6.2 Gaussian quadrature

Gaussian quadrature is a powerful numerical integration technique that uses carefully chosen points and weights to approximate definite integrals with high accuracy. It's a key topic in advanced integration methods, offering superior precision for smooth functions while minimizing function evaluations.

This method builds on orthogonal polynomials and their roots to optimize quadrature point placement. Various formulas, like Gauss-Legendre and Gauss-Hermite, cater to different integration problems. While highly effective for smooth integrands, it has limitations with discontinuities and high dimensions.

Fundamentals of Gaussian quadrature

• Numerical integration technique uses carefully chosen points and weights to approximate definite integrals with high accuracy
• Plays crucial role in Numerical Analysis II enhances understanding of advanced integration methods and their applications in scientific computing

Definition and purpose

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• Approximates definite integrals by evaluating the integrand at specific points (nodes) and multiplying by corresponding weights
• Achieves exact results for polynomials up to degree 2n-1 using n points
• Minimizes function evaluations while maximizing accuracy compared to other quadrature methods
• Utilizes orthogonal polynomials determines optimal placement of quadrature points

Historical background

• Developed by Carl Friedrich Gauss in the early 19th century revolutionized numerical integration techniques
• Built upon earlier work on orthogonal polynomials by Adrien-Marie Legendre
• Evolved through contributions from mathematicians (Christoffel, Stieltjes, Hermite)
• Gained prominence with the advent of computers enabled efficient implementation and widespread use

Comparison with Newton-Cotes formulas

• Gaussian quadrature offers superior accuracy for smooth functions with fewer function evaluations
• Newton-Cotes formulas use equally spaced points while Gaussian quadrature optimizes point placement
• Gaussian methods provide exact results for higher-degree polynomials than Newton-Cotes formulas of the same order
• Newton-Cotes formulas may suffer from Runge's phenomenon for high-degree polynomials avoided in Gaussian quadrature

Mathematical foundations

• Builds upon theory of orthogonal polynomials fundamental to understanding Gaussian quadrature
• Utilizes properties of special functions and polynomial roots enhances numerical integration techniques

Orthogonal polynomials

• Set of polynomials satisfying orthogonality condition with respect to a given weight function
• Form basis for constructing Gaussian quadrature formulas
• Include families (Legendre, Chebyshev, Hermite, Laguerre) each suited for different integration problems
• Satisfy three-term recurrence relation enables efficient computation of polynomial values and derivatives

Legendre polynomials

• Orthogonal polynomials on interval [-1, 1] with weight function w(x) = 1
• Defined by Rodrigues' formula: $P_n(x) = \frac{1}{2^n n!} \frac{d^n}{dx^n}[(x^2-1)^n]$
• Satisfy orthogonality condition: $\int_{-1}^1 P_m(x)P_n(x)dx = \frac{2}{2n+1}\delta_{mn}$
• Used in Gauss-Legendre quadrature widely applied for integrals over finite intervals

Roots and weights

• Roots of orthogonal polynomials serve as quadrature points in Gaussian quadrature
• Weights calculated using properties of orthogonal polynomials and their derivatives
• For Legendre polynomials, weights given by formula: $w_i = \frac{2}{(1-x_i^2)[P'_n(x_i)]^2}$
• Symmetry of roots and weights about origin simplifies calculations and implementation

Gaussian quadrature formulas

• Family of numerical integration methods based on different orthogonal polynomials
• Each formula tailored to specific weight functions and integration intervals

Gauss-Legendre quadrature

• Uses roots of Legendre polynomials as quadrature points
• Optimal for integrals over finite intervals with no weight function
• Approximates integral as: $\int_{-1}^1 f(x)dx \approx \sum_{i=1}^n w_i f(x_i)$
• Achieves exact results for polynomials up to degree 2n-1 using n points

Gauss-Chebyshev quadrature

• Based on Chebyshev polynomials of the first kind
• Suited for integrals with weight function w(x) = 1/√(1-x^2) on interval [-1, 1]
• Quadrature points given by: $x_i = \cos\left(\frac{2i-1}{2n}\pi\right)$
• Weights calculated as: $w_i = \frac{\pi}{n}$

Gauss-Hermite quadrature

• Utilizes Hermite polynomials for integrals over (-∞, ∞) with weight function w(x) = e^(-x^2)
• Particularly useful for integrals involving Gaussian distributions
• Approximates integral as: $\int_{-\infty}^{\infty} e^{-x^2}f(x)dx \approx \sum_{i=1}^n w_i f(x_i)$
• Roots and weights symmetric about origin simplifies implementation

Gauss-Laguerre quadrature

• Based on Laguerre polynomials for integrals over [0, ∞) with weight function w(x) = e^(-x)
• Suitable for improper integrals and problems in quantum mechanics
• Approximates integral as: $\int_0^{\infty} e^{-x}f(x)dx \approx \sum_{i=1}^n w_i f(x_i)$
• Requires careful handling of large arguments prevents numerical instability

Implementation and algorithms

• Focuses on practical aspects of implementing Gaussian quadrature in numerical software
• Addresses challenges in accurate computation of nodes and weights essential for reliable integration

Node and weight calculation

• Golub-Welsch algorithm uses eigenvalues and eigenvectors of tridiagonal matrix to compute nodes and weights
• Newton's method with good initial guesses finds roots of orthogonal polynomials efficiently
• Tabulated values for low-order quadratures provide quick reference for common cases
• Arbitrary precision arithmetic ensures accuracy for high-order quadratures prevents loss of significance

Adaptive Gaussian quadrature

• Dynamically adjusts number of quadrature points based on estimated error
• Subdivides integration interval recursively until desired accuracy achieved
• Combines advantages of Gaussian quadrature with flexibility of adaptive methods
• Efficiently handles integrands with varying behavior across integration domain

Error estimation techniques

• Uses difference between quadrature results of different orders estimates error magnitude
• Extrapolation methods (Richardson extrapolation) improve accuracy and provide error estimates
• Peano kernel theorem bounds error in terms of higher derivatives of integrand
• Practical error estimators balance computational cost with reliability of error bounds

Applications in numerical integration

• Demonstrates versatility of Gaussian quadrature in solving diverse integration problems
• Highlights importance in fields (physics, engineering, finance) rely on accurate numerical integration

Definite integrals

• Evaluates definite integrals over finite intervals with high accuracy
• Handles smooth functions efficiently requires fewer function evaluations than other methods
• Particularly effective for integrals of products of polynomials and other well-behaved functions
• Can be combined with change of variables extends applicability to wider range of integrands

Improper integrals

• Gauss-Laguerre and Gauss-Hermite quadratures handle infinite integration limits
• Suitable for integrals involving exponential decay or Gaussian distributions
• Transforms semi-infinite integrals to finite intervals using change of variables
• Requires careful consideration of convergence and tail behavior of integrand

Multidimensional integrals

• Extends Gaussian quadrature to higher dimensions using tensor product of one-dimensional rules
• Curse of dimensionality limits effectiveness for high-dimensional problems
• Sparse grid methods alleviate computational cost for moderate dimensions
• Monte Carlo methods often preferred for very high-dimensional integrals due to dimension-independent convergence rate

Advantages and limitations

• Evaluates strengths and weaknesses of Gaussian quadrature in context of numerical integration
• Guides selection of appropriate integration method based on problem characteristics

Accuracy vs computational cost

• Achieves high accuracy with relatively few function evaluations for smooth integrands
• Computational cost increases rapidly with number of quadrature points
• Optimal for low to moderate dimensions becomes less efficient in high dimensions
• Pre-computed nodes and weights amortize setup cost over multiple integrations

Comparison with other methods

• Outperforms Newton-Cotes formulas for smooth functions in terms of accuracy per function evaluation
• Adaptive quadrature methods offer more robustness for integrands with localized features
• Monte Carlo integration scales better to high dimensions but converges more slowly for smooth integrands
• Spectral methods provide exponential convergence for analytic functions may be preferable in some cases

Handling discontinuities

• Performance degrades for integrands with discontinuities or sharp peaks
• Composite Gaussian quadrature subdivides interval improves handling of non-smooth integrands
• Adaptive methods automatically refine integration near discontinuities
• Specialized quadrature rules (Gauss-Radau, Gauss-Lobatto) can incorporate known endpoint behavior

Advanced topics

• Explores extensions and variations of Gaussian quadrature enhance its applicability and efficiency
• Addresses limitations of standard Gaussian quadrature provides more versatile integration tools

Gauss-Kronrod quadrature

• Extends Gaussian quadrature by adding interleaving points
• Reuses function evaluations from lower-order rule efficient for adaptive integration
• Provides embedded error estimate without additional function evaluations
• Widely used in adaptive quadrature algorithms (QUADPACK library)

Clenshaw-Curtis quadrature

• Uses Chebyshev points instead of Gaussian points
• Nearly as accurate as Gaussian quadrature for many integrands
• Allows for efficient nested quadrature rules
• Particularly effective for periodic functions and Fourier transforms

Sparse grid methods

• Combines one-dimensional quadrature rules in way that reduces curse of dimensionality
• Achieves similar accuracy to tensor product quadrature with significantly fewer points in higher dimensions
• Based on Smolyak's construction generalizes to different families of univariate quadrature rules
• Particularly useful for moderate-dimensional problems (3-10 dimensions)

Practical considerations

• Addresses real-world aspects of implementing and using Gaussian quadrature in numerical software
• Provides guidance on selecting appropriate quadrature methods for specific problem types

Software implementations

• Numerical libraries (QUADPACK, GNU Scientific Library) offer robust implementations of Gaussian quadrature
• Symbolic computation systems (Mathematica, SymPy) provide high-precision quadrature rules
• Custom implementations require careful attention to numerical stability and accuracy
• Vectorized implementations leverage modern hardware accelerators (GPUs, SIMD instructions)

Choice of quadrature points

• Selection of quadrature rule depends on integrand behavior and integration domain
• Gauss-Legendre suitable for finite intervals with smooth integrands
• Gauss-Chebyshev effective for integrals with singularities at endpoints
• Gauss-Hermite and Gauss-Laguerre handle infinite domains with appropriate weight functions

Integration over complex domains

• Contour integration techniques extend Gaussian quadrature to complex plane
• Conformal mapping transforms complex domains to standard intervals
• Multidimensional integration over non-rectangular domains requires domain decomposition or coordinate transformations
• Adaptive methods help handle integrable singularities and branch cuts in complex integrands