Glossary

6.2 Gaussian quadrature

7 min read•august 21, 2024

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-, 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

Top images from around the web for Definition and purpose

  • Gaussian integral - Wikipedia View original

    Is this image relevant?

  • Legendre-Gauss Quadrature formula - Knowino View original

    Is this image relevant?

  • Gaussian integral - Wikipedia View original

    Is this image relevant?

1 of 3

Top images from around the web for Definition and purpose

  • Gaussian integral - Wikipedia View original

    Is this image relevant?

  • Legendre-Gauss Quadrature formula - Knowino View original

    Is this image relevant?

  • Gaussian integral - Wikipedia View original

    Is this image relevant?

1 of 3
  • Approximates definite integrals by evaluating the integrand at specific points () 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 in the early 19th century revolutionized numerical integration techniques
  • Built upon earlier work on orthogonal polynomials by
  • Evolved through contributions from mathematicians (, , 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: Pn(x)=12nn!dndxn[(x2−1)n]P_n(x) = \frac{1}{2^n n!} \frac{d^n}{dx^n}[(x^2-1)^n]
  • Satisfy orthogonality condition: ∫−11Pm(x)Pn(x)dx=22n+1δmn\int_{-1}^1 P_m(x)P_n(x)dx = \frac{2}{2n+1}\delta_{mn}
  • Used in 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: wi=2(1−xi2)[Pn′(xi)]2w_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 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: ∫−11f(x)dx≈∑i=1nwif(xi)\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: xi=cos⁡(2i−12nπ)x_i = \cos\left(\frac{2i-1}{2n}\pi\right)
  • Weights calculated as: wi=πnw_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: ∫−∞∞e−x2f(x)dx≈∑i=1nwif(xi)\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: ∫0∞e−xf(x)dx≈∑i=1nwif(xi)\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

  • 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 () improve accuracy and provide error estimates
  • 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
  • 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

Key Terms to Review (27)

Adrien-Marie Legendre: Adrien-Marie Legendre was a French mathematician known for his significant contributions to number theory, statistics, and the development of mathematical analysis. His work laid the groundwork for various numerical methods, particularly in approximation techniques, where he is most notably recognized for the Legendre polynomials, which play a crucial role in Gaussian quadrature and numerical integration.
Approximation error: Approximation error refers to the difference between the exact value of a mathematical quantity and its approximate value derived from numerical methods. This concept is crucial in evaluating the accuracy and reliability of numerical techniques, as it helps determine how close an approximation is to the true value. Understanding approximation error allows for better assessments of algorithms, guiding improvements in their design and implementation.
Carl Friedrich Gauss: Carl Friedrich Gauss was a prominent German mathematician and physicist, known for his contributions to various fields, including number theory, statistics, and numerical analysis. His work laid the foundation for several important algorithms and methods that are widely used today, influencing techniques in solving equations, approximating functions, and performing numerical integration.
Christoffel: In the context of numerical analysis, Christoffel symbols are used in the formulation of Gaussian quadrature, which is a method for approximating the definite integral of a function. These symbols help in defining how to optimally choose sample points and weights for numerical integration to achieve maximum accuracy. They relate closely to polynomial interpolation and the geometry of the space in which the functions reside.
Clenshaw-Curtis Quadrature: Clenshaw-Curtis quadrature is a numerical integration technique that uses Chebyshev polynomials to approximate the integral of a function. It provides an efficient way to compute integrals over finite intervals, especially when the integrand has endpoints at -1 and 1. By utilizing Chebyshev nodes, it achieves high accuracy with fewer evaluation points compared to traditional methods like Gaussian quadrature.
Continuous functions: Continuous functions are mathematical functions where small changes in the input produce small changes in the output, meaning there are no abrupt jumps or breaks in their graphs. This property is crucial for various mathematical techniques, especially in numerical integration, as it ensures that the function behaves predictably across its domain.
Degree of precision: Degree of precision refers to the accuracy with which a numerical integration method can approximate the definite integral of a function. It indicates how well the method can capture the behavior of the integrand, especially for functions that exhibit varying degrees of smoothness or oscillation. Understanding the degree of precision is crucial as it helps in selecting appropriate numerical methods for integration based on the properties of the function being integrated.
Evaluating definite integrals: Evaluating definite integrals involves calculating the exact value of the integral of a function over a specified interval. This process gives the net area under the curve of the function between two points, allowing for the computation of quantities like total distance or total accumulation. It's fundamental in various applications such as physics, engineering, and probability theory, as it provides precise numerical results from continuous data.
Gauss-Chebyshev quadrature: Gauss-Chebyshev quadrature is a numerical integration technique that uses the roots of Chebyshev polynomials as sample points to evaluate integrals, specifically for functions weighted by the Chebyshev weight function. This method is particularly effective for approximating integrals over the interval [-1, 1], making it a powerful tool for dealing with oscillatory functions or those that exhibit singular behavior at the endpoints. By combining the properties of Gaussian quadrature with Chebyshev polynomials, this technique optimizes accuracy and efficiency in numerical integration.
Gauss-Hermite Quadrature: Gauss-Hermite Quadrature is a numerical integration technique specifically designed to approximate integrals of the form $$\int_{-\infty}^{\infty} e^{-x^2} f(x) \, dx$$, where the weight function is $$e^{-x^2}$$. This method utilizes strategically chosen sample points, known as nodes, and corresponding weights to achieve high accuracy for functions that exhibit Gaussian-like behavior. It's particularly useful in probabilistic contexts and applications involving normal distributions.
Gauss-Kronrod Quadrature: Gauss-Kronrod quadrature is a numerical integration technique that extends the Gaussian quadrature method by introducing additional nodes and weights to improve the accuracy of the approximation. It allows for the calculation of integrals with a higher degree of precision by combining points from both Gaussian quadrature and an additional set of Kronrod points, making it particularly useful when evaluating integrals that exhibit rapid changes or singularities. This method provides an efficient way to achieve adaptive integration, which can enhance performance in various numerical applications.
Gauss-Laguerre Quadrature: Gauss-Laguerre quadrature is a numerical integration technique specifically designed for evaluating integrals of the form $$\int_0^{\infty} e^{-x} f(x) \, dx$$, where $f(x)$ is a well-behaved function. This method utilizes the roots of Laguerre polynomials as the points at which the function is evaluated, along with corresponding weights that ensure high accuracy in the approximation of the integral. Its strength lies in its efficiency when dealing with functions that rapidly decay to zero, making it particularly useful for applications in physics and engineering.
Gauss-Legendre Quadrature: Gauss-Legendre Quadrature is a numerical integration technique that approximates the definite integral of a function using strategically chosen points and weights. This method is particularly effective for polynomials, as it leverages the roots of Legendre polynomials to maximize accuracy with fewer evaluation points compared to other methods. By selecting specific nodes within the integration interval, it minimizes the error associated with the approximation.
Gaussian weights and nodes: Gaussian weights and nodes are specific points and associated weights used in Gaussian quadrature to approximate the definite integral of a function. These nodes represent the locations at which the function is evaluated, while the weights determine how much each function value contributes to the total integral approximation. This method is particularly effective for functions that are smooth and can be closely approximated by polynomials.
Golub-Welsch Algorithm: The Golub-Welsch algorithm is a numerical method used to compute the nodes and weights for Gaussian quadrature, which allows for efficient approximation of definite integrals. This algorithm specifically targets polynomial interpolation and provides a systematic way to find the optimal points for evaluating functions, enhancing accuracy in numerical integration. Its foundation lies in leveraging orthogonal polynomials, particularly associated with the weight function of the integral being approximated.
Hermite: In numerical analysis, Hermite refers to a family of interpolation methods that not only use function values but also derivatives at specified points to achieve higher accuracy. These methods are particularly useful in approximating functions where both function values and their rates of change are known, enhancing the precision of polynomial approximations.
Integrating Polynomial Functions: Integrating polynomial functions involves calculating the area under the curve of a polynomial equation over a specified interval. This process is essential in numerical analysis, particularly for approximating definite integrals when analytical solutions are complex or impossible to obtain. The method used can significantly affect the accuracy and efficiency of the integration, especially when applied to higher-degree polynomials or in conjunction with techniques like Gaussian quadrature.
L2 space: The l2 space, also known as the space of square-summable sequences, is a vector space consisting of all infinite sequences whose squared absolute values sum to a finite number. This concept is essential in functional analysis and has significant applications in numerical methods, particularly in error analysis and approximation theory, where it helps to quantify the convergence of numerical solutions.
Nodes: In numerical analysis, nodes refer to specific points at which a function is evaluated, particularly in the context of interpolation and numerical integration. These points play a critical role in determining how well the function can be approximated by a polynomial or how accurately an integral can be estimated. The selection and placement of nodes directly influence the accuracy and efficiency of numerical methods.
Orthogonality of polynomials: Orthogonality of polynomials refers to the property where two different polynomials are orthogonal if their inner product is zero. This concept is essential in numerical analysis, particularly in techniques like Gaussian quadrature, which relies on choosing points and weights to approximate integrals effectively. The orthogonality condition ensures that certain polynomial sets can provide optimal approximation properties, making computations more efficient and accurate.
Peano Kernel Theorem: The Peano Kernel Theorem is a fundamental result in numerical analysis that provides a method for approximating continuous functions using polynomial interpolation. It highlights the existence of a kernel function, which can be used to express the error of an approximation and ensures that certain types of quadrature formulas can achieve higher accuracy. This theorem is especially important when dealing with Gaussian quadrature as it underpins the way integrals can be estimated through polynomial approximations.
Quadrature Formula: A quadrature formula is a mathematical expression used to approximate the definite integral of a function, providing a way to estimate the area under a curve. These formulas rely on selecting specific points (nodes) and corresponding weights to achieve an accurate approximation, often leveraging the properties of polynomials. Gaussian quadrature is a popular type of quadrature formula that optimally selects nodes and weights based on the function being integrated, making it highly efficient for polynomial approximations.
Richardson Extrapolation: Richardson extrapolation is a technique used to improve the accuracy of numerical approximations by combining results from different discretization levels. It works on the principle that if you have an approximation to a value that has a known error term, you can refine that approximation by using another one with a smaller step size, effectively eliminating lower-order error terms. This approach is relevant across various numerical methods and is particularly useful in enhancing the precision of numerical integration and root-finding processes.
Sparse grid methods: Sparse grid methods are numerical techniques used to approximate high-dimensional integrals and functions efficiently, by combining the advantages of both quadrature and interpolation. They leverage a carefully constructed grid that requires significantly fewer points than a full tensor product grid, thus reducing computational costs while maintaining accuracy. These methods are particularly effective in scenarios where the dimensionality is high, as they help overcome the curse of dimensionality.
Stieltjes: Stieltjes refers to a type of integral, known as the Stieltjes integral, which generalizes the Riemann integral by integrating a function with respect to another function, typically a monotonically increasing function. This concept plays a crucial role in numerical analysis, particularly in Gaussian quadrature, where the Stieltjes integral allows for more flexibility in approximating integrals of functions that can be expressed in terms of a weight function.
Truncation Error: Truncation error is the difference between the exact mathematical solution and the approximation obtained through numerical methods. This error arises when an infinite process is approximated by a finite process, leading to discrepancies in calculated values, especially in methods that involve approximating derivatives or integrals.
Weight functions: Weight functions are mathematical tools used in numerical integration, particularly in methods like Gaussian quadrature. They help assign different levels of importance to various points in the domain of integration, allowing for more accurate approximation of the integral by emphasizing certain intervals or values. The choice of weight function can significantly influence the convergence and accuracy of numerical integration methods.
© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.
Back
Practice QuizGlossary
Practice QuizGlossary
Next guide