Hermite interpolation takes interpolation to the next level by matching both function values and derivatives at data points. This results in smoother, more accurate approximations of complex functions, making it ideal for computer-aided design and solving differential equations.

Unlike Lagrange interpolation, Hermite requires derivative information, leading to higher-degree polynomials. While this can improve accuracy, it also increases computational complexity and the risk of overfitting. Careful point selection is crucial for stability and effectiveness.

Hermite Interpolation vs Lagrange Interpolation

Key Differences and Applications

• Hermite interpolation matches both function values and derivative values at given data points while Lagrange interpolation only matches function values
• Higher degree polynomial results from Hermite interpolation compared to Lagrange for same number of data points due to inclusion of derivative information
• Smoother approximation of original function provided by Hermite interpolation by accounting for function's rate of change at interpolation points
• Particularly useful in computer-aided geometric design and numerical solutions of differential equations where both function values and derivatives are known
• Generalization of Lagrange interpolation considers each data point with multiplicity to account for derivative information

Polynomial Characteristics and Constraints

• Requires knowledge of function values and derivative values at each interpolation point (may not always be available in practice)
• Number of conditions to satisfy equals sum of multiplicities of all interpolation points determining resulting polynomial degree
• Interpolation points must be distinct but same point can be used multiple times with different derivative orders to increase smoothness
• Choice of points and multiplicities affects stability and accuracy (equally spaced points often lead to Runge's phenomenon for high-degree polynomials)
• Can extend to include higher-order derivatives (Hermite-Birkhoff interpolation) imposing additional constraints on interpolating polynomial

Conditions for Hermite Interpolation

Data Requirements and Point Selection

• Function values and derivative values needed at each interpolation point (f(x), f'(x), f''(x), etc.)
• Distinct interpolation points required (x₀, x₁, ..., xₙ where xᵢ ≠ xⱼ for i ≠ j)
• Same point can have multiple multiplicities to increase smoothness (x₀ with f(x₀), f'(x₀), f''(x₀))
• Total number of conditions determines polynomial degree (sum of multiplicities across all points)
• Careful selection of points and multiplicities improves stability (Chebyshev nodes often preferred over equally spaced)

Constraint Implications

• Higher degree polynomial compared to Lagrange interpolation (2n+1 degree for n+1 points with function and first derivative)
• Increased smoothness at interpolation points due to derivative matching
• Potential for overfitting with too many conditions or high multiplicities
• Trade-off between accuracy and computational complexity as degree increases
• Consideration of function behavior crucial for effective point and multiplicity choices (more points in rapidly changing regions)

Derivation of Hermite Polynomials

Basis Polynomial Construction

• Hermite basis polynomials designed to satisfy interpolation conditions at each point
• General form of Hermite interpolation polynomial: $H(x) = \sum_{i=0}^n \left(f(x_i)h_{i,0}(x) + f'(x_i)h_{i,1}(x)\right)$
• Basis polynomials $h_{i,0}(x)$ and $h_{i,1}(x)$ constructed to meet specific conditions:
  • $h_{i,0}(x_j) = \delta_{ij}$ (Kronecker delta)
  • $h'_{i,0}(x_j) = 0$
  • $h_{i,1}(x_j) = 0$
  • $h'_{i,1}(x_j) = \delta_{ij}$
• Example for two points: $H(x) = f(x_0)h_{0,0}(x) + f'(x_0)h_{0,1}(x) + f(x_1)h_{1,0}(x) + f'(x_1)h_{1,1}(x)$

Coefficient Determination Methods

• Linear system of equations solved to determine coefficients of basis polynomials
• Newton's divided difference formula extended for Hermite interpolation:
  • Treat each point with its multiplicity
  • Use generalized divided differences
• Barycentric form provides efficient and numerically stable evaluation: $H(x) = \frac{\sum_{i=0}^n \left(w_{i,0}f(x_i) + w_{i,1}f'(x_i)\right) / (x - x_i)}{\sum_{i=0}^n \left(w_{i,0} + w_{i,1}/(x - x_i)\right) / (x - x_i)}$
  • $w_{i,0}$ and $w_{i,1}$ are precomputed weights
• Lagrange form can be adapted for Hermite interpolation using osculatory polynomials

Error and Convergence of Hermite Interpolation

Error Analysis and Bounds

• Generalized error formula for Hermite interpolation: $f(x) - H(x) = \frac{f^{(2n+2)}(\xi)}{(2n+2)!} \prod_{i=0}^n (x - x_i)^2$
  • $\xi$ is some point in the interpolation interval
• Error bounds depend on maximum value of higher-order derivative over interpolation interval
• Distribution of interpolation points affects error magnitude
• Runge's phenomenon can occur for high-degree polynomials with equally spaced points (less severe than Lagrange)
• Example: For function $f(x) = \sin(x)$ on [-π, π], Hermite interpolation with 5 equally spaced points yields max error ≈ 10⁻⁵, compared to ≈ 10⁻³ for Lagrange

Convergence Properties and Improvements

• Faster convergence rate than Lagrange interpolation due to additional derivative information
• Convergence speed depends on function smoothness (faster for infinitely differentiable functions)
• Node distribution significantly impacts convergence and stability:
  • Chebyshev nodes: $x_i = \cos\left(\frac{2i+1}{2n+2}\pi\right)$, i = 0, 1, ..., n
  • Leja points: iteratively chosen to maximize product of distances from previous points
• Adaptive Hermite interpolation techniques enhance accuracy:
  • Adjust distribution and multiplicity of points based on local function behavior
  • Concentrate points in regions of rapid variation or near singularities
• Error can be reduced by increasing number of interpolation points or using higher-order derivatives