🔢Numerical Analysis I Unit 7 – Newton's Divided Differences Interpolation

What's the Big Idea?

• Newton's Divided Differences Interpolation constructs a polynomial that passes through a given set of data points
• Interpolates values between known data points by fitting a polynomial curve
• Useful when the underlying function is unknown or difficult to evaluate directly
• Approximates complex functions using simpler polynomial expressions
• Degree of the interpolating polynomial is one less than the number of data points
  • For example, with 5 data points, the interpolating polynomial will be of degree 4
• Polynomial coefficients are determined using divided differences
• Provides a way to estimate function values at points not included in the original data set

Key Concepts to Grasp

• Interpolation involves constructing a curve that fits a set of known data points
• Divided differences are the key computational tool in Newton's interpolation method
  • Divided differences are calculated recursively using a triangular table
  • Each level of divided differences represents the coefficients of the interpolating polynomial
• Newton's interpolating polynomial is unique for a given set of data points
• The interpolating polynomial exactly matches the function values at the interpolation points
• Error in the interpolation increases as the distance from the known data points increases
• Runge's phenomenon can occur when using high-degree polynomials for interpolation
  • Oscillations and large errors can appear near the edges of the interpolation interval
• Newton's interpolation formula is expressed in terms of divided differences and the product of linear factors

The Math Behind It

• Given a set of $n+1$ data points $(x_0, y_0), (x_1, y_1), \ldots, (x_n, y_n)$, Newton's interpolating polynomial $P_n(x)$ is defined as:

$P_n(x) = f[x_0] + f[x_0, x_1](x - x_0) + f[x_0, x_1, x_2](x - x_0)(x - x_1) + \ldots + f[x_0, x_1, \ldots, x_n](x - x_0)(x - x_1)\ldots(x - x_{n-1})$

• $f[x_i]$ represents the function value at $x_i$, and $f[x_i, x_{i+1}, \ldots, x_{i+j}]$ denotes the $j$-th divided difference
• Divided differences are calculated recursively using the formula:

$f[x_i, x_{i+1}, \ldots, x_{i+j}] = \frac{f[x_{i+1}, x_{i+2}, \ldots, x_{i+j}] - f[x_i, x_{i+1}, \ldots, x_{i+j-1}]}{x_{i+j} - x_i}$

• The first divided difference $f[x_i, x_{i+1}]$ is defined as:

$f[x_i, x_{i+1}] = \frac{f[x_{i+1}] - f[x_i]}{x_{i+1} - x_i}$

• The divided differences can be arranged in a triangular table for easy computation
• The interpolating polynomial $P_n(x)$ has the property that $P_n(x_i) = f(x_i)$ for $i = 0, 1, \ldots, n$

Step-by-Step Walkthrough

1. Given a set of $n+1$ data points $(x_0, y_0), (x_1, y_1), \ldots, (x_n, y_n)$

2. Arrange the data points in a table with $x_i$ values in the first row and $y_i$ values in the second row

3. Calculate the first divided differences $f[x_i, x_{i+1}]$ using the formula:

   $f[x_i, x_{i+1}] = \frac{f[x_{i+1}] - f[x_i]}{x_{i+1} - x_i}$

   Place these values in the third row of the table

4. Calculate the second divided differences $f[x_i, x_{i+1}, x_{i+2}]$ using the formula:

   $f[x_i, x_{i+1}, x_{i+2}] = \frac{f[x_{i+1}, x_{i+2}] - f[x_i, x_{i+1}]}{x_{i+2} - x_i}$

   Place these values in the fourth row of the table

5. Continue calculating higher-order divided differences until you reach the $n$-th divided difference, which will be a single value

6. Construct Newton's interpolating polynomial $P_n(x)$ using the divided differences:

   $P_n(x) = f[x_0] + f[x_0, x_1](x - x_0) + f[x_0, x_1, x_2](x - x_0)(x - x_1) + \ldots + f[x_0, x_1, \ldots, x_n](x - x_0)(x - x_1)\ldots(x - x_{n-1})$

7. Evaluate $P_n(x)$ at any desired point $x$ to obtain an interpolated value

Real-World Applications

• Curve fitting and data analysis in various fields such as engineering, physics, and economics
  • Constructing smooth curves that pass through experimental or observed data points
  • Estimating values between known data points for further analysis or prediction
• Numerical integration and differentiation
  • Interpolating polynomials can be used to approximate integrals and derivatives of functions
  • Useful when the original function is not easily integrable or differentiable analytically
• Computer graphics and animation
  • Interpolation techniques are used to generate smooth curves and surfaces
  • Helps in creating realistic animations and visual effects
• Signal and image processing
  • Interpolation is used to resample signals or images at different resolutions
  • Allows for zooming, scaling, and enhancing digital media
• Numerical solution of differential equations
  • Interpolation methods are employed in numerical schemes for solving ODEs and PDEs
  • Provides a way to approximate solutions at intermediate points within the computational domain

Common Pitfalls and How to Avoid Them

• Oscillations and instability for high-degree interpolating polynomials (Runge's phenomenon)
  • Avoid using polynomials of very high degree, especially when interpolating over a large interval
  • Use piecewise interpolation methods (e.g., splines) for better stability and accuracy
• Extrapolation beyond the range of known data points
  • Interpolating polynomials are most accurate within the interval of interpolation points
  • Be cautious when extrapolating values outside the known data range, as errors can quickly accumulate
• Unevenly spaced or clustered data points
  • Interpolation accuracy may suffer when data points are not well-distributed
  • Consider using weighted interpolation methods or data preprocessing techniques to mitigate the impact of uneven spacing
• Numerical instability and roundoff errors
  • Be mindful of the numerical precision and stability of the computations
  • Use appropriate data types and algorithms to minimize the accumulation of roundoff errors
• Misinterpretation of interpolated values
  • Interpolation provides estimates based on the given data points
  • Understand the limitations and uncertainties associated with interpolated values
  • Clearly communicate the assumptions and limitations when presenting interpolated results

Practice Problems

1. Given the data points $(1, 2), (2, 5), (4, 1)$, find the interpolating polynomial $P_2(x)$ using Newton's divided differences method.
2. Evaluate the interpolating polynomial from problem 1 at $x = 3$ to estimate the function value at that point.
3. Construct the divided differences table for the data points $(-1, 4), (0, 1), (2, -1), (3, 2)$.
4. Find the interpolating polynomial $P_3(x)$ using the divided differences from problem 3.
5. Given the data points $(0, 1), (1, e), (2, e^4)$, find the interpolating polynomial $P_2(x)$ and evaluate it at $x = 0.5$.
6. Use Newton's divided differences method to find the interpolating polynomial for the data points $(-2, 4), (-1, -2), (1, 2), (3, -4)$. Evaluate the polynomial at $x = 0$.
7. Construct the divided differences table for the data points $(0, 1), (1, 0), (2, 1), (3, 0), (4, 1)$.
8. Find the interpolating polynomial $P_4(x)$ using the divided differences from problem 7 and evaluate it at $x = 2.5$.

Going Beyond Newton

• Hermite interpolation
  • Interpolates both function values and derivatives at the interpolation points
  • Useful when derivative information is available or required for a smooth interpolant
• Spline interpolation
  • Constructs a piecewise polynomial function that passes through the data points
  • Ensures continuity and smoothness at the joining points (knots) between polynomial pieces
  • Commonly used spline types include cubic splines and B-splines
• Least-squares approximation
  • Finds a polynomial that minimizes the sum of squared differences between the polynomial and the data points
  • Useful when the data points are subject to noise or measurement errors
  • Provides a best-fit polynomial that may not necessarily pass through all the data points
• Chebyshev interpolation
  • Uses Chebyshev points as the interpolation nodes to minimize the maximum interpolation error
  • Chebyshev points are chosen as the roots of Chebyshev polynomials
  • Provides good stability and accuracy properties for polynomial interpolation
• Rational interpolation
  • Interpolates using rational functions (ratios of polynomials) instead of polynomials
  • Can handle poles and singularities in the data
  • Useful for interpolating functions with vertical asymptotes or rapidly changing behavior
• Adaptive interpolation methods
  • Dynamically adjust the interpolation points or the degree of the interpolating polynomial based on local error estimates
  • Helps in capturing local features and maintaining accuracy in regions with rapid variations
• Multivariate interpolation
  • Extends interpolation techniques to functions of multiple variables
  • Includes methods such as bilinear interpolation, bicubic interpolation, and trilinear interpolation
  • Useful in applications involving multidimensional data, such as image processing and scientific simulations