Lagrange Interpolation: Step-by-Step Numerical Method

Newton's Divided Difference Form is an alternative polynomial interpolation method that addresses Lagrange interpolation's limitations. Unlike Lagrange, which requires matrix inversion and complete recalculation when new data points are added, Newton's method allows incremental updates. The general form is P(x) = a₀ + a₁(x-x₀) + a₂(x-x₀)(x-x₁) + ... + aₙ(x-x₀)...(x-xₙ₋₁), where coefficients aᵢ are divided differences calculated from the data points. This method is computationally efficient for dynamic datasets where new points are frequently added.

Newton Divided Difference Interpolation is a numerical method that constructs interpolating polynomials recursively by calculating divided difference coefficients from discrete data points; unlike Lagrange polynomials, it allows efficient addition of new data points without recalculating previous coefficients, making it faster and more practical for dynamic datasets.

The Newton divided difference formula provides an alternative approach to polynomial interpolation using divided differences instead of the Lagrange method. The formula involves terms where each term is the product of the divided difference at a specific point and a product of differences between the variable x and all previous x-values. This triangular structure allows for incremental updates when adding new data points, making it computationally efficient for dynamic datasets. The instructor explains that both Lagrange and Newton methods are mathematically equivalent, producing identical polynomials for the same data points. Understanding both methods provides flexibility in solving interpolation problems, with the choice between them depending on computational efficiency and the specific characteristics of the data being interpolated.

The Lagrange interpolation polynomial constructs a polynomial passing through given points using a sum of basis polynomials. Each basis polynomial is a product of linear factors divided by differences between points. A key limitation is that adding new data points requires recalculating all previous terms. Newton's divided differences method provides an efficient alternative by expressing the polynomial in a nested form using coefficients called divided differences. The first divided difference represents the slope between two points, while higher-order differences involve multiple points. This approach allows reuse of previously computed values when adding new data points, making it computationally more efficient for dynamic datasets.

This segment compares two approaches to linear interpolation. The direct method uses the form y = a₀ + a₁x, requiring solution of two simultaneous linear equations to find coefficients a₀ and a₁. In contrast, Newton's divided difference method uses the form f₁(x) = b₀ + b₁(x - x₀), where coefficients are found sequentially: b₀ = f(x₀) and b₁ = [f(x₁) - f(x₀)]/(x₁ - x₀). The key advantage of Newton's method is computational efficiency—it avoids solving simultaneous equations, making it more practical for higher-order interpolations where simultaneous equation solving would require complex numerical methods.

Cubic spline interpolation addresses the problem of polynomial wiggle by constructing a smooth curve that passes through all data points through piecewise cubic polynomials, where each interval between consecutive points is fitted with a cubic polynomial having four unknown coefficients; the method enforces continuity of the function value, first derivative, and second derivative at each interior point, requiring n+1 interpolation constraints plus n-1 derivative continuity constraints for n intervals, which results in an underdetermined system that needs two additional boundary conditions (typically zero second derivatives at endpoints) to yield a unique solution, though splines are only suitable for interpolation within the data range and not for extrapolation.

Cubic spline interpolation is a method for constructing smooth interpolating functions that pass through given data points. Unlike linear interpolation which connects points with straight lines, cubic spline interpolation uses piecewise cubic polynomials to create a much smoother curve. The method involves defining n piecewise cubic polynomials g₀(x), g₁(x), ..., gₙ₋₁(x), each defined on an interval [xᵢ, xᵢ₊₁]. Each polynomial has four coefficients (a, b, c, d), resulting in 4n unknowns that must be determined. The challenge is to find these coefficients while ensuring the interpolating function passes through all data points and maintains smoothness across the entire domain.

Cubic spline interpolation is a mathematical technique that constructs a smooth curve through a set of data points by dividing the interval into segments and fitting cubic polynomials to each segment, ensuring continuity of the function, its first derivative, and second derivative at the interior points, with natural boundary conditions setting the second derivative to zero at the endpoints.

Cubic spline interpolation is a piecewise polynomial method that fits a third-degree polynomial between each pair of consecutive data points, ensuring continuity of the function, its first derivative, and its second derivative at the knot points, with the additional condition that the second derivative equals zero at the endpoints for natural cubic splines.

This section covers numerical integration formulas derived from polynomial interpolation. Newton-Cotes formulas are: ∫ₐᵇ f(x)dx = h × Σᵢ₌₀ⁿ wᵢ f(xᵢ). The Trapezoidal Rule (n=1) is: (b-a)/2 × [f(a) + f(b)], with error O(h³). Simpson's 1/3 Rule (n=2) is: (b-a)/6 × [f(a) + 4f((a+b)/2) + f(b)], with error O(h⁵). Simpson's 3/8 Rule (n=3) is: (b-a)/8 × [f(a) + 3f(a+h) + 3f(a+2h) + f(a+3h)], with error O(h⁵). Weddle's Rule (n=6) is: (3h/10) × [f(a) + 5f(a+h) + f(a+2h) + 6f(a+3h) + f(a+4h) + 5f(a+5h) + f(a+6h)], with error O(h⁷).

The Newton-Cotes quadrature formula is derived by replacing the integrand f(x) with its Newton's forward interpolation polynomial and integrating term-by-term, yielding the general formula ∫ₐᵇ f(x)dx = nh[y₀ + (n/2)Δy₁ + (n/12)(2n-3)Δ²y₁ + (n/24)(n-2)Δ³y₁ + ...], which produces specific rules including the trapezoidal rule (n=1), Simpson's 1/3 rule (n=2), and Simpson's 3/8 rule (n=3) when substituted with different values of n.

Numerical integration approximates definite integrals without analytical methods. The interval [a, b] is divided into n subintervals of width h. Substituting x = x₀ + ph transforms the integral into ∫₀ⁿ f(x₀ + ph)h dp. Applying Newton's forward difference expansion and term-by-term integration yields the Newton-Cotes formula: ∫ₐᵇ f(x)dx ≈ H[y₀ + (n/2)Δy₀ + (n(2n-3)/12)Δ²y₀ + ...]. This general framework enables systematic numerical approximation of integrals through polynomial interpolation.

Newton-Cotes formulae approximate definite integrals by replacing the integrand with Lagrange interpolation polynomials and integrating those polynomials; the trapezoidal rule uses first-order polynomials (linear approximation) to yield ∫ₐᵇ f(x)dx ≈ (b-a)/2 × [f(a) + f(b)], while Simpson's one-third rule uses second-order polynomials (quadratic approximation) to give ∫ₐᵇ f(x)dx ≈ (b-a)/6 × [f(a) + 4f((a+b)/2) + f(b)], and Simpson's 3/8 rule uses third-order polynomials (cubic approximation) with ∫ₐᵇ f(x)dx ≈ (3(b-a))/8 × [f(a) + 3f((a+b)/2) + 3f((a+3b)/4) + f(b)], with errors proportional to h³ and h⁵ respectively, though higher-order methods may introduce oscillations that increase error.

This comprehensive section covers the foundational concepts of numerical integration. The instructor introduces Newton-Cotes Quadrature Formula derivation by substituting x = x₀ + ph into the definite integral and applying Newton's forward interpolation formula. The general formula is: h × [y₀ + (n/2)y₁ + (n/2 - 1)y₂ + ... + yₙ]. Three specific rules are derived: Trapezoidal Rule (n=1): h/2 × [y₀ + y₁]; Simpson's 1/3 Rule (n=2): h/3 × [y₀ + 4y₁ + y₂]; Simpson's 3/8 Rule (n=3): 3h/8 × [y₀ + 3y₁ + 3y₂ + y₃]. Key requirements: n must be even for Simpson's 1/3 Rule, and multiple of 3 for Simpson's 3/8 Rule. Memory techniques are provided for each formula. The instructor then demonstrates practical application by solving ∫₀¹ e^(-x) dx using Trapezoidal Rule with 6 intervals, showing step-by-step table creation and formula application. Error is calculated by comparing with exact value 1 - e^(-1) ≈ 0.6321. The instructor then applies Simpson's 1/3 Rule (n=4) and Simpson's 3/8 Rule (n=6), demonstrating that accuracy improves with more intervals: Trapezoidal ≈ 1.4103, Simpson's 1/3 ≈ 1.3662, Simpson's 3/8 ≈ 1.3571.