🔢Numerical Analysis I Unit 8 – Interpolation – Hermite Interpolation

What's Hermite Interpolation?

• Hermite interpolation is a polynomial interpolation method that uses function values and derivatives at interpolation points to construct a polynomial approximation
• Interpolates both the function values and the derivatives at the given data points
• Produces a smoother and more accurate interpolation compared to methods that only use function values (Lagrange interpolation)
• Requires the values of the function and its derivatives at the interpolation points as input
• The resulting Hermite interpolating polynomial matches the function values and derivatives at each interpolation point
• Hermite interpolation is named after Charles Hermite, a French mathematician who developed the method in the late 19th century
• Hermite interpolation has a unique polynomial of the least possible degree that satisfies the given interpolation conditions

Key Concepts and Terminology

• Interpolation: the process of constructing new data points within the range of a discrete set of known data points
• Interpolation points: the known data points used to construct the interpolating polynomial
• Hermite interpolating polynomial: the resulting polynomial that matches the function values and derivatives at the interpolation points
• Divided differences: a recursive method used to calculate the coefficients of the Hermite interpolating polynomial
• Hermite basis functions: a set of polynomials used to construct the Hermite interpolating polynomial
  • Defined in terms of the interpolation points and the order of the derivatives
• Continuity: the property of a function being continuous (no gaps or breaks) at the interpolation points
  • Hermite interpolation ensures continuity of the function and its derivatives at the interpolation points
• Smoothness: the property of a function having continuous derivatives up to a certain order
  • Hermite interpolation produces a smoother interpolation compared to methods that only use function values

How It's Different from Other Interpolation Methods

• Hermite interpolation uses both function values and derivatives at the interpolation points, while other methods (Lagrange, Newton) only use function values
• Provides a smoother and more accurate interpolation compared to methods that only use function values
• Requires the values of the function and its derivatives at the interpolation points, which may not always be readily available or easily computable
• The resulting Hermite interpolating polynomial has a higher degree compared to interpolating polynomials obtained from methods that only use function values
  • Degree of Hermite interpolating polynomial: $2n-1$, where $n$ is the number of interpolation points
  • Degree of Lagrange interpolating polynomial: $n-1$
• Hermite interpolation ensures continuity of the function and its derivatives at the interpolation points, while other methods may not guarantee this property
• Computationally more expensive than methods that only use function values due to the additional information required (derivatives)

The Math Behind Hermite Interpolation

• Given a set of $n$ interpolation points $\{(x_i, y_i, y'_i)\}_{i=0}^{n-1}$, where $y_i = f(x_i)$ and $y'_i = f'(x_i)$, the Hermite interpolating polynomial $H(x)$ is defined as:
  • $H(x) = \sum_{i=0}^{n-1} (y_i H_{i,0}(x) + y'_i H_{i,1}(x))$
  • $H_{i,0}(x) = (1 - 2(x - x_i)L'_i(x_i))L_i^2(x)$
  • $H_{i,1}(x) = (x - x_i)L_i^2(x)$
  • $L_i(x) = \prod_{j=0, j\neq i}^{n-1} \frac{x - x_j}{x_i - x_j}$
• The Hermite basis functions $H_{i,0}(x)$ and $H_{i,1}(x)$ are defined using the Lagrange basis polynomials $L_i(x)$
• The coefficients of the Hermite interpolating polynomial can be calculated using divided differences
  • Divided differences for Hermite interpolation involve both function values and derivatives
• The error in Hermite interpolation can be estimated using the following formula:
  • $|f(x) - H(x)| \leq \frac{M}{(2n)!} \prod_{i=0}^{n-1} (x - x_i)^2$
  • $M = \max_{x \in [a,b]} |f^{(2n)}(x)|$, where $[a,b]$ is the interval containing the interpolation points

Step-by-Step Process

1. Given a set of $n$ interpolation points $\{(x_i, y_i, y'_i)\}_{i=0}^{n-1}$, where $y_i = f(x_i)$ and $y'_i = f'(x_i)$
2. Calculate the Lagrange basis polynomials $L_i(x)$ for each interpolation point:
   • $L_i(x) = \prod_{j=0, j\neq i}^{n-1} \frac{x - x_j}{x_i - x_j}$
3. Calculate the Hermite basis functions $H_{i,0}(x)$ and $H_{i,1}(x)$ for each interpolation point:
   • $H_{i,0}(x) = (1 - 2(x - x_i)L'_i(x_i))L_i^2(x)$
   • $H_{i,1}(x) = (x - x_i)L_i^2(x)$
4. Construct the Hermite interpolating polynomial $H(x)$:
   • $H(x) = \sum_{i=0}^{n-1} (y_i H_{i,0}(x) + y'_i H_{i,1}(x))$
5. Evaluate the Hermite interpolating polynomial $H(x)$ at any desired point $x$ within the range of the interpolation points
6. If needed, estimate the error in the interpolation using the error formula:
   • $|f(x) - H(x)| \leq \frac{M}{(2n)!} \prod_{i=0}^{n-1} (x - x_i)^2$
   • $M = \max_{x \in [a,b]} |f^{(2n)}(x)|$, where $[a,b]$ is the interval containing the interpolation points

Real-World Applications

• Computer graphics and animation: Hermite interpolation is used to create smooth curves and surfaces for 3D modeling and animation
  • Hermite splines are commonly used to define smooth paths for camera movements and object trajectories
• Numerical methods: Hermite interpolation is used in various numerical algorithms to approximate functions and their derivatives
  • Examples include root-finding methods, numerical integration, and solving differential equations
• Data analysis and visualization: Hermite interpolation can be used to interpolate and visualize data points with known function values and derivatives
  • Useful for creating smooth curves and surfaces from discrete data sets
• Image processing: Hermite interpolation is used in image scaling and resizing algorithms to maintain smoothness and reduce artifacts
• Computer-aided design (CAD): Hermite interpolation is used to create smooth curves and surfaces for designing objects and components
  • Hermite curves are commonly used in CAD software for modeling and drafting
• Signal processing: Hermite interpolation can be used to interpolate and reconstruct signals from discrete samples
  • Useful for upsampling and downsampling signals while maintaining smoothness and reducing aliasing

Common Pitfalls and How to Avoid Them

• Oscillations and overshooting: Hermite interpolation may produce oscillations or overshoot near the interpolation points, especially when the function has high curvature or rapid changes
  • To avoid this, use a higher number of interpolation points or consider using piecewise Hermite interpolation
• Ill-conditioned interpolation points: If the interpolation points are too close together or poorly distributed, the Hermite interpolating polynomial may be ill-conditioned and produce inaccurate results
  • To avoid this, ensure that the interpolation points are well-distributed and not too close together
• Inaccurate or unavailable derivatives: The accuracy of Hermite interpolation depends on the accuracy of the function values and derivatives at the interpolation points
  • If the derivatives are inaccurate or not available, the interpolation may produce poor results
  • To avoid this, ensure that the derivatives are accurately computed or consider using numerical differentiation techniques
• Computational cost: Hermite interpolation is computationally more expensive than methods that only use function values due to the additional information required (derivatives)
  • To mitigate this, consider using efficient algorithms for calculating divided differences and Hermite basis functions
  • If the computational cost is a concern, consider using simpler interpolation methods (Lagrange, Newton) when appropriate

Practice Problems and Examples

1. Given the interpolation points $\{(0, 1, 0), (1, 2, 1), (2, 5, 4)\}$, find the Hermite interpolating polynomial $H(x)$.
2. For the function $f(x) = \sin(x)$, find the Hermite interpolating polynomial using the interpolation points $\{(0, 0, 1), (\frac{\pi}{2}, 1, 0), (\pi, 0, -1)\}$. Compare the interpolation with the original function on the interval $[0, \pi]$.
3. Given the interpolation points $\{(-1, 2, -3), (0, 1, 0), (1, 2, 3)\}$, find the value of the Hermite interpolating polynomial $H(x)$ at $x = 0.5$.
4. The position of an object moving along a straight line is given by the interpolation points $\{(0, 0, 1), (1, 3, 2), (2, 8, 5)\}$, where the first value in each pair represents time (in seconds) and the second value represents position (in meters). Find the Hermite interpolating polynomial describing the object's motion and determine its position at $t = 1.5$ seconds.
5. Implement a function in a programming language of your choice that takes a set of interpolation points $\{(x_i, y_i, y'_i)\}_{i=0}^{n-1}$ and returns the Hermite interpolating polynomial $H(x)$ as a function that can be evaluated at any point $x$ within the range of the interpolation points.