Newton interpolation is a powerful technique for approximating functions using polynomials. It builds on divided differences, allowing easy addition of new data points without recalculating the entire polynomial. This method is particularly useful for constructing interpolating polynomials of increasing degree.

Newton interpolation offers computational advantages over other forms, like Lagrange interpolation. It's commonly used in numerical analysis, computer graphics, and signal processing. The method provides a natural way to estimate interpolation error and is efficiently implemented using nested multiplication.

Newton Interpolation

Polynomial Representation and Structure

• Newton form of interpolating polynomial uses divided differences and basis of polynomials
• General form: $P_n(x) = a_0 + a_1(x - x_0) + a_2(x - x_0)(x - x_1) + ... + a_n(x - x_0)(x - x_1)...(x - x_{n-1})$
• Coefficients $a_i$ determined by divided differences of given data points
• Allows easy addition of new data points without recalculating entire polynomial
• Equivalent to other forms (Lagrange form) but offers computational advantages
• Particularly useful for constructing interpolating polynomials of increasing degree
• Basis polynomials in Newton form: $1, (x - x_0), (x - x_0)(x - x_1), ..., (x - x_0)(x - x_1)...(x - x_{n-1})$
• Each term in Newton form represents contribution of additional data point

Applications and Implementations

• Commonly used in numerical analysis for approximating functions
• Efficiently implemented using nested multiplication (Horner's method)
• Provides natural error estimation through remainder term
• Used in computer graphics for curve fitting and surface reconstruction
• Applied in signal processing for interpolation of discrete signals
• Utilized in scientific computing for solving differential equations
• Implemented in various programming languages and numerical libraries (MATLAB, NumPy)

Divided Differences for Interpolation

Calculation and Properties

• Divided differences serve as recursive calculations for Newton form coefficients
• First-order divided difference: $f[x_i, x_{i+1}] = (f(x_{i+1}) - f(x_i)) / (x_{i+1} - x_i)$
• Higher-order divided differences computed recursively: $f[x_i, ..., x_{i+k}] = (f[x_{i+1}, ..., x_{i+k}] - f[x_i, ..., x_{i+k-1}]) / (x_{i+k} - x_i)$
• Coefficients $a_i$ in Newton form correspond to diagonal elements of divided difference table
• Divided differences remain unchanged under permutation of their arguments
• Represent slopes of secant lines through data points
• Higher-order divided differences approximate higher-order derivatives of interpolated function
• Provide measure of non-linearity in data

Construction and Evaluation

• To construct Newton interpolating polynomial:
  • Compute divided differences up to desired order
  • Use computed divided differences as coefficients in Newton form
  • Multiply each coefficient by appropriate product of $(x - x_j)$ terms
• Efficient evaluation using nested multiplication (Horner's method)
• Algorithm for constructing divided difference table:
  • Start with function values $f(x_i)$ in first column
  • Compute successive columns using recursive formula
  • Diagonal elements form coefficients of Newton polynomial
• Example: Given data points $(0, 1), (1, 2), (2, 4)$, construct divided difference table and Newton polynomial
• Divided difference table provides insight into behavior of interpolated function

Newton vs Other Interpolation Methods

Advantages of Newton Interpolation

• Allows easy addition of new data points without recalculating entire polynomial
• Computationally efficient for constructing polynomials of increasing degree
• Provides natural way to estimate interpolation error
• Well-suited for numerical implementations and computer algorithms
• Offers good numerical stability for moderate number of points
• Allows for incremental construction of interpolating polynomial
• Efficient for evaluating polynomial at multiple points using Horner's method

Disadvantages and Limitations

• May suffer from Runge's phenomenon for high-degree polynomials with equally spaced points
• Can be less stable numerically compared to barycentric Lagrange interpolation for large numbers of points
• Requires storage of divided difference table, potentially increasing memory usage
• Not as suitable for interpolation on periodic domains
• May exhibit oscillations near endpoints of interpolation interval
• Sensitive to roundoff errors in divided difference calculations for ill-conditioned data

Comparison with Other Methods

• Lagrange interpolation:
  • Newton form more efficient for adding new points, Lagrange form more symmetric
  • Both forms mathematically equivalent but different computational characteristics
  • Lagrange form easier to express explicitly, Newton form better for incremental construction
• Spline interpolation:
  • Newton interpolation produces single polynomial, spline interpolation uses piecewise polynomials
  • Spline interpolation generally provides better stability and avoids Runge's phenomenon
  • Splines offer better control over smoothness and local behavior of interpolant
• Hermite interpolation:
  • Newton form can be extended to include derivative information (Hermite-Birkhoff interpolation)
  • Hermite interpolation provides better accuracy when derivative data available
• Choice between methods depends on specific problem requirements (computational efficiency, stability, ease of implementation)