4.6 Gaussian quadrature

Gaussian quadrature is a powerful method for numerical integration. It uses carefully chosen points and weights to approximate integrals with high accuracy, often outperforming simpler methods like the trapezoidal rule or Simpson's rule.

This technique fits perfectly into the chapter on numerical integration. It showcases how advanced mathematical principles can be applied to create efficient computational methods for solving complex integrals in various fields.

Gaussian Quadrature for Integration

Concept and Advantages

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• Gaussian quadrature approximates definite integrals using carefully chosen evaluation points (nodes) and corresponding weights
• Based on orthogonal polynomials principle achieving higher accuracy with fewer function evaluations compared to other methods
• Exactly integrates polynomials of degree 2n-1 or less using only n points (n = number of nodes used in quadrature rule)
• Particularly effective for smooth, well-behaved functions achieving high accuracy even with relatively low integration points
• Adaptable to different interval ranges and weight functions allowing for specialized rules for specific integral types
• Efficiency makes it valuable in computational physics, engineering, and financial mathematics where rapid and accurate integration proves crucial
• Outperforms simpler methods like trapezoidal rule or Simpson's rule for many types of integrals (oscillatory functions, infinite intervals)
• Can handle integrals with singularities or rapid variations by appropriate choice of weight function

Mathematical Formulation

• General form of Gaussian quadrature approximation: $\int_{a}^{b} w(x)f(x)dx \approx \sum_{i=1}^{n} w_i f(x_i)$
  • w(x) represents weight function
  • w_i denotes weights
  • x_i signifies nodes
• Nodes (x_i) determined as roots of orthogonal polynomials corresponding to weight function of integral
• Weights (w_i) calculated using properties of orthogonal polynomials ensuring exactness for polynomials up to certain degree
• Error in Gaussian quadrature approximation generally proportional to (b-a)^(2n+1) * f^(2n)(ξ) for some ξ in [a,b]
  • Indicates rapid convergence for smooth functions as n increases
• Composite Gaussian quadrature divides integration interval into subintervals applying quadrature rule to each
  • Useful for more complex or oscillatory functions

Applying Gauss-Legendre and Gauss-Hermite Rules

Gauss-Legendre Quadrature

• Designed for integrals over finite intervals typically [-1, 1]
• Adaptable to other finite intervals [a, b] through linear transformation: $x = \frac{(b-a)t + (b+a)}{2}$
• Nodes are roots of Legendre polynomials
• Weight function w(x) = 1 for x ∈ [-1, 1]
• Commonly used for integrals without specific weight functions
• Example application integrating f(x) = x^3 + 2x^2 - x + 1 over [-1, 1]:
  • 2-point Gauss-Legendre approximation: $\int_{-1}^{1} (x^3 + 2x^2 - x + 1) dx \approx w_1f(x_1) + w_2f(x_2)$
    • x_1 = -1/√3, x_2 = 1/√3 (nodes)
    • w_1 = w_2 = 1 (weights)

Gauss-Hermite Quadrature

• Tailored for integrals of form $\int_{-\infty}^{\infty} e^{-x^2} f(x) dx$
• Particularly useful in quantum mechanics (harmonic oscillator wavefunctions) and statistics (normal distribution)
• Nodes are roots of Hermite polynomials
• Weight function w(x) = e^(-x^2)
• Example application calculating expectation of x^2 for standard normal distribution: $E[X^2] = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} x^2 e^{-x^2/2} dx$
  • Can be transformed to Gauss-Hermite form and approximated using appropriate nodes and weights

Implementation and Error Estimation

• Select appropriate rule based on integral's domain and weight function
• Apply pre-computed nodes and weights from mathematical software packages or specialized libraries
• Variable substitution techniques transform integrals not directly matching standard forms
• Error estimation methods:
  • Compare results from different orders of quadrature
  • Use theoretical error bounds based on function's properties
• Adaptive quadrature techniques automatically determine optimal number and distribution of nodes
  • Based on error estimates and function's behavior in different integration domain regions

Optimal Nodes and Weights for Gaussian Quadrature

Determination of Nodes and Weights

• Nodes (x_i) found as roots of orthogonal polynomials corresponding to integral's weight function
  • Gauss-Legendre uses Legendre polynomial roots
  • Gauss-Hermite employs Hermite polynomial roots
• Weights (w_i) calculated using orthogonal polynomial properties
  • Ensure quadrature exactness for polynomials up to certain degree
• Number of nodes (n) selection depends on:
  • Desired accuracy
  • Integrand smoothness
  • Higher n generally provides better accuracy at increased computation cost
• Golub-Welsch algorithm computes custom nodes and weights for non-standard weight functions or domains
  • Utilizes recurrence relations of appropriate orthogonal polynomials

Practical Considerations and Tools

• Specialized software libraries and mathematical packages (MATLAB, SciPy, Mathematica) provide pre-computed nodes and weights
• Tables of nodes and weights available in numerical analysis textbooks for common quadrature rules
• For high-precision calculations consider arbitrary-precision arithmetic libraries
• Symmetry of nodes and weights can be exploited to reduce storage and computation requirements
  • Gauss-Legendre nodes symmetric about origin, weights symmetric pairs

Adaptive and Custom Quadrature Techniques

• Adaptive quadrature automatically determines optimal node number and distribution
  • Uses error estimates and function behavior in different integration domain regions
• Gauss-Kronrod quadrature extends Gaussian quadrature with additional points
  • Allows for error estimation and adaptive refinement
• Clenshaw-Curtis quadrature uses Chebyshev polynomial roots as nodes
  • Often comparable to Gaussian quadrature in performance
  • Nested rule structure allows for efficient adaptive algorithms
• Custom quadrature rules can be developed for specific weight functions
  • Example Gauss-Laguerre quadrature for integrals with weight function w(x) = e^(-x) on [0, ∞)