximation can sometimes provide a good estimate, but it may not capture the exact behavior of the function, leading to potential inaccuracies in the Hermite interpolant. These inaccuracies are mostly pronounced when the function has highly variable second derivatives.

Divided Differences

Divided differences are a recursive division process utilized in the mathematical field to interpolate polynomials given a set of data points. The simplest form of a divided difference is the first divided difference used in linear interpolation. It can be extended to higher orders which correspond to quadratic, cubic, or higher-degree polynomial interpolations.

For Hermite interpolation, divided differences serve as a practical means to estimate the unknown derivative values at interior points. Specifically, the divided difference \( f[t_{i-1}, t_{i+1}] \) is calculated as:
\[ \frac{f(t_{i+1}) - f(t_{i-1})}{t_{i+1} - t_{i-1}} \]
This estimate captures the average rate of change of the function over the interval \( [t_{i-1}, t_{i+1}] \), which is then used as an approximation for the derivative \( f'(t_i) \). While this estimate is simple and convenient, it may introduce errors, especially if the actual derivative fluctuates significantly within the interval.

Interpolation Methods

Interpolation methods are algorithms that construct new data points within a set of known data points. The goal is to formulate a function that closely aligns with the available data and can predict values at unseen locations. Some common interpolation methods include linear, polynomial, and spline interpolation.

Cubic spline interpolation aims to piece together a set of cubic polynomials between consecutive data points, with restrictions on the function and its first two derivatives for smoothness. A complete cubic spline ensures that both the function and its derivatives (first and second) are continuous across the full domain.

In summary, interpolation methods are essential techniques in numerical analysis, providing tools to estimate values and model complex relationships from a finite set of data points. The choice of method often balances between computational efficiency, desired smoothness, and the fidelity of approximation for the underlying function.